On the L -polynomials of curves over finite fields

Abstract

We discuss, in a non-Archimedean setting, the distribution of the coefficients of L-polynomials of curves of genus g over Fq\mathbb{F}_q . Among other results, this allows us to prove that the Q\mathbb{Q} -vector space spanned by such characteristic polynomials has dimension g + 1. We also state a conjecture about the Archimedean distribution of the number of rational points of curves over finite fields.

Full text

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1–49, 2025
DOI:10.1017/prm.2025.7
On the L-polynomials of curves over finite fields
Francesco Ballini
Mathematical Institute, University of Oxford, Andrew Wiles Building,
Radcliffe Observatory Quarter (550), Woodstock Road, Oxford, United
Kingdom (Francesco.Ballini@maths.ox.ac.uk)
Davide Lombardo
Dipartimento di Matematica, Universit`a di Pisa, Largo Bruno
Pontecorvo 5, Pisa, Italy (davide.lombardo@unipi.it)
Matteo Verzobio
Institute of Science and Technology Austria (ISTA), Am Campus 1,
Klosterneuburg, Austria (matteo.verzobio@gmail.com) (corresponding
author)
(Received 27 October 2024; revised 6 January 2025; accepted 15 January 2025)
We discuss, in a non-Archimedean setting, the distribution of the coefficients of
L-polynomials of curves of genus gover Fq. Among other results, this allows us to
prove that the Q-vector space spanned by such characteristic polynomials has
dimension g+ 1. We also state a conjecture about the Archimedean distribution of
the number of rational points of curves over finite fields.
Keywords: curves; equidistribution; rational points; L-polynomials; Katz-Sarnak
theory
2010 Mathematics Subject Classification: Primary 11G20
Secondary 11G10; 14G10
1. Introduction
Let Fqbe a finite field of characteristic pand order q=pf. For every g≥1,
we let Mg(Fq) be the set of smooth projective curves of genus gover Fq, up to
isomorphism over Fq. Recall that, given a (smooth projective) curve C/Fq, one may
introduce its zeta function
Z(C/Fq, s) = exp 
X
m≥1
#C(Fqm)
mq−ms
,
©The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society
of Edinburgh. This is an Open Access article, distributed under the terms of the Creative Commons
Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted
re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
1
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2F. Ballini, D. Lombardo and M. Verzobio
and that by work of Schmidt [46] and Weil [51] we know that Z(C/Fq, s) is a
rational function of t:= q−s. More precisely, we can write
Z(C/Fq, s) = PC(t)
(1 −t)(1 −qt),
where PC(t) is a polynomial (often called the L-polynomial of C) that satisfies the
following:
Lemma 1.1.
(1) PC(t)has integral coefficients and PC(0) = 1;
(2) deg PC(t) = 2g, where g=g(C)is the genus of C;
(3) writing PC(t) = P2g
i=0 aitiwe have the symmetry relations ag+i=qiag−i
for every i= 0, . . . , g.
Our main object of interest in this article is the set of L-polynomials of all the
curves of a given genus over a finite field Fq:
Definition 1.2. Given a finite field Fqand a positive integer gwe define
Pg(Fq) := {PC(t)C∈ Mg(Fq)}.
We will focus in particular on the non-Archimedean distribution of these L-
polynomials. For a fixed integer N≥2, upon reduction modulo None obtains
from Pg(Fq) a set Pg,N (Fq) of polynomials in (Z/NZ)[t]. Considering this set of
reduced polynomials both for a fixed value of qand in the limit q→ ∞, we obtain
results in three different but related directions:
1. We adapt results of Katz–Sarnak from the Archimedean to the non-
Archimedean setting, obtaining equidistribution statements for Pg,N (Fq) as
q→ ∞ (theorem 2.1). While special instances of this result appear in the lit-
erature (especially for the case of elliptic curves, see [13,23]), the general case
does not seem to have been explored previously—though see [2] for related
results.
2. The previous result allows us to disprove a recent conjecture by
Bergstr¨om–Howe–Lorenzo Garc´ıa–Ritzenthaler [9, conjecture 5.1] about the
Archimedean distribution of the number of rational points of non-hyperelliptic
curves over finite fields (see proposition 3.6 and the discussion before it).
Theorem 2.1, combined with the general Lang–Trotter philosophy, leads us
to propose a new conjecture (conjecture 3.4), which seems both more natural
(in view of the general principles that seem to regulate statistical phenomena
in arithmetic) and in better accord with the numerical evidence (see §3.2).
3. Finally, theorem 2.1 easily implies that, for a fixed genus gand for qg1,
the set Pg(Fq) spans a Q-vector space of dimension g+ 1 (remark 2.9). By
considering more carefully the set Pg,2(Fq) for every fixed value of q, we
are able to prove that this statement does, in fact, hold for all pairs (g,q)
(theorem 1.4), thus confirming a conjecture of Kaczorowski and Perelli [28,
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On the L-polynomials of curves over finite fields 3
remark 8]. The proof is based on properties of L-polynomials modulo 2 which
have also recently been explored, with different aims, in [16]. Using theorem
2.1, we can also obtain an asymptotic result for non-linear relations among
the coefficients of elements of Pg(Fq), see theorem 6.1.
Recently, much attention has been devoted to questions close to those that we
consider here: in addition to the aforementioned [9], we also refer the reader to [4],
as well as [3,42], and [49]. We discuss some relations between our work and these
latter articles in remark 3.15. We believe that different parts of the mathematical
community are approaching the same questions we discuss in this article from
complementary perspectives, and we hope that the present work will also encourage
a fruitful exchange of ideas between these different points of view.
For this introduction, we focus more specifically on our contributions. The
non-Archimedean behaviour of the L-polynomials is closely related to the (geo-
metric version of the) Chebotarev density theorem, in the following sense. Let
Cπ
−→ S→Spec Zbe a versal family of curves of genus g, that is, a family in which
every isomorphism class of curves of genus gappears at least once (we use the
tri-canonically embedded family, see §2for details). Considering the N-torsion sec-
tions of Jac C → Sgives rise to a Galois cover S0→Swhose Galois group GNis a
subgroup of GL2g(Z/NZ)—essentially, S0is the minimal cover of Sover which all
the N-torsion sections of Jac Care defined. For every closed point s∈S, we have a
curve Cs, defined over the finite field κ(s), and a Frobenius element Frobs,N ∈GN.
Note that this Frobenius is an element of the Galois group of the cover and is
determined by the property of inducing the finite-field Frobenius t7→ t(#κ(s)) on
the residue field at a point s0∈S0lying over s. As usual, Frobs,N is only well
defined up to conjugacy, or equivalently, up to the choice of the point s0∈S0lying
over s. The reduction modulo Nof the L-polynomial of Csis determined by the
characteristic polynomial of Frobs,N , so equidistribution results for Frobs,N trans-
late into equidistribution results for PCmod N. We make this precise in §2, using
Deligne and Katz’s equidistribution theorem instead of Chebotarev’s.
Having precise control over the non-Archimedean distribution of L-polynomials
is sufficient to show that the values of Fq(t) = #{C:C∈ Mg(Fq),#C(Fq) = t}
show significant local oscillations—consecutive values of t∈Ncan correspond to
wildly different values of Fq(t). As already mentioned, we use this to disprove [9,
conjecture 5.1].
We propose a new conjecture that takes these local oscillations into account
to compute Fq(t) (we achieve this by introducing a suitable product of local fac-
tors). Here we give an informal statement: for a precise version, see conjecture 3.4
and remark 3.8 for an interpretation of the quantity ν`(q, t). See also the remarks
after conjecture 3.4 for a more extended discussion of the motivation behind this
conjecture.
Conjecture 1.3. Let g≥1 and qbe a prime power. Let H0(q, t) be the ‘probabil-
ity’ that a curve C/Fqof genus ghas q+ 1 −trational points. Given a prime
`define ν`(q, t) as the ‘normalized probability that a matrix M∈GSp2g(Z`)
with multiplier qhas trace t’ (see Eqs. (7) and (8) for a precise definition). Let
ν∞(q, t) = STg(t/√q), where STgis the Sato–Tate measure in dimension g. Let
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4F. Ballini, D. Lombardo and M. Verzobio
ν0(q, ·) be the measure c·ν∞(q , ·)Q`<∞ν`(q, ·), where cis the normalization con-
stant that ensures that ν0has total mass 1 (i.e., that it is a probability measure).
The L1-distance between H0(q, ·) and ν0(q, ·) tends to 0 as q→ ∞.
Finally, theorem 1.4 answers the following natural question: does lemma 1.1
capture all the (linear) relations among the coefficients of the polynomials PC(t)?
In other words, what is the dimension of the Q-vector subspace of Q[t] spanned by
the polynomials in Pg(Fq)? As a consequence of lemma 1.1, it is immediate to see
that this space has dimension at most g+ 1. Equality holds if and only if all the
linear relations among the coefficients are already listed in lemma 1.1. We show
that equality does in fact hold for all genera and all finite fields: this extends work
of Birch [10] for curves of genus 1 and of Howe–Nart–Ritzenthaler [27] for curves
of genus 2 and confirms the aforementioned conjecture of Kaczorowski and Perelli
[28, remark 8]:
Theorem 1.4. Let pbe a prime, let f≥1, and denote by Fqthe finite field with
q=pfelements. Let Pg(Fq)be as in definition 1.2 and let Lg(Fq)be the Q-vector
subspace of Q[t]spanned by Pg(Fq). We have
dimQLg(Fq) = g+ 1.
The proof is based on the following observation: in order to establish the linear
independence of a set of polynomials with integral coefficients, it is certainly enough
to show that they are linearly independent modulo 2. In the case of the L-polynomial
of a curve C, the reduction modulo 2 can be read off the action of Galois on the set
of 2-torsion points of the Jacobian of C. In turn, when Cis hyperelliptic, this action
is easy to write down explicitly in terms of a defining equation of C: one can then
find g+ 1 curves whose L-polynomials form a basis of Lg(Fq). Since the properties
of the 2-torsion points are slightly different depending on whether the characteristic
is odd or even, we split our proof into two parts, one for the case podd and one for
the case p= 2. We remark in particular that our proof is constructive: we explicitly
give g+ 1 curves whose L-polynomials form a basis of Lg(Fq), see corollary 5.4 for
odd pand the proof in §5.2 for p= 2.
We conclude this introduction by briefly describing the structure of the article.
In §2, we prove an equidistribution result for Pg,N (see theorem 2.1). In §3, we
state our conjecture on the probability that a curve has a given number of rational
points (see conjecture 3.4). We also explain why we believe this conjecture to be
true and present some numerical evidence that supports it. We further discuss the
difficulties that arise in formally defining the quantities involved in the conjecture
(see, in particular, remark 3.14). This justifies the work of §4, where we prove some
technical results necessary to even state conjecture 3.4. Finally, in §5, we prove
theorem 1.4 and in §6we study non-linear relations among the coefficients of the
polynomials in Pg(Fq).
1.1. Notation and classical results
We fix our notation for symplectic groups:
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On the L-polynomials of curves over finite fields 5
Definition 1.5. Let g≥1and let Rbe a commutative ring with identity. Fix a
non-degenerate alternating bilinear form on R2g, represented by the matrix Ω(note
that the form is non-degenerate if and only if det Ω ∈R×). The group GSp2g(R)
is by definition
GSp2g(R) = {M∈GL2g(R) : ∃λ∈R×such that tMΩM=λΩ}.
The multiplier of a matrix M∈GSp2g(R)is the uniquely determined λ∈R×such
that tMΩM=λΩ. We denote it by mult(M). Given q∈R, we further let GSpq
2g(R)
be the subset of GSp2g(R)consisting of those matrices that have multiplier equal to
q(equality in the group R×).
Remark 1.6. We will mostly be interested in the cases R=Z/`nZ,Z`or Q`,
where `is prime. By definition, the group GSp2g(R) depends on the choice of Ω,
but when Ris a local ring, different choices of Ω lead to isomorphic groups [33].
It follows easily that the same is true for R=Z/NZfor any integer N≥2.
When R∈ {Z/`nZ,Z`,Q`,Z/NZ}, we will therefore refer to GSp2g(R) without
necessarily specifying the choice of anti-symmetric form.
It will be useful to recall the well-known connection between the L-polynomial of
a (smooth projective) curve Cof genus gand the Galois representations attached
to the Jacobian Jof C. Let pbe a prime, let qbe a power of p, and let Cbe a
curve of genus gdefined over Fq. Denote by Jthe Jacobian of C. Let `be any
prime different from pand let T`Jbe the `-adic Tate module of J, that is,
T`J:= lim
←−
n
J(Fq)[`n].
There is a natural action of Gal(Fq/Fq) on T`J(induced by the action of Gal(Fq/Fq)
on the torsion points of J), and it can be shown that T`Jis a free Z`-module of rank
2g. Fixing a Z`-basis of T`J, we thus obtain a representation ρ`∞: Gal(Fq/Fq)→
GL2g(Z`) whose image is contained in GSp2g(Z`); the relevant antisymmetric bilin-
ear form is given by the Weil pairing. Since Gal(Fq/Fq) is procyclic, generated by
the Frobenius automorphism Frobq, we are mostly interested in the action of Frobq
on T`J, which is captured by its characteristic polynomial
fC,`∞(t) = det(tId −ρ`∞(Frobq)) ∈Z`[t].
The matrix representing the action of Frobenius is symplectic with multiplier q.
Notice that we also have an action of Gal(Fq/Fq) on the `-torsion points of J(Fq),
which form an F`-vector space of dimension 2g; we can thus obtain a mod-`
representation ρ`: Gal(Fq/Fq)→GL2g(F`) and a corresponding characteristic
polynomial fC,`(t) = det(tId −ρ`(Frobq)) ∈F`[t]. It is clear from the definitions
that fC,`(t) is nothing but the reduction modulo `of fC,`∞(t). We can now recall
the connection between PC(t) and fC,`∞(t):
Theorem 1.7. (Grothendieck–Lefschetz formula, [17]). The equality PC(t) =
t2gfC,`∞(1/t)holds for every prime `6=p. In particular, the polynomial fC,`∞(t)∈
Z`[t]has integer coefficients and does not depend on `.
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6F. Ballini, D. Lombardo and M. Verzobio
2. The distribution of L-polynomials modulo an integer N
In this section, we adapt [29,§10] to the problem of the distribution of charac-
teristic polynomials of Frobenius modulo a fixed integer N≥2 (as opposed to the
distribution of the coefficients with respect to the Archimedean metric which is
considered in [29]). Fix a genus g≥2 and a finite field Fqof characteristic p>0
(not dividing N). We denote by Mgthe stack of smooth projective curves of genus
g, so that Mg(Fq) denotes the set of Fq-isomorphism classes of smooth projective
curves of genus gover Fq. We see Mg(Fq) as a probability space by endowing it
with one of the following two natural measures:
•the ‘naive’ counting measure Pnaive
g,q , which assigns equal measure to every
singleton {C}, and which we normalize by requiring Pnaive
g,q (Mg(Fq)) = 1.
•the ‘intrinsic’ measure Pintr
g,q such that
Pintr
g,q ({C}) = α1
# Aut(CFq),
where Aut(CFq) is the group of automorphisms of Cdefined over Fqand
α=
X
C∈Mg(Fq)
1
# Aut(CFq)

−1
is the uniquely determined normalization constant that ensures
X
C∈Mg(Fq)
Pintr
g,q ({C}) = Pintr
g,q (Mg(Fq)) = 1.
Note that αis simply the inverse of the (groupoid) cardinality of Mg(Fq).
In other words, it is the inverse of the number of points of the moduli space
of curves of genus gover Fq, when these are counted with the correct weight
(given by the inverse of the size of their automorphism group).
Our objective in this section is to study the random variable
charpol : Mg(Fq)→Z[t]
C7→ fC,`∞(t),
where `is any auxiliary prime different from pthat we use to compute the char-
acteristic polynomial of the Frobenius acting on Jac(C). More precisely, we will
consider the (infinitely many) random variables
charpolN:Mg(Fq)→Z/NZ[t]
C7→ fC,`∞(t) mod N
obtained from charpol by reducing the characteristic polynomials modulo N, for
all N6≡ 0(mod p). For simplicity, since charpol(C) is always a monic polynomial
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On the L-polynomials of curves over finite fields 7
of degree 2g, we restrict the codomain to be the finite set Z/NZ[t]≤2g, the addi-
tive group of polynomials with coefficients in Z/NZand degree at most 2g. For
each positive integer Nnot divisible by p, we obtain a measure µq
Non Z/NZ[t]≤2g
as follows. Consider the finite set GSpq
2g(Z/NZ) and its natural counting mea-
sure µGSpq
2g(Z/NZ), normalized so that the total mass is 1. Concretely, this is
given by
µGSpq
2g(Z/NZ)(X) = #X
# GSpq
2g(Z/NZ)∀X⊆GSpq
2g(Z/NZ).
The map
charpol : GSpq
2g(Z/NZ)→Z/NZ[t]≤2g
that sends each matrix in GSpq
2g(Z/NZ) to its characteristic polynomial allows us
to define the measure
µq
N:= (charpol)∗µGSpq
2g(Z/NZ).
We will show:
Theorem 2.1. Let N, g be positive integers with g≥2. With the notation above, as
q→ ∞ along prime powers with (q, N )=1, the measures (charpolN)∗Pnaive
g,q −µq
N
and (charpolN)∗Pintr
g,q −µq
Nconverge weakly to 0.
Remark 2.2. For g= 1, very precise results about the distribution of characteristic
polynomials modulo arbitrary integers Nare proven in [13]. In particular, the
results of that article describe a very explicit measure ˜µq
Nand show that for g= 1
the difference (charpolN)∗Pnaive
1,g −˜µq
Nconverges to zero with an error of size at
most ON(q−1/2). Thus, the case g= 1 is very well understood. For this reason, and
since theorem 2.4 below does not apply in genus 1, we exclude the case g= 1 from
our discussion.
We begin by recalling a version of Deligne’s equidistribution theorem, as extended
by Katz and Katz–Sarnak. We partially follow the presentation in [6,§2]. We fix an
integer N≥2 and a geometrically connected, smooth, finite-type Z[1/N]-scheme
Uwhose fibres are all geometrically connected of the same dimension. Denote by
ηthe generic point of Uand by ηa corresponding geometric generic point. Let F
be a local system of symplectic free Z/NZ-modules of rank 2gon U—equivalently,
a representation
ρF:π1(U, η)→GSp2g(Z/NZ)∼
=GSp(Fη)⊂Aut(Fη).
Given a finite field kof characteristic not dividing N, there is a unique map Spec k→
Spec Z[1/N]. As in the introduction, a classical construction associates with every
u∈U(k) a (conjugacy class of) Frobenius Frobu,k ∈π1(U, η).
Theorem 2.3. In the situation above, suppose that the following holds. For
every finite field k(of characteristic not dividing N) and for the unique map
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8F. Ballini, D. Lombardo and M. Verzobio
(1)
Spec k→Spec Z[1/N], denote by ηka geometric generic point of Ukand write
πgeom
1(Uk, ηk) = π1Uk, ηk. The representation ρFfits in a commutative diagram
where ρgeom
Fis surjective and ρk
Fsends the canonical generator Frobkof Gal k/k
to #k. Suppose furthermore that the restriction of ρFto π1(UQ, η)has image in
Sp2g(Z/NZ).
There is a constant C(depending at most on U,F, and N) such that, for
any union of conjugacy classes W⊆GSp2g(Z/NZ)and any finite field kof
characteristic not dividing N, we have

#{u∈U(k) : ρF(Frobu,k)∈W}
#U(k)−#(W∩GSpγ(k)
2g(Z/NZ))
# Sp2g(Z/NZ)≤C
√#k,
where γ(k) = #kis the image of the canonical generator of Gal(k/k)under ρk
F.
The deduction of this result from the work of Katz–Sarnak [29] is certainly well
known to experts, but it is difficult to find details in print: see for example [12,
principle 2], where a similar result is labelled Principle ‘because no complete proof
of this statement has appeared in the literature to date’. We thus prefer to provide
a short proof.
Proof. This is a special case of [29, theorem 9.7.13]. More precisely, we fix an
auxiliary prime `dividing Nand a faithful Q`-representation Λ : GSp2g(Z/NZ)→
GL(V) for some Q`-vector space V, and apply [29, theorem 9.7.13] to the `-adic
sheaf F0corresponding to the representation ρ:= Λ ◦ρF. In the notation of [29,§
9.7.1], we further take S= Spec Z[1/N] and X=U.
We check that these data satisfy assumptions (1)–(4) of [29,§9.7.2]; set Garith =
Λ(GSp2g(Z/NZ)) ∼
=GSp2g(Z/NZ) and G= Λ(Sp2g(Z/NZ)) ∼
=Sp2g(Z/NZ) (we
identify these finite groups with constant algebraic subgroups of GL(V)).
1. The fact that ρ(π1(U, η)) ⊂Garith(Q`) is true by definition. The Zariski
density of ρ(π1(U, η)) in Garith(Q`) is equivalent to the fact that Λ ◦ρFsur-
jects onto Garith, or equivalently, that ρFsurjects onto GSp2g(Z/NZ). The
image of ρFcontains the image of ρgeom
F(for any finite field kof character-
istic prime to N), which is Sp2g(Z/NZ) by assumption. On the other hand,
by the commutative diagram in the statement, the image of mult◦ρFcon-
tains ρk
F(Frobk) = #kfor any finite field kof characteristic prime to N. By
Dirichlet’s theorem, the quantity #krealizes all invertible classes modulo N,
hence the image of mult ◦ρFcontains all of (Z/NZ)×. Taken together, these
facts imply that the image of ρFis GSp2g(Z/NZ).
2. The inclusion ρ(π(UQ, η )) ⊆Λ(Sp2g(Z/NZ)) is true by assumption.
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On the L-polynomials of curves over finite fields 9
3. We have to check that for every finite field kand every k-valued point sof
Z[1/N], the geometric monodromy group of F|Usis Λ(Sp2g(Z/NZ)). This is
precisely the assumption that ρgeom
Fis surjective for every finite field k.
4. The image of Λis finite. Thus, all eigenvalues of any matrix in its image are
roots of unity. This implies that Fis ι-pure of weight 0, for any embedding
ιof Q`into C. See also the proof of [15, theorem 4.1].
Since Garith is a finite group (which implies that Karith =Garith is finite, in
the notation of [29, theorem 9.7.13], see [29, remark 9.7.11]), the conclusion fol-
lows from [29, theorem 9.7.13]. Note that here we also use the obvious fact that
# GSpq
2g(Z/NZ) = Sp2g(Z/NZ) for any qprime to N.
Let Cπ
−→ U→Spec Z[1/N] be a smooth, irreducible family of projective curves
of genus g≥1, with the property that the map U→Spec Z[1/N] has geometrically
irreducible fibres, all of the same dimension. The ´etale sheaf F=FC,N := Jac(C)[N]
is a sheaf of Z/NZ-free symplectic modules of rank 2gwhose fibre at a geometric
point x∈Uis the N-torsion of the Jacobian Jac(Cx)[N]. Theorem 2.3 applies to this
situation provided that ρgeom
Fis surjective for every finite field kof characteristic not
dividing N. The existence of a commutative diagram as in (1) is automatic thanks to
well-known properties of the Weil pairing. The assumption ρF(π1(UQ, η)) ⊆Sp(Fη)
is also automatically satisfied, again by the properties of the Weil pairing. We will
say that the family of curves C → Uhas full N-monodromy if the associated
representation ρF:πgeom
1(Uk, ηk)→Sp2g(Z/NZ) is surjective for every finite field
kof characteristic not dividing N.
For the proof of theorem 2.1, we will rely on the functor Mg,3Kof tri-canonically
embedded curves. Referring the reader to [29,§10.6] and [19] for more details, we
recall that for a field kone has
Mg,3K(k) = 




(C/k, α) :
C/k is a smooth pro jective
curve of genus g
αis a basis of H0C, (Ω1
C/k )⊗3




/k-isomorphism.
The functor Mg,3Kwas extensively studied by Mumford [43] and Deligne–Mumford
[19]. We will need the following results:
Theorem 2.4. Deligne–Mumford [19,§5], see also [29, Theorem 10.6.10] Let g≥2.
The following hold:
1. The functor Mg ,3Kis represented by a smooth Z-scheme of relative dimension
3g−3 + (5g−5)2, with geometrically connected fibres.
2.Mg,3Kis a fine moduli space: there exists a universal curve Cg,3K→ Mg,3K.
There is an obvious forgetful functor Mg,3K→ Mg,which on field-valued points
is given by
Mg,3K(k)→ Mg(k)
(C/k, α)7→ C/k.
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10 F. Ballini, D. Lombardo and M. Verzobio
This map is surjective for every field k, and, when kis finite, the fibre over any
C/k ∈ Mg(k) has cardinality # GL5g−5(k)
# Aut(C/k)[29, lemma 10.6.8]. As an immedi-
ate consequence [29, lemma 10.7.8], the intrinsic measure Pintr
g,q on Mg(Fq) can be
described as
1
#Mg,3K(Fq)X
(C,α)∈Mg,3K(Fq)
δC,(2)
where δCis the characteristic function of the singleton {C}. By theorem 2.4 (2),
we have that the sum P(C,α)∈Mg,3K(Fq)δCcan be replaced by
X
u∈Mg,3K(Fq)
δ(Cg,3K)u,(3)
where (Cg,3K)uis the fibre over u∈ Mg,3K(Fq) of the universal curve Cg,3K→
Mg,3K. We will apply theorem 2.3 to U= (Mg,3K)Z[1/N]and F=FCg,3K,N . For
g≥2 and p-N, this family has full N-monodromy by [19, 5.12] (see also the
discussion in [35,§5]). We are almost ready to prove theorem 2.1, but before doing
so, we need a few estimates on the size of Mg(Fq):
Lemma 2.5. For every g≥3, the following hold:
1.#Mg(Fq) = PC∈Mg(Fq)1 = q3g−3(1 + Og(q−1/2));
2.PC∈Mg(Fq)1
# Aut(CFq)=q3g−3(1 + Og(q−1/2));
3.#nC∈ Mg(Fq) : # Aut(CFq)≥2o=Og(q3g−3−1).
For g=2, one has
10.#M2(Fq) = PC∈M2(Fq)1 = q3(1 + O(q−1/2));
20.PC∈M2(Fq)1
# Aut(CFq)=1
2q3(1 + O(q−1/2));
30.#nC∈ Mg(Fq) : # Aut(CFq)>2o=O(q2).
Proof. For g≥3, all the statements follow from [29, lemmas 10.6.23, 10.6.25,
and 10.6.26], together with the obvious asymptotic relation # GL5g−5(Fq)∼
q(5g−5)2(1 + Og(q−1)). For g= 2, one can adapt the proof of the same lemmas
in [29], simply taking into account that the open subset U≤2of M2parametrizing
curves whose geometric automorphism group has order 2 meets every geometric
fibre of M2,3K/Z[29, lemma 10.6.13, remark 10.6.20]. In particular, the generic
value of # Aut(CFq) for (smooth projective) curves of genus 2 is 2. Note that when
the group Aut(CFq) has order 2 it is generated by the hyperelliptic involution. 
Corollary 2.6. For all g≥2, we have X
C0∈Mg(Fq)Pnaive
g,q ({C0})−Pintr
g,q ({C0})=
Og(q−1/2).
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On the L-polynomials of curves over finite fields 11
Proof. For g≥3, using the definition of Pnaive
g,q and Pintr
g,q and lemma 2.5 (1), (2),
and (3) we obtain
X
C0∈Mg(Fq)Pnaive
g,q ({C0})−Pintr
g,q ({C0})
=X
C0∈Mg(Fq)
1
#Mg(Fq)−1/# Aut(C0
Fq)
PC∈Mg(Fq)1/# Aut(CFq)
=X
C0∈Mg(Fq)
q3−3g(1 + Og(q−1/2)) −q3−3g(1 + Og(q−1/2))
# Aut(C0
Fq)
=X
C0∈Mg(Fq)
# Aut(C0
Fq)=1
Og(q3−3g−1/2) + X
C0∈Mg(Fq)
# Aut(C0
Fq)≥2
Og(q3−3g)
=Og

#nC∈ Mg(Fq) : # Aut(CFq)≥2o
q3g−3
+Og#Mg(Fq)
q3g−3q−1/2
=Og(q−1) + Og(q−1/2) = Og(q−1/2).
The same proof applies, with minimal changes, also to g= 2, simply using (1’), (2’),
and (3’) of lemma 2.5 instead of (1), (2), and (3). 
Proof of theorem 2.1.By definition, the weak convergence in the statement means
that—for every continuous bounded function fon Z/NZ[x]≤2g—the integral of f
with respect to (charpolN)∗Pnaive
g,q −µq
Nconverges to 0 as q→ ∞ and similarly for
the sequence of measures (charpolN)∗Pintr
g,q −µq
N. We begin by treating the case of
the measures (charpolN)∗Pintr
g,q −µq
N. Since any function f:Z/NZ[x]≤2g→Ris a
linear combination of characteristic functions of singletons, it suffices to show the
result when fis of the form
f(h(t)) = 


1,if h(t) = h0(t)
0,otherwise
for some polynomial h0(t)∈Z/NZ[t]≤2g. Fix h0(t). The condition charpol(M) =
h0(t)∈Z/NZ[t] defines a (possibly empty) union of conjugacy classes Wh0⊆
GSp2g(Z/NZ). For a curve C/Fq, we denote by ρC,N the natural representation
of Gal Fq/Fqon the N-torsion of Jac(C). We regard Mg,3Kas a Z[1/N]-
scheme. It will play the role of the scheme Uof our general discussion of the
Deligne–Katz–Sarnak equidistribution theorem. We take as curve C → Uthe
universal curve Cg,3Kover Mg,3K.
Recall that we introduced the sheaf F=FCg,3K,N and that the universal family
over Mg,3Khas full N-monodromy [19, 5.12]. Given a curve Cuin the family C, lying
over an Fq-rational point uof U=Mg,3K, the definitions imply that ρCu,N (Frobq)
and ρF(Frobu,Fq) represent the same conjugacy class.
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12 F. Ballini, D. Lombardo and M. Verzobio
For any fixed q, using Eqs. (2) and (3), we have
ZMg(Fq)
f(charpol(C) mod N)dPintr
g,q (C)
=1
#Mg,3K(Fq)X
(C,α)∈Mg,3K(Fq)
f(charpolN(C))
=1
#Mg,3K(Fq)X
u∈Mg,3K(Fq)
1ρCu,N (Frobq)∈Wh0
=1
#Mg,3K(Fq)X
u∈Mg,3K(Fq)
1ρF(Frobu,Fq)∈Wh0
=#{u∈ Mg,3K(Fq) : ρF(Frobu,Fq)∈Wh0}
#Mg,3K(Fq).
(4)
We now apply theorem 2.3 to rewrite the above as
ZMg(Fq)
f(charpolN(C)) dPintr
g,q (C) = #Wh0∩GSpq
2g(Z/NZ)
# Sp2g(Z/NZ)+Og ,N (q−1/2).
(5)
On the other hand, by definition, we have
ZZ/NZ[t]≤2g
f(h(t)) dµq
N(h) = ZGSpq
2g(Z/NZ)
f(charpol(M)) dµGSpq
2g(Z/NZ)(M)
=ZGSpq
2g(Z/NZ)
1charpol(M)=h0dµGSpq
2g(Z/NZ)(M)
=#(Wh0∩GSpq
2g(Z/NZ))
# GSpq
2g(Z/NZ)
=#(Wh0∩GSpq
2g(Z/NZ))
# Sp2g(Z/NZ).
(6)
The claim follows upon comparing Eqs. (5) and (6). We now show that
(charpolN)∗Pnaive
g,q −µq
Nconverges weakly to 0. We have already established that
(charpolN)∗Pintr
g,q −µq
Nweakly converges to 0. Thus, it suffices to show that
(charpolN)∗(Pintr
g,q −Pnaive
g,q ) converges weakly to 0, which in turn is implied by the
following statement: for every ε > 0, there exists q0such that, for all q > q0and all
subsets Aof Mg(Fq), one has |Pintr
g,q (A)−Pnaive
g,q (A)|< ε. This follows immediately
from corollary 2.6, because
|Pintr
g,q (A)−Pnaive
g,q (A)|=X
C0∈APintr
g,q ({C0})−Pnaive
g,q ({C0})
≤X
C0∈A|Pintr
g,q ({C0})−Pnaive
g,q ({C0})|=Og(q−1/2).
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On the L-polynomials of curves over finite fields 13

Remark 2.7. Note that the measure µq
Nonly depends on qmod N. In particular,
if we take a sequence of prime powers qisuch that qimod Nis constant (say equal to
rmod N), Theorem 2.1 shows that the measures (charpolN)∗Pintr
qi,g converge weakly
to µr
N. As a special case, taking N= 2, this applies to any choice of qithat are not
powers of 2.
Remark 2.8. Continuing from remark 2.7, we take N= 2, let qibe the sequence
of all odd primes, and apply the weak convergence of measures to the function
f=1Tr≡0 (mod 2), where
Tr(x2g−a2g−1x2g−1+· ·· +a0) = a2g−1.
In this way, if Cis a curve over Fq,
f(charpol2(C)) = 


1,if Tr(C) := q+ 1 −#C(Fq)≡0 (mod 2)
0,otherwise.
Note that this means f(charpol2(C)) = 1 if and only if #C(Fq) is even. Applying
theorem 2.1 to the case of the naive measure Pnaive
g,q , we obtain the convergence of
1
#Mg(Fq)X
C∈Mg(Fq)
f(charpol2(C)) = #{C∈ Mg(Fq) : Tr(C)≡0 (mod 2)}
#Mg(Fq)
to
µ1
2{M∈GSp2g(Z/2Z) : Tr(M)≡0 (mod 2)}
=#{M∈GSp2g(Z/2Z) : Tr(M)≡0 (mod 2)}
# GSp2g(Z/2Z).
Thus, we have proven
lim
q→∞
#{C∈ Mg(Fq) : Tr(C)≡0 (mod 2)}
#Mg(Fq)
=#{M∈GSp2g(Z/2Z) : Tr(M)≡0 (mod 2)}
# GSp2g(Z/2Z),
where the limit is taken along the sequence of odd primes (or of their powers).
Remark 2.9. Theorem 2.1 implies theorem 1.4, at least when the order qof the
finite field is sufficiently large compared to g. For simplicity, we only discuss the
case of odd q. Using [32], or equivalently [45, theorem A.1] (see also proposition 6.3
and remark 6.4), one checks that the set of characteristic polynomials of matrices
in GSp2g(F2) is the F2-vector space of reciprocal polynomials (which has dimen-
sion g+ 1). Theorem 2.1 with N= 2 implies that, if qg1, all characteristic
polynomials of elements in GSp2g(F2) are also the reduction modulo 2 of the
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14 F. Ballini, D. Lombardo and M. Verzobio
characteristic polynomial of Frobenius corresponding to some curve C/Fq. This
immediately implies that the Q-vector space Lg(Fq) of theorem 1.4 has dimension
at least g+ 1.
3. A conjecture on the distribution of #C(Fq)
In this section, we describe a heuristic (motivated by the Lang–Trotter philosophy
and by results of Gekeler [23] in genus 1) that gives precise predictions for the
number of (smooth projective) curves over a finite field with a given number of
rational points. We define the trace of a curve C/Fqby the formula
Tr(C/Fq) = Tr(C) = q+ 1 −#C(Fq);
by the Hasse–Weil bound, Tr(C) is an integer in the interval [−2g√q, 2g√q].
We begin by recalling the definition of the Sato–Tate measure on the real interval
[−2g, 2g]. Consider the complex Lie group GSp2g(C) and let USp2gbe the maximal
compact subgroup of GSp2g(C) given by unitary symplectic matrices. The group
USp2g, being compact, is canonically equipped with a unique Haar measure µUSp2g
normalized so that µUSp2g(USp2g) = 1.
The trace map tr : USp2g→Chas image contained in the real interval [−2g, 2g].
We denote by dSTg:= tr∗µUSp2gthe pushforward of the Haar measure of USp2g
along the trace map, and we call it the Sato–Tate measure in dimension g. It can
be shown (for example using [48, lemma 8.5]) that dSTgis absolutely continuous
with respect to the Lebesgue measure, so we also denote by STg: [−2g, 2g]→R
the density function of dSTg.
Remark 3.1. Explicit expressions for the function ST2(x) can be found in [34],
see in particular theorem 5.2 of op. cit. We discuss the computation of STg(x) for
general gin remark 3.16.
Let g≥2 and let q=pnbe an odd prime power. We now introduce certain local
factors, both at infinity and for each finite prime. We motivate the choice of these
factors in remarks 3.5 and 3.8. First we need some notation: for an integer tand a
prime `6=p, we define
Xq
t(Z`) = {M∈GSp2g(Z`) : mult M=q, tr M=t}.
Similarly, for any prime `(including `=p), we define
GSpq
2g,Q`(Q`) = {M∈GSp2g(Q`) : mult M=q}
and
Xq
t(Q`) = {M∈GSp2g(Q`) : mult M=q, tr M=t}.
These notations are compatible with our later general definition of GSpq
2g,R and
Xq
t, see notation 4.1 and definition 4.3. We are now ready to introduce our local
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On the L-polynomials of curves over finite fields 15
factors. Given an integer t, we set
ν∞(q, t) = STg(t/√q).
For each prime `6=p, we define
ν`(q, t) = lim
k→∞
# Im Xq
t(Z`)→GSp2g(Z/`kZ)
# GSp2g(Z/`kZ)/(`kϕ(`k)) ,(7)
while for `=pwe set
νp(q, t) = lim
k→∞
# Im Xq
t(Qp)∩Mat2g(Zp)→Mat2g(Z/pkZ)
# Im GSpq
2g,Qp(Qp)∩Mat2g(Zp)→Mat2g(Z/pkZ)/pk.(8)
In these formulas, Xq
t(Z`)→GSp2g(Z/`kZ) and Mat2g(Zp)→Mat2g(Z/pkZ) are
the natural reduction maps modulo `k(or pk), and Im denotes the image of a
function.
Remark 3.2. The limit in the definition of ν`(q, t), including for `=p, exists
thanks to [44, th´eor`eme 2] (see also [47, equation (62), p. 348, Section 3]). Indeed,
the Q`-variety defined by {˜
M∈GSp2g(Q`) : Tr( ˜
M) = t, mult ˜
M=q}has
dimension d:= dim GSp2g,Q`−2, so by Oesterl´e’s theorem [44, th´eor`eme 2] the
numerators of (7) and (8) are asymptotic to c`dk for some constant c. For a similar
reason, the denominators also admit an asymptotic of the form c0`dk for some con-
stant c0(this is also easy to prove directly, at least for the case `6=p). Therefore,
the ratio converges when k→ ∞. We justify the definition given in Eq. (8) in
remark 3.8.
We will work under the assumption that q > 4g2−1; see remark 3.9 for a
discussion of what happens when qis small with respect to g. Let
ν(q, t) = ν∞(q, t)Y
`<∞
ν`(q, t).(9)
Notice that ν∞(q, t) = 0 for t /∈[−2g√q, 2g√q] and in particular ν(q, t) is non-zero
for finitely many t(for a fixed q). The fact that the product (9) converges for all t
is far from obvious. We will show this in §4. Define
ν0(q, t) = ν(q, t)
Pt∈Zν(q, t).(10)
The denominator is non-zero, as we will show in lemma 4.9. By definition, we have
X
t∈Z
ν0(q, t) = 1.
Definition 3.3. Let g≥2, let qbe an odd prime power, and let tbe an integer.
Denote by H(q, t)the number of isomorphism classes of (smooth projective) curves
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16 F. Ballini, D. Lombardo and M. Verzobio
of genus gdefined over Fqwith trace t, that is, for which q+ 1 −#C(Fq) = t.
Define
H0(q, t) = H(q, t)
Pt∈ZH(q, t)=H(q, t)
#Mg(Fq)=Pnaive
g,q ({C∈ Mg(Fq) : Tr(C) = t}).(11)
Thus, H0(q, t)is the ‘naive probability’ that a curve of genus g, defined over Fq,
has trace t.
Notice that H0(q, t) = 0 for t /∈[−2g√q, 2g√q]. We conjecture that, for fixed g,
as q→ ∞ the measures ν0(q, t) and H0(q , t) converge to one another. To make this
precise, we use the L1-norm on the space of probability measures on Z: since Zis
countable, we define the L1distance d(µ1, µ2) between two probability measures as
d(µ1, µ2) := X
t∈Z|µ1(t)−µ2(t)|.
By [41, proposition 4.2], the L1distance is equal up to a factor of 2 to another
natural distance on the space of probability measures, namely the total variation
distance
dtot.var.(µ1, µ2) = sup
A⊆Z|µ1(A)−µ2(A)|.
We can now formulate our conjecture: we phrase it in terms of d, but clearly we
obtain an equivalent statement by replacing dwith dtot.var.
Conjecture 3.4. Fix an integer g≥2. As q→ ∞ along prime powers, we have
d(H0(q, ·), ν 0(q, ·)) →0,(12)
where H0(q, ·) and ν0(q, ·) are considered as probability measures on Z.
Remark 3.5. We now give our reasons for believing in conjecture 3.4. First of all,
notice that by corollary 2.6 one may as well state conjecture 3.4 using the intrinsic
measure Pintr
g,q .
1. For the case of elliptic curves and the intrinsic measure Pintr
g,q , the analogue
of our conjecture has been proved in [23, theorem 5.5], at least when qis a
prime number. In the proof, the author computes the value of ν0(q, t) (see
[23, corollary 4.8]) and shows that it is equal to H0(q, t), which is computed
in [20].
2. Let Cbe a curve of genus gdefined over Fq. The trace tof Cmodulo `kis
equal to the trace of the matrix M∈GSp2g(Z/`kZ) that represents the action
of the Frobenius Frobqon the `k-torsion points of the Jacobian of C. Notice
that there exists ˜
M∈GSp2g(Z`) such that ˜
M≡M(mod `k) with tr( ˜
M) = t
and mult( ˜
M) = q: indeed, it suffices to take as ˜
Mthe matrix representing
the action of Frobenius on the full Tate module T`Jac(C)∼
=Z2g
`. Hence, by
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On the L-polynomials of curves over finite fields 17
theorem 2.1, as q→ ∞ the probability that a curve Chas trace tmodulo `k
converges to
#M∈GSp2g(Z/`kZ) : tr(M) = t, mult(M) = q
# GSp2g(Z/`kZ)/(`kϕ(`k)) .
Taking the limit k→ ∞,ν`(q, t) should represent the probability that, given a
random curve C, the Frobenius endomorphism acts on the `∞-torsion points
of the Jacobian of the curve with trace t. (The numerator of ν`(q, t) counts
those matrices in GSp2g(Z/`kZ) with trace tand multiplier qwhich can be
lifted to Xq
t(Z`). See remark 3.14 for this condition and remark 3.8 for the
case `=p).
Our conjecture can then be seen as a minimalist one: we are essentially
claiming that the distributions of the trace of Frobenius in Z`for differ-
ent primes `are independent of each other (which we know is the case by
theorem 2.1, at least for `6=p), and that (as q→ ∞) they also become
independent of the distribution of the absolute value of Tr(Frob) ∈R. To put
it in another way, conjecture 3.4 is the simplest joint distribution that repro-
duces the correct (known) ‘marginal’ distributions for Tr(C) mod Nand for
|Tr(C)|
|√q|∈[−2g, 2g].
3. The ‘minimalist’ philosophy just outlined is, of course, the same that underlies
the widely believed Lang–Trotter conjecture [36, Part I, Section 3].
4. Finally, numerical evidence points in the direction of the conjecture being
true, see §3.2.
Our conjecture should be contrasted with [9, conjecture 5.1], which makes a
different prediction for H0(q, t). The authors of [9] define (the analogue of our)
ν(q, t) purely in terms of the Sato–Tate density ν∞. We believe that—as happens
for g= 1—one should also take into account the measures ν`for all finite `. In fact,
even though we cannot prove conjecture 3.4, the results of §2are enough to show
that [9, conjecture 5.1] is not correct. The proof of this fact is a bit technical: [9,
conjecture 5.1] refers only to non-hyperelliptic curves and replaces t/√qwith the
nearest integer, both of which introduce formal difficulties. However, the key idea
is comparatively simple, so we isolate it in the next proposition, which shows that
the measures ν∞and H0are substantially different infinitely often. Intuitively, this
contradicts [9, conjecture 5.1]. A complete argument showing that [9, conjecture
5.1] does not hold is given in the preprint version of this article [8]. In particular,
in [8, Appendix A], we prove all the technical details necessary to show that an
argument very similar to that of proposition 3.6 disproves [9, conjecture 5.1]. For
the sake of brevity, and since that proof does not add much to the mathematical
content of the article, we decided to omit it here. The following proposition is stated
for g= 3, but we suspect it should hold for all g≥3.
Proposition 3.6. Let g=3. There exists ε > 0such that for all odd prime powers
qbigger than a constant q0>0there exists t∈[−2g√q , 2g√q]∩Zsuch that
√qPnaive
g,q (Tr C/Fq=t)−STg(t/√q)≥ε.
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18 F. Ballini, D. Lombardo and M. Verzobio
Proof. We denote simply by Pthe naive probability measure Pnaive
g,q on Mg(Fq). We
assume that
∀ε > 0∀q0>0∃q > q0odd prime power such that ∀t∈[−2g√q, 2g√q]∩Z
one has

P(Tr C/Fq=t)−STg(t/√q)
√q
<ε
√q(13)
and aim for a contradiction. Fix ε > 0 and let pbe an odd prime. Let q=pn. We
have
P(Tr(C/Fq)≡0 (mod 2))
=X
t∈[−2g√q,2g√q]∩Z
t≡0 (mod 2)
P(Tr(C/Fq) = t)−STg(t/√q)
√q+STg(t/√q)
√q
=X
t∈[−2g√q,2g√q]∩Z
t≡0 (mod 2)
STg(t/√q)
√q
+X
t∈[−2g√q,2g√q]∩Z
t≡0 (mod 2)
P(Tr C/Fq=t)−STg(t/√q)
√q
=1
√qX
t∈[−2g√q,2g√q]∩Z
t≡0 (mod 2)
STg(t/√q) + E,
with
|E| ≤ (4g+ 1)√q·ε
√q≤(4g+ 1)ε(14)
by (13). On the other hand, some basic analysis shows that (since STgis Riemann-
integrable)
1
√qX
t∈[−2g√q,2g√q]∩Z
t≡0 (mod 2)
STg(t/√q)
converges, as q=pngoes to infinity, to
1
2√qZ2g√q
−2g√q
STg(t/√q)dt =1
2Z2g
−2g
STg(t)dt =1
2.
Therefore,

P(Tr(C/Fq)≡0 (mod 2)) −1
2≤ |E|+ε(15)
for q=pnlarge enough. Let
L1(g) := #{M∈GSp2g(F2) : Tr M≡0 (mod 2),mult M=q≡1 (mod 2)}
# GSp2g(F2).
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On the L-polynomials of curves over finite fields 19
Note that the condition mult M=q≡1 (mod 2) is actually automatic, since
1 is the only invertible element in F2. By remark 2.8, as q→ ∞ we have
|L1(g)−P(Tr(C/Fq)≡0 (mod 2))|=o(1),and in particular, for qlarge enough,
we have
|L1(g)−P(Tr(C/Fq)≡0 (mod 2))|< ε. (16)
We now prove that the initial claim does not hold for g= 3. It seems likely
that a similar strategy can be applied for every g>3. By direct computation,
L1(3) = 1436
2835 ≈0.5065 . . . is strictly greater than 1/2. Fix 0 < ε < |L1(g)−1/2|
8gfor
g= 3. For q=pnlarge enough, by Eqs. (14), (15), and (16), we get

L1(3) −1
2≤|L1(3) −P(Tr(C/Fq)≡0 (mod 2))|
+
P(Tr(C/Fq)≡0 (mod 2)) −1
2
≤ |E|+ 2ε≤(4g+ 3)ε < 
L1(3) −1
2
,
contradiction. 
3.1. Further remarks on conjecture 3.4
In this section, we collect several other remarks on conjecture 3.4 and the possible
limits of its validity. As all the material in this section is speculative, we do not
go into much detail, but we hope that this discussion will encourage others to
investigate the issues raised here.
Since the statistics of the distribution of the trace of principally polarized abelian
varieties (PPAVs) of a fixed dimension gover finite fields are the same as those
of Jacobians (equivalently, of curves of genus g), it seems reasonable to extend
conjecture 3.4 to the family of all PPAVs of a fixed dimension. More precisely and
more generally, we formulate the following conjecture, of which conjecture 3.4 is a
special case.
Conjecture 3.7. Let Ube a scheme of finite type over Zand let A → Ube a
family of g-dimensional, PPAVs with full monodromy. For a prime power q, let
H0(q, t) = #{u∈U(Fq) : q+ 1 −t= #Au(Fq)}
#U(Fq),
seen as a measure on Z. Let ν0be as in (10). As q→ ∞ along prime powers, we
have d(H0(q, ·), ν0(q, ·)) →0, where H0(q, ·) and ν0(q, ·) are considered as probability
measures on Z.
In particular, Gekeler’s results [23] should perhaps be interpreted in this light.
From this perspective, one should perhaps ask if conjecture 3.4 could not be
upgraded to an actual equality for fixed q(as opposed to an asymptotic state-
ment for q→ ∞) when one considers the better-behaved family of all PPAVs. We
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20 F. Ballini, D. Lombardo and M. Verzobio
will see that, while the measures H0(q, t) and ν0(q, t)cannot be equal in general,
even for abelian varieties (remark 3.9), this point of view can still be helpful.
In this section, we mostly focus on conjecture 3.4, but—with minimal
modifications—similar comments also apply to conjecture 3.7. Given the limited
evidence we have in support of conjecture 3.4, it seems safer to restrict our discus-
sion to the special case of the family of all curves (but we have no reason to expect
a substantially different behaviour for any other family of abelian varieties with full
monodromy).
Remark 3.8. Local factor at pWe justify the choice of the local factor (8).
Observe first that the more general formula
ν`(q, t) = lim
k→∞
# Im Xq
t(Q`)∩Mat2g(Z`)→Mat2g(Z/`kZ)
# Im GSpq
2g,Q`(Q`)∩Mat2g(Z`)→Mat2g(Z/`kZ)/`k
reduces to (7) and (8) respectively when `6=pand `=p. The denominator of
this formula is essentially the average over t∈ {0, . . . , `k−1}of the numerator,
so the ratio measures the deviation from the average of the number of symplectic
matrices with a given trace. For g= 1, Gekeler shows [23] that this formula does
give the correct local factor at p. For g>1, at least when the field of definition is
the prime field Fp, one can consider the action of Frobenius on rigid (or crystalline)
cohomology, which is a free W(Fp) = Zp-module of rank 2g: in this way, Frobenius
acts symplectically on a 2g-dimensional Qp-vector space (the cohomology group
tensored with Qp) preserving a Zp-lattice, so it defines a matrix with entries in Zp
and multiplier q(any such matrix does not lie in GSp2g(Zp), because the multiplier
is not invertible in Zp—in fact, such a matrix does not even lie in GL2g(Zp)).
Note that we cannot simply consider the Frobenius action on the Tate module Tp,
because this has rank at most g, so it doesn’t provide a good p-adic analogue of
T`for `6=p. It seems likely that an equidistribution result similar to theorem 2.3
should also hold in rigid cohomology (see [26,30]), which would lead to the local
factor (8), just like theorem 2.3 leads to (7), see remark 3.5.
Remark 3.9. (qsmall with respect to g). Notice that ν0(q, t) can be positive also
for values of tsuch that q+ 1 −t < 0. Of course, this does not make sense, because
q+ 1 −tshould represent the number of Fq-rational points of a curve. The point is
that the support of ν0(q, t) is the full interval [−2g√q, 2g√q], and when qis small
with respect to git may well happen that q+ 1 −2g√q < 0.
There are also subtler issues. The Sato–Tate distribution arises as the push-
forward via the trace map of the Haar measure on USp2g. Suppose that M∈
USp2gcorresponds to the unitarized Frobenius FrobC/Fq
√q, where C/Fqis a smooth
projective curve of genus g. Then, for every m≥1 one has
#C(Fqm) = qm+ 1 −qm/2tr(Mm),
and in particular, for all integers m1|m2, we must have
#C(Fqm1) = qm1+ 1 −qm1/2tr(Mm1)≤qm2+ 1 −qm2/2tr(Mm2) = #C(Fqm2).
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On the L-polynomials of curves over finite fields 21
When qis small with respect to g, there are matrices in USp2gand integers m1|m2
for which this inequality does not hold. In this regime, one should perhaps replace
the usual Sato–Tate measure with the following. Let Xbe the subset of USp2g
consisting of those matrices that satisfy all the inequalities
0≤qm1+ 1 −qm1/2tr(Mm1)≤qm2+ 1 −qm2/2tr(Mm2) = #C(Fqm2)
for all m1|m2. A candidate to replace STgis the pushforward via the trace of the
restriction of the Haar measure to the set X(renormalized so as to have mass 1).
Recall that we are fixing gand sending qto infinity, so this issue does not affect
our conjecture 3.4.
Remark 3.10. Asymmetry of the distribution H0(q, t) An advantage of working
with PPAVs rather than curves is that the former always admit quadratic twists,
which implies that the distribution of their traces is always symmetric around 0.
This is further indication that perhaps conjecture 3.4 is more natural for the family
of PPAVs. In fact, we remark that while ν0(q, t) is symmetric (that is, ν0(q, −t) =
ν0(q, t)), this is not necessarily the case for H0(q, t) as soon as g≥3, as one can
see for example in [9, figure 4], or below in our own figure 3. See also [9,§5] for a
more extensive discussion of the asymmetry of H0(q, t). In particular, we note again
that one cannot have an exact equality H0(q, t) = ν0(q, t) for general g, because the
right-hand side is easily seen to be symmetric. All the same, we expect the two
measures to be arbitrarily close in the limit q→ ∞.
Remark 3.11. (Speed of convergence). The limit in conjecture 3.4 cannot con-
verge too quickly. We briefly show why. Given a measure µon Z, let (−1)∗µ(·)
be the measure defined as (−1)∗µ(t) = µ(−t) for all t∈Z. By definition,
(−1)∗ν0(q, ·)−ν0(q, ·) = 0 since ν0(q, ·) is symmetric. In particular, the moments
of (−1)∗(√qν0(q, ·)) −(√qν0(q, ·)) are 0 for all q. Assume that d(H0(q, ·), ν0(q, ·))
converges to zero sufficiently quickly (for example, assume that the difference is
O(q−k−1) for some k≥0): the first 2kmoments of (−1)∗(√qH0(q, ·))−(√qH0(q, ·))
then also converge to zero as qgoes to infinity. By [9, corollary 5.3], the n-th moment
of (−1)∗(√qH0(q, ·))−(√qH0(q , ·)) converges, for nodd, to a real number bnand bn
is non-zero for nlarge enough (see [9, proposition 2.3]). Hence, for nlarge enough,
the n-th moment of (−1)∗(√qH0(q, ·)) −(√qH0(q, ·)) does not tend to zero as q
goes to infinity. If bn6= 0 and 2k≥n, this is a contradiction.
We thank Christophe Ritzenthaler and Elisa Lorenzo Garc´ıa for their comments
that led to this remark.
Remark 3.12. (Jacobians among PPAVs). We again take the view that con-
jecture 3.4 should be a shadow of a (possibly sharper) statement for the family
of PPAVs of a given dimension. From this point of view, it is important to note
that—asymptotically—100% of PPAVs of dimension 2 are Jacobians (those that
are not are either products of PPAVs of lower dimension or Weil restrictions of
elliptic curves). Thus, for g=2, the two conjectures that one can formulate (for
curves of genus 2 and principally polarized abelian surfaces) are equivalent. For
g= 3, 100% of PPAVs are either Jacobians or quadratic twists of Jacobians (this is
explained by the so-called Serre obstruction, see, e.g., Serre’s appendix to [38]), so
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22 F. Ballini, D. Lombardo and M. Verzobio
conjectures 3.4 and 3.7 for g= 3 are still closely related. As the dimension grows,
conjecture 3.4 can be interpreted as saying that Jacobians are ‘typical’ among
PPAVs—the distribution of the trace on the subfamily of Jacobians is the same
as the distribution among all PPAVs. While we believe that conjecture 3.4 holds
for all genera g, we should point out that it is very hard to get numerical evidence
when the genus/dimension is 4 or more. This is precisely the threshold above which
the difference between PPAVs that are geometrically Jacobians and general PPAVs
becomes (asymptotically) relevant, so it would be interesting to study this regime
more closely. See figure 5 for an example in which we show the difference between
taking into account only Jacobians or all PPAVs.
Remark 3.13. (Principally polarized abelian surfaces with trace zero). In dimen-
sion 2, PPAVs that are not Jacobians are either products of elliptic curves (with the
product polarization) or Weil restrictions of elliptic curves defined over a quadratic
extension. In particular, over the finite field with qelements, there are q2abelian
surfaces that are Weil restrictions of elliptic curves defined over Fq2, but not over
Fq. The Galois representation attached to A:= ResFq2/Fq(E) is the induction
from Gal(Fq/Fq2) to Gal Fq/Fqof the representation attached to E/Fq2, which
implies that the Frobenius trace of Ais zero for any such Weil restriction. Since
the total number of genus-2 curves over Fqis of order q3(see lemma 2.5), we
expect that the proportion of PP abelian surfaces with trace 0 should be sig-
nificantly higher than the proportion of genus-2 curves with trace 0 (both the
number of genus-2 curves and the number of PP abelian surfaces is ≈q3. The
number of PP abelian surfaces with trace 0 is q2more than the corresponding
number of curves. In particular, we expect the proportion of PP abelian surfaces
of trace 0 to be 1/q more than the corresponding proportion of curves). If
we interpret conjecture 3.4 as a prediction for the distribution of the number of
points of PPAVs, this helps in explaining the peak at 0 in figure 5 (this peak
is particularly noticeable since for q= 37 the quantity 1/q is not at all negligi-
ble). Similar comments apply in higher dimensions, but the proportion of PPAVs
having trace zero for geometric reasons becomes less significant as the dimension
increases.
Remark 3.14. (Lift to Z`). Equation (7) requires that we only count those
matrices M∈GSp2g(Z/`kZ) with trace tand multiplier qthat lift to a matrix
˜
M∈Xq
t(Z`). While this condition is natural in our setting (since Frobenius is in
fact represented by an `-adic matrix with the given trace and multiplier), we believe
that omitting this condition should lead to the same result, that is, we conjecture
that
˜ν`(q, t) := lim
k→∞
#M∈GSp2g(Z/`kZ) : tr(M) = t, mult(M) = q
# GSp2g(Z/`kZ)/(`kϕ(`k))
coincides with ν`(q, t). It is not hard to check that this holds for g= 1, but we
have been unable to prove the result in general. The difficulties that arise lie
in understanding the singularities of the variety Xq
t, that is, the Z`-subscheme
of GSpm
2g,Z`defined by the equation Tr(M) = t. When Xq
tis smooth over
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On the L-polynomials of curves over finite fields 23
Z`, an application of Hensel’s lemma shows that ν`(q, t) and ˜ν`(q , t) both
coincide with
#M∈GSp2g(F`) : tr(M) = t, mult(M) = q
# GSp2g(F`)/(`ϕ(`)) .
Note that, without any information on the singularities of a variety X/Z`, it is
very hard to control the point-counts #X(Z/`nZ): for example, for the reduced
variety defined by the equation x4=`y4in the affine plane, we have dimX= 1
and #X(Z/`nZ)`3/2n, with the point-count dominated by the singular points
with x≡y≡0 (mod `n/4). Without control on the singularities of X, it seems
to us that no version of Hensel’s lemma can be applied to understand the ratio
#X(Z/`nZ)/`ndim Xas n→ ∞.
Remark 3.15. (Comparison to other recent work). The recent preprint [49] relates
the moments
Mn(g, q) = EPintr
g,q [#A(Fq)n]
of the random variable ‘number of rational points of A’ (here Ais drawn at random
from Ag(Fq) using a suitable intrinsic measure) to the higher cohomology of certain
moduli spaces, see [49, p. 2]. This yields explicit formulas for these moments for
small gand n[49, corollaries 4.3 and 5.4] and it would be interesting to compare
these results with the predictions of conjecture 3.4. It may be possible to carry out
this comparison by using the techniques of [3,5].
In particular, [3, theorem A] comes near to proving conjecture 3.4 in the context
of PPAVs. However, we point out that to establish conjecture 3.4 one would still
need to overcome several obstacles: the formula of [3, theorem A] only applies to
certain isogeny classes of abelian varieties and involves Tamagawa numbers that
would have to be averaged; even more substantially, it is not clear how one would
isolate Jacobians among all abelian varieties. Finally, even though this is perhaps
only a technical problem, the existence of the limits (7) and (8) seems substantially
easier to prove in the context of [3, theorem A] than it is in the general case we
consider here (essentially because in the setting of [3, theorem A] the expression
appearing under the limit sign in (7) is constant for k0, which is not necessarily
true in our generality).
3.2. Numerical evidence
In this section, we report on numerical experiments that seem to support conjecture
3.4. The data are computed using MAGMA [11]. All the MAGMA scripts to verify
our data are available online [7].
In the graphs below (see figures 1,2,3,4,5), we plot the distribution t7→ H0(q, t)
for various values of gand q. These distributions are obtained by directly counting
all isomorphism classes of curves of the given genus over the given finite field (the
data for q= 53, g = 3 are taken from [40]). In addition, on the same graphs, we also
plot an approximation of the Sato–Tate density and of ν0(q, t). We briefly explain
how we obtain these approximations, starting with a general technique to compute
the Sato–Tate density in arbitrary dimension.
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24 F. Ballini, D. Lombardo and M. Verzobio
Figure 1. Case g= 2 and q= 1009. The red dots are the values of H0. The black stars are
the values of the approximation of ν0(q, t). The blue graph is the approximation of the
Sato–Tate density. In this case, d(H0, ν0)≈0.00439 and d(H0, ν∞)≈0.15528.
Figure 2. Case g= 2 and q= 101. The red dots are the values of H0. The black stars are
the values of the approximation of ν0(q, t). The blue graph is the approximation of the
Sato–Tate density. In this case, d(H0, ν0)≈0.01117 and d(H0, ν∞)≈0.15166.
Remark 3.16. (Computation of STg(x) for arbitrary g). For general g, the density
STg(x) can be calculated up to arbitrary precision by using a technique due to
Kedlaya-Sutherland [31] and Lachaud [34]. One can first use [31, Section 4.1] to
compute the moments of STg, that is,
mn=Z2g
−2g
xndSTg(x).
Once the moments (or at least, sufficiently many moments) are known, we can
recover STg(x) as follows. Let Ln(x) be the Legendre polynomials, which form a
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On the L-polynomials of curves over finite fields 25
Figure 3. Case g= 3 and q= 53. The red dots are the values of H0. The black stars are
the values of the approximation of ν0(q, t). The blue graph is the approximation of the
Sato–Tate density. In this case, d(H0, ν0)≈0.03842 and d(H0, ν∞)≈0.03940.
Figure 4. Case g= 2 and q= 5. As pointed out in remark 3.9, there is an issue when
q+ 1 −t < 0 (for example when t= 7). Indeed, H0(q, 7) = 0 because q+ 1 −trepresents
the number of Fq-rational points of a curve. Instead, both ν0(q, 7) ≈0.0009 and ν∞(q, 7) ≈
0.0011 are strictly positive.
complete orthogonal basis of L2([−1,1]). By rescaling, the polynomials
˜
Ln(x) := Z2g
−2g
Ln(x/2g)2−1/2
Ln(x/2g)
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26 F. Ballini, D. Lombardo and M. Verzobio
Figure 5. This graph shows the difference between considering all PPAVs or only Jacobians
of curves (see remark 3.12). We take g= 2 and q= 37. We plot in red the distribution H0
and in black (an approximation of) the distribution ν0(q, t). The green dots represent
the probabilities of the various traces when we take into account all principally polarized
abelian surfaces over Fq. Call this distribution H00. The distance between the distributions
H0and ν0(q, t) is ≈0.02673. The distance between H00 and ν0(q, t) is ≈0.00777. Note in
particular the considerable difference between the data at t= 0, where the inclusion of
all PPAVs gives a much better agreement with our prediction. An explanation for this
phenomenon is given in remark 3.13.
form an orthonormal basis of L2([−2g, 2g]). From the explicit expression of ˜
Ln(x) =
Pn
i=0 an,ixias a polynomial, one can easily compute
cn=Z2g
−2g
˜
Ln(x)dSTg(x) =
n
X
i=0
an,imi.
Finally, we have the convergent expansion in L2([−2g, 2g])
STg(x) = X
n≥0
cn˜
Ln(x),(17)
which allows the computation of STg(x) to arbitrary precision. In our numerical
experiments, we use this technique to approximate ST3(x).
In our numerical experiments, we approximate the Sato–Tate density with the
value of the series in Eq. (17) truncated at n≤100. For ν0(q, t), we approximate
the value of ν(q, t) (see Eq. (9)) by considering the product of ν`(q, t) for `≤100
and `=∞. To compute an approximation of ν`(q, t) for `prime, we compute the
value of the expression appearing under the limit sign in Eq. (7) for k= 1 or 2. To
compute an approximation of ν∞(q, t), we use our approximation of the Sato–Tate
density.
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On the L-polynomials of curves over finite fields 27
Let
H0
intr(q, t) = Pintr
g,q ({C∈ Mg(Fq) : Tr(C) = t}).
We compute the value of H0
intr(q, t) by direct enumeration of all the curves of genus
gdefined over Fq.
Finally, below each graph, we also give the distance dbetween the measures
H0:= H0
intr(q, ·) and ν0:= ν0(q , ·), as well as the distance between H0and the
Sato–Tate measure. Our conjecture predicts that d(H0, ν0) should go to 0 as qgoes
to infinity. As a consequence of [9, conjecture 5.1], d(H0, ν∞) should go to 0. We
proved in proposition 3.6 that the conjecture does not hold.
4. Well-posedness of Eq. (9)
In this section, we prove that the quantity ν(q, t) is well defined. We have already
observed (remark 3.2) that ν`(q, t) is well defined for all `≤ ∞, so it suffices to
show that, as `→ ∞ among the prime numbers, we have ν`(q, t) = 1 + O(`−2).
This suffices to ensure that the product (9) converges.
As a preparation for the proof, we introduce the following notation and make
some remarks.
Notation 4.1. Let Rbe a (commutative unitary) ring and let m∈R×be a
fixed element. We define GSpm
2g,R as the subscheme of GSp2g,R cut by the equation
mult(M) = m.
Remark 4.2. Let us fix the antisymmetric form 0 Idg
−Idg0!. The matrix
Mm:=











m
...
m
1
...
1











is in GSp2g(R) and has multiplier m. Multiplication by Mmgives an algebraic
isomorphism between the R-schemes Sp2g,R and GSpm
2g,R. The same applies for
any matrix Mm∈GSp2g(R) with multiplier m. In particular, GSpm
2g,R is smooth
for any value of m. If Ris a field, the dimension of GSpm
2g,R is equal to dim Sp2g,R.
In what follows we will be interested in the subschemes of GSpm
2g,R defined by the
equation Tr(M) = tfor a fixed value of t∈R. We will mostly work with R=Z`
and R=F`.
Definition 4.3. For m∈R×, t ∈R, we define the R-scheme Xm
tas the subscheme
of GSpm
2g,R defined by the equation Tr(M) = t.
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28 F. Ballini, D. Lombardo and M. Verzobio
Notice that, if we fix m∈Z\{0}, then mis invertible in Z[1/m], and hence Xm
t
makes sense as a scheme over Spec Z[1/m]. We will be able to reduce this scheme
modulo any prime that does not divide m.
4.1. Number of points of Xm
tover finite fields
In this section, we study the number of F`-points of Xt
m(theorem 4.4 and lemma
4.5) and show that a large proportion of them correspond to smooth points of
Xt
m(lemma 4.6). For the first objective, our approach is inspired by [39]. More
precisely, the main result of [39] gives a formula counting the number of elements
in GSp2g(F`) with given trace and determinant. The same strategy allows us to
prove the following version, where we count matrices with given trace and multiplier.
Before stating the result, we remind the reader that the q-binomial coefficient "n
r#q
is defined as Qr−1
j=0
qn−j−1
qr−j−1. For ease of comparison with [39], we adopt the same
notation as in op. cit.
Theorem 4.4. Let qbe a prime power, ζ∈F×
q, and η∈Fq. Let
Tm(ζ, η ) = qX
α1,...,αm∈F×
q
tα1+ζα−1
1+· ·· +αm+ζα−1
m−(q−1)m,
where
t(x) = 


1,if x=η
0,otherwise,
and the sum is regarded as t(0) for m=0. Let
C(ζ, η ) := {g∈GSp2n(Fq)mult g=ζ, tr g=η}=Xζ
η(Fq).
We have the following exact formula for C(ζ , η):
C(ζ, η ) = qn2−1
n
Y
j=1 q2j−1+E, (18)
where
E=qn2−1bn/2c
X
b=0 
qb2+b"n
2b#q
b
Y
j=1
(q2j−1−1)
×bn/2−bc
X
l=0
qlR(n−2b+ 1, l)Tn−2b−2l(ζ , η)
,(19)
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On the L-polynomials of curves over finite fields 29
R(m, l)denotes
R(m, l) = X
0<j1<···<jl<m−l
l
Y
ν=1
(qm−ν−jν−1),
and we set by convention R(m, 0) = 1.
Proof. The proof is virtually identical to that of [39, theorem 1]: if one simply
replaces every occurrence of det with mult in the proof of [39, theorem 1] everything
goes through without difficulty. More precisely, let
e(x) = 


1 if x=ζ,
0 otherwise.
Throughout the proof, several instances of det(dα) = αnare replaced by mult(dα) =
α, where dα= Idn0
0αIdn!. In particular, the sums Pα∈F×
qe(αn) are replaced
by Pα∈F×
qe(α). In the proof of [39, theorem 1], the sum Pα∈F×
qe(αn) evaluates
to the number Sof nth roots of ζin F×
q; in our case, the sum Pα∈F×
qe(α) simply
evaluates to 1 for all ζ∈F×
q.
We will think of the expression Eappearing in Eq. (19) as an error term. We
now proceed to bound this error. We work with a fixed value of n: this implies in
particular that the number of summands (resp. factors) in the sum (resp. products)
appearing in (19) is O(1). We then have the following estimates (where the implicit
constants may depend on n, but not on q):
1. "n
r#q
=Qr−1
j=0
qn−j−1
qr−j−1Qr−1
j=0
qn−j
qr−j=Qr−1
j=0 qn−r=qnr−r2, and hence in
particular "n
2b#qq2bn−4b2.
2. Qb
j=1(q2j−1−1) ≤Qb
j=1 q2j−1=qPb
j=1(2j−1) =qb2.
3. We claim that R(m, l)qml−l(l+1) for m≤n. To see this, notice that the
length of the sum defining R(m, l) is O(1), so it suffices to estimate the largest
summand. (The length of the sum is O(1) because it is bounded by a function
of m, and mis bounded in terms of n.) Clearly, the condition jk> jk−1for
k= 2, . . . , l yields jν≥ν, so qm−ν−jν≤qm−2ν. We can then estimate
R(m, l)
l
Y
ν=1
qm−2ν=qml−l(l+1),
as claimed.
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30 F. Ballini, D. Lombardo and M. Verzobio
4. We also claim that |Tm(ζ, η )|  qm. To show this, we first remark that, for
fixed α1, . . . , αm−1∈F×
q, the equation
α1+ζα−1
1+· ·· +αm+ζα−1
m=η
has at most 2 solutions αm∈F×
q. We can then rewrite and estimate |Tm(ζ, η)|
as follows:
qX
α1,...,αm−1∈F×
qX
αm∈F×
q
α1+ζα−1
1+···+αm+ζα−1
m=η
1−(q−1)m
≤q·(q−1)m−1·2+(q−1)mqm,
as desired.
We now give an upper bound for the quantity |E|, with Eas in Eq. (19).
According to our previous estimates,

bn/2−bc
X
l=0
qlR(n−2b+ 1, l)Tn−2b−2l(ζ , η)bn/2−bc
X
l=0
qlq(n−2b+1)l−l(l+1)qn−2b−2l
qn−2bbn/2−bc
X
l=0
q(n−2b−1)l−l2.
Notice again that the length of this sum is O(1), so it suffices to give an upper bound
for its largest summand. For a fixed value of b, the exponent (n−2b−1)l−l2is
maximal for l=n−2b−1
2(which might not be an integer, but still provides an upper
bound for the value of the exponent). We thus get

bn/2−bc
X
l=0
qlR(n−2b+ 1, l)Tn−2b−2l(ζ , η)qn−2bqn−2b−1
22
.
We now consider the expression

qb2+b"n
2b#q
b
Y
j=1
(q2j−1−1) bn/2−bc
X
l=0
qlR(n−2b+ 1, l)Tn−2b−2l(ζ , η)
qb2+bq2bn−4b2qb2qn−2bqn−2b−1
22
,
corresponding to a fixed value of bin the sum (19). The exponent of qon the
right-hand side is again a quadratic function of b(to be precise, it is given by
−b2+bn +1
4n2+1
2n+ 1/4), which is easily seen to achieve its maximum for
b=n/2. This maximum value is given by 1
2n2+1
2n+1
4. Thus, q(1/2)n2+(1/2)n+1/4
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On the L-polynomials of curves over finite fields 31
is an upper bound for each summand. Keeping once again in mind that the length
of the sum is O(1), we have proved that
|E|  qn2−1q1
2n2+1
2n+1
4=q3
2n2+1
2n−3
4.
We can finally prove:
Lemma 4.5. For all g≥2, all primes `, and all mwith (m, `) = 1 we have
#Xm
t(F`)
# GSp2g(F`)/(`ϕ(`)) =#{M∈GSp2g(F`) : Tr(M) = t, mult M=m}
# GSp2g(F`)/(`ϕ(`)) (20)
= 1 + O(`−2),
where the constant implicit in the big-Osign depends only on g.
Proof. The numerator of (20) is given by (18) (with n=g,q=`,ζ=mand η=t).
Note that `g2−1Qg
j=1 `2j−1is exactly # GSp2g(F`)
`ϕ(`). Thus, the ratio in (20) is
given by
1 + E
1
`(`−1) # GSp2g(F`).
Since
1
`(`−1)# GSp2g(F`) = 1
`(`−1)(`−1)# Sp2g(F`) = `g2−1
g
Y
j=1
(`2j−1) `2g2+g−1,
we obtain that (20) is
1 + O`3
2g2+1
2g−3
4−(2g2+g−1)= 1 + O`−1
2g2−1
2g+1
4,
which is 1 + O(`−2) for all g≥2. 
Lemma 4.6. Fix t, m ∈Zand let `≥3be a prime number not dividing m. Let
X:= (Xm
t)F`= GSp2g,F`∩{Tr = t}∩{mult = m},
considered as a variety over F`. Write Xsmooth for the smooth locus of X. The
singular locus Xsing has codimension at least 3in X. We have #Xsing(F`) =
O(`2g2+g−4)and
#Xsmooth(F`) = # GSp2g,F`(F`)
`ϕ(`)(1 + O(`−2)).
The implied constants depend on tand m, but not on `.
Proof. We view Xas a subvariety of the affine space A(2g)2
F`, considered as the space
of matrices of size 2g×2g. The variety Xis the intersection of GSpm
2g,F`∼
=Sp2g,F`
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32 F. Ballini, D. Lombardo and M. Verzobio
with the hyperplane Hdefined by the condition Tr(M) = t. The hyperplane section
GSpm
2g,F`∩His smooth at a point x∈X(F`) unless the (tangent space to the)
hyperplane Hcontains the tangent space of GSpm
2g,F`at the point x. Take any
point x∈X(F`). Since xhas multiplier m, left multiplication by x∈GSp2g(F`)
gives an isomorphism Lxbetween Sp2g,F`and GSpm
2g,F`. The differential of Lxgives
an isomorphism between the tangent space at Id and the tangent space at x. If
we identify both tangent spaces to subspaces of the tangent space to A(2g)2
F`(that
is, to matrices of size 2g×2g), the differential in question is simply multiplication
by xitself. Thus, we may view the tangent space at xas the image via xof the
tangent space at Id, which is the Lie algebra of Sp2g,F`. This can be written down
explicitly: choose the anti-symmetric bilinear form represented by the matrix
Ω := 0 Idg
−Idg0!.
Differentiating the condition tMΩM= Ω, we find that the Lie algebra of Sp2g,F`is
given by those matrices Mthat satisfy tMΩ + ΩM= 0. Writing Min block form,
we obtain that Lie Sp2g,F`is the vector space of F`-matrices
A B
C D!
with tB=B, tC=C, tD=−A(see [22,§16.1] for the identical calculation over
the complex numbers). From the previous arguments, it follows that xcan only be
a singular point if
xLie(Sp2g,F`)⊆ {Tr = 0},
which is to say
Tr(xL) = 0 ∀L∈Lie(Sp2g,F`).
Write x= α β
γ δ!and L= A B
C D!with B, C symmetric and D=−tA. This
easily gives Tr(βC) = Tr(γB) = 0 for all symmetric B, C (which implies that β, γ
are anti-symmetric) and
Tr(αA −δ·tA) = Tr(αA −A·tδ) = Tr(αA −tδ·A) = 0
for all A(which implies α=tδ).
Thus, the locus of non-smooth points is contained in the linear space defined by
the equations
tβ=−β, tγ=−γ , tδ=α.
This linear space has dimension g2+ 2g(g−1)
2= 2g2−g, and hence codimension
at least 2g−1≥3 in X, each of whose irreducible components has dimension
at least dim GSpm
2g,F`−1 = dim Sp2g,F`−1=2g2+g−1 (at least one irreducible
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On the L-polynomials of curves over finite fields 33
component has exactly this dimension). We now observe that by the Lang–Weil
estimates [37, theorem 1] we have #Xsing(F`) = O(`dim Xsing ) = O(`dim X−3),
with an implicit constant that depends only on Xand not `. Taking into account
the obvious decomposition Xsmooth(F`)FXsing (F`) = X(F`) and the fact that
#X(F`) = # GSp2g,F`(F`)
`ϕ(`)(1 + O(`−2)) by lemma 4.5, we obtain the desired estimate
#Xsmooth(F`) = # GSp2g,F`(F`)
`ϕ(`)(1 + O(`−2)).
4.2. Convergence of the infinite product (9)
Lemma 4.7. Let g≥2,qbe a prime power, and t∈Z. Let `≥3be a prime
that does not divide q. We have ν`(q, t) = 1 + O(`−2), where the implied constant
depends on g,q, and t.
Proof. Let X:= Xq
t. We apply [44, Property (U), p. 326] to
X(Z`) = {M∈GSp2g(Z`) : Tr M=t, mult M=mq}
m= 1, N = (2g)2, n =n, B =x0+`Z(2g)2
`
where x0mod `is a matrix lying in Xsing (F`). We first assume that XZ`is irre-
ducible. Considering Xas a scheme over the spectrum of the DVR Z`, [1,lemma
0B2J] shows that XF`is equidimensional of some dimension d, and Oesterl´e’s result
gives
#{closed balls Aof radius `−n:A∩X6=∅and A⊆B} ≤ C`dim X(n−1)
for a constant Cthat depends only on the degree in dimension d[44,§0.6] of XF`,
which is clearly bounded independently of `. On the other hand, we have
#{closed balls A of radius `−n:A∩X6=∅and A⊆B}
= # 




M∈GSp2g(Z/`nZ) : ∃˜
M∈X(Z`)
˜
M≡M(mod `n)
M≡x0(mod `)





.
Hence, summing over the points x0∈Xsing(F`), we obtain
#




M∈GSp2g(Z/`nZ) : ∃˜
M∈X(Z`)
˜
M≡M(mod `n)
Mmod `∈Xsing (F`)




≤C#Xsing(F`)`(n−1) dim X.
(21)
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34 F. Ballini, D. Lombardo and M. Verzobio
If XZ`is not irreducible, we can repeat the above argument with each irreducible
component Xi. If Ciis the constant that corresponds to the component Xi, applying
the previous argument to Xiand summing over iwe obtain
#




M∈GSp2g(Z/`nZ) : ∃˜
M∈X(Z`)
˜
M≡M(mod `n)
Mmod `∈Xsing (F`)





≤ X
i
Ci!#Xsing(F`)`(n−1) dim X.
Note that the number of irreducible components is bounded independently of `,
and so is the constant (PiCi) (because the degrees are bounded in terms of the
equations of X, which are independent of `). The conclusion is that there exists a
constant Csuch that (21) holds for all nand all but finitely many `.
Recall now the definition of ν`(q, t) from Eq. (7): it is the limit over kof the
ratio
# Im X(Z`)→GSp2g(Z/`kZ)
# GSp2g(Z/`kZ)/(`kϕ(`k)) .(22)
Clearly, a matrix Mcounted in the numerator of this expression in particular
reduces modulo `to a point in X(F`). For a fixed x0∈X(F`), denote by N(x0, k)
the quantity
N(x0, k) = # M∈Im X(Z`)→GSp2g(Z/`kZ):M≡x0(mod `).
When x0is a smooth point of X(F`), Hensel’s lemma shows that x0has precisely
`(k−1) dim XF`lifts to X(Z/`kZ), and each of these further lifts to a point in X(Z`)
(note that a smooth point necessarily lies on a component of dimension equal to
dim XF`: indeed, Xis a hyperplane section of a smooth variety, so every smooth
point lies on a component of maximal dimension). Therefore, we have N(x0, k) =
`(k−1) dim XF`for such x0. On the other hand, Eq. (21) and lemma 4.6 show that
Px0∈Xsing(F`)N(x0, k ) = O(`kdimXF`−3).
Thus, the numerator of (22) is given by
X
x0∈X(F`)
N(x0, k) = X
x0∈Xsmooth(F`)
N(x0, k) + X
x0∈Xsing(F`)
N(x0, k)
= #Xsmooth(F`)`(k−1) dim XF`+O(`kdim XF`−3)
=# GSp2g(F`)
`ϕ(`)(1 + O(`−2)) ·`(k−1) dim XF`+O(`kdim XF`−3),
where in the last equality we have applied lemma 4.6. Using dim XF`=
dim GSp2g,F`−2 and dividing by
# GSp2g(Z/`kZ)
`kϕ(`k)=# GSp2g(F`)
`ϕ(`)`(k−1) dim XF`,
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On the L-polynomials of curves over finite fields 35
we obtain that (22) is 1 + O(`−2). The claim follows upon passing to the limit
in k.
Theorem 4.8. Let qbe a prime power and t∈Z. The infinite product
ν(q, t) = ν∞(q, t)Y
`<∞
ν`(q, t)
converges.
Proof. By lemma 4.7, we have ν`(q, t) = 1 + O(`−2) as `ranges over primes `≥3
that do not divide q. The factors ν∞(q, t), ν2(q, t) and νp(q , t) are well defined, as
already argued. It follows that the infinite product Q`<∞ν`(q, t) converges. 
We conclude this section by proving that ν(q, t) is strictly positive for t∈Z
lying in the interval (−2g√q, 2g√q). This also proves that the denominator in Eq.
(10) is non-zero and that ν0(q, t) is strictly positive for t∈Zlying in the interval
(−2g√q, 2g√q).
Lemma 4.9. Let tbe an integer in the open interval (−2g√q, 2g√q). The quantity
ν(q, t)is non-zero (hence strictly positive).
Proof. Since the infinite product defining ν(q, t) converges, it suffices to show that
each factor in this product is non-zero. This is well known to be true for the infinite
factor ν∞(q, t), whose support is the interval [−2g√q, 2g√q]. To show that ν`(q, t)
is non-zero (including for `=p) we proceed as follows. Let Xq
tbe as in definition
4.3 (for the ring R=Q`) and let for simplicity Xq:= GSpq
2g,Q`. We rewrite the
definition of ν`(q, t) in the form of remark 3.8, namely,
ν`(q, t) = lim
k→∞
# Im Xq
t(Q`)∩Mat2g(Z`)→Mat2g(Z/`kZ)
# Im (Xq(Q`)∩Mat2g(Z`)→Mat2g(Z/`kZ)) /`k.
Set d:= dim GSp2g,Q`−2 = 2g2+g−1 and multiply both numerator and denomina-
tor by `−kd. We see both Xqand Xq
tas subschemes of A(2g)2
Q`, so that their Q`-points
are subsets of Q(2g)2
`. Let Yq
t:= Xq
t(Q`)∩Z(2g)2
`and Yq:= Xq(Q`)∩Z(2g)2
`. The
sets Yq
tand Yqare closed analytic subsets of Z(2g)2
`. Note that Xqis smooth and
irreducible of dimension d+ 1, hence Xq
t—which is a subscheme of Xqdefined by
a single non-trivial equation—has dimension d: slicing with a hyperplane makes
the dimension drop at most by 1; on the other hand, the dimension must drop (if
Xq
thad a component of dimension d+ 1, by the irreducibility of Xqwe would have
Xq
t⊇Xq, which is not the case). More precisely, by the same argument, every
irreducible component of Xq
thas dimension d. We can thus write
ν`(q, t) = lim
k→∞
`−dk# Im(Yq
t→(Z/`kZ)(2g)2)
`−(d+1)k# Im(Yq→(Z/`kZ)(2g)2).(23)
Recall from [44,§3] the notion of measure in dimension d of a closed analytic
subset Yof Z(2g)2
`of dimension ≤d(denoted by µd(Y)). By [44, th´eor`eme 2], the
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36 F. Ballini, D. Lombardo and M. Verzobio
numerator and denominator of (23) admit limit as k→ ∞, and these limits are
given by µd(Yq
t) and µd+1(Yq), respectively. Hence, ν`(q, t) = µd(Yq
t)
µd+1(Yq).
To conclude, it suffices to show that µd+1(Yq) and µd(Yq
t) are both strictly
positive. Note that Yqis open in Xq(Q`) for the `-adic topology, since it is the
intersection of Xq(Q`) with the `-adically open set (Z`)(2g)2; a similar comment
applies to Xq
t. We claim that to check the positivity of µd+1(Yq) and µd(Yq
t) it
suffices to show that Yq, Y q
tcontain at least one smooth point of Xq(Q`), Xq
t(Q`)
respectively. To show this implication, we argue as follows (we discuss the case
of Xq
t, but the case of Xqis completely analogous, and in fact easier since Xq
is smooth). The `-adic analytic variety Xq
t(Q`) is of pure dimension d, and its
smooth points (Xq
t)smooth form an `-adically open set, so the intersection Yq
t∩
(Xq
t)smooth is `-adically open (recall that Yq
tis `-adically open). In particular, if
Yq
tcontains at least one smooth point of Xq
t(Q`), then it contains an open set
of smooth points. The local dimension at each smooth point of Xq
t(Q`) is d. By
construction of the measure µd(see again [44,§3]), an open subset of Xq
t(Q`)
consisting of smooth points has positive measure: indeed, in the case of constant
dimension dthat we are considering here, µdis constructed locally by taking an
analytic isometry between a ball in (Xq
t)smooth and an open ball in Qd
`, and pulling
back the Haar measure νof Qd
`, normalized by ν(Zd
`) = 1. It is then clear that any
open set in (Xq
t)smooth has positive measure with respect to µd, and we have shown
that Yq
tcontains an open set of smooth points of Xq
t(Q`) as soon as it contains one.
We are thus reduced to checking that Yq, Y q
tcontain at least one smooth point of
Xq(Q`), Xq
t(Q`) respectively.
For Xq, which is smooth, this amounts to constructing a symplectic matrix
with coefficients in Z`and given multiplier; this follows immediately from
either proposition 6.3 and remark 6.5 or from remark 4.2 after observing that
the identity matrix lies in Sp2g(Z`). For Xq
t, we construct the relevant point
explicitly.
We observe that Xq
tarises as a fibre of the trace map:
trace : Xq→A1
i.e., Xq
t= trace−1(t). A sufficient condition for a point P∈Xq
tto be smooth is
the existence of a curve C⊆Xqcontaining Psuch that the restriction of the trace
map
trace : C→A1
has non-vanishing differential at P. To see this, notice that the dimension of the
tangent space at Pin Xq
tis the dimension of the tangent space at Pin Xqminus
the dimension of the image of the differential of the trace map (restricted to Xq)
at P. Let us fix the symplectic form
Ω = 0 Idg
−Idg0!.
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On the L-polynomials of curves over finite fields 37
We consider the curve Ma, parametrized by a∈A1, given by
Ma=




a z a −q z
tz q Idg−1tz0g−1
1z1z
tz0g−1tzIdg−1





where zis the 1 ×(g−1) vector (0, . . . , 0). One checks that Ma∈Xq(Q`):
up to a suitable change of basis, the symplectic form is represented by
diag 0 1
−1 0!, . . . , 0 1
−1 0!!, and in the same basis Mabecomes the matrix
diag a a −q
1 1 !, 1 0
0q!, . . . , 1 0
0q!!, which is manifestly symplectic since
every 2 ×2 block has determinant q. Moreover, trace(Ma) = a+qg −q+g; the
composition
a→Ma→trace(Ma) = a+qg −q+g
is just a translation of A1, which implies that the differential of the trace map at
Mais surjective. Therefore, the point Mt−qg+q−g∈Xq
tis smooth and its entries
are elements of Z`. This concludes the proof. 
5. Proof of theorem 1.4
The goal of this section is to show that the set Pg(Fq) of definition 1.2 spans a Q-
vector space of dimension g+ 1 for all pairs (g,q). For a fixed genus gand qg1,
this follows from theorem 2.1 (see remark 2.9). Studying more precisely the set
Pg,2(Fq) for every fixed value of q, we prove the statement for all qand g. Recall
that Pg(Fq) is defined in definition 1.2 and Pg,2(Fq) is its reduction modulo 2.
As we pointed out in the introduction, we split our proof of theorem 1.4 into two
parts, one for the case podd and one for the case p= 2, since the properties of the
2-torsion points are slightly different when the characteristic is odd or even.
5.1. Proof of theorem 1.4:podd
Throughout this section, the prime p= char(Fq) is assumed to be odd. Thanks to
theorem 1.7, it makes sense to define fC(t)∈Z[t] as fC,`∞(t), where `is any prime
different from p; from now on, we shall choose `= 2. This choice has the additional
advantage that working modulo 2 makes the connection between the L-polynomial
and the characteristic polynomial of Frobenius particularly simple:
Corollary 5.1. We have PC(t)≡fC(t) (mod 2).
Proof. Write PC(t) = P2g
i=0 aiti∈Z[t] and fC(t) = P2g
i=0 biti. By theorem 1.7, we
have the equality bi=a2g−i, and since qis odd we also have bi=a2g−i=qg−iai≡
ai(mod 2). 
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38 F. Ballini, D. Lombardo and M. Verzobio
We now recall a concrete description for the vector space of 2-torsion points of a
hyperelliptic Jacobian, at least in the case when the hyperelliptic model is given by
a polynomial of odd degree. Let f(x)∈Fq[x] be a separable polynomial of degree
2g+ 1 and let C/Fqbe the unique smooth projective curve birational to the affine
curve y2=f(x).Furthermore, let J/Fqbe the Jacobian of Cand {α1, . . . , α2g+1}
be the set of roots of f(x) in Fq. Then for i= 1, . . . , 2g+ 1, we have a point
(αi,0) ∈C(Fq); also notice that C, being given by an odd-degree model, has a
unique point at infinity, which we denote by ∞. We denote by Ri= [(αi,0) − ∞]
the classes of the divisors Qi= (αi,0) − ∞ in J(Fq). We then have the following
well-known description for the 2-torsion of J(see for example [25, Section 4]):
Lemma 5.2. The following hold:
1. Each of the divisor classes Ri∈J(Fq)represents a point of order 2.
2. The classes Rispan J[2].
3. The only linear relation satisfied by the Riis R1+· ·· +R2g+1 = 0.
We can now compute the action of Frobenius on the 2-torsion points of C. A
similar result appeared independently in [16, proposition 2.4].
Lemma 5.3. With notation as above, write f(x) = Qr
i=1 fi(x)for the factorization
of f(x) as a product of irreducible polynomials in Fq[x], and let di= deg(fi).
Let ρ2: Gal(Fq/Fq)→AutF2(J[2]) be the Galois representation attached to the
2-torsion points of J. Then
fC,2(t) = det(tId −ρ2(Frob)) = (t−1)−1
r
Y
i=1
(tdi−1) ∈F2[t].
Proof. As above, let ∞be the unique point at infinity of C, and for i= 1, . . . , 2g+1
let Qi= (αi,0) − ∞ ∈ DivC(Fq). Write Pifor the image of Qiin the F2-vector
space DivC(Fq)⊗F2, and let Vbe the (2g+ 1)-dimensional F2-vector subspace of
DivC(Fq)⊗F2spanned by the Pi. There is a natural action of Gal(Fq/Fq) on V,
which we consider as a representation ρ: Gal(Fq/Fq)→GL(V). By Galois theory,
it is clear that Frob acts on the set {αi}2g+1
i=1 with rorbits, one corresponding to each
irreducible factor of f(x). The lengths of the orbits are given by the degrees diof the
factors fi(x). This means that, in the natural basis of Vgiven by the Pi, the action
of Frobenius is given by a permutation matrix corresponding to a permutation of
cycle type (d1, d2, . . . , dr). It follows immediately that the characteristic polynomial
of ρ(Frob) is
det(tId −ρ(Frob)) = (td1−1) ··· (tdr−1) ∈F2[t].
On the other hand, by lemma 5.2, there is a Galois-equivariant exact sequence
0→F2→V→J[2] →0,
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On the L-polynomials of curves over finite fields 39
where the first map is given by 1 →P1+P2+··· P2g+1 and the action of Frob on
the sum P1+· ·· +P2g+1 is trivial. This implies that
det(tId −ρ(Frob)) = det(tId −ρ2(Frob))(t−1),
which, combined with our previous determination of the characteristic polynomial
of ρ(Frob), concludes the proof. 
Thanks to the previous lemma, it is easy to obtain the reduction modulo 2 of
the L-polynomial of any given hyperelliptic curve with an odd degree model. In the
next corollary, we use this to produce curves whose L-polynomials have particularly
simple reductions modulo 2.
Corollary 5.4. Let f0(x) = 1 and, for d= 1, . . . , 2g+ 1, let fd(x)∈Fq[x]be an
irreducible polynomial of degree d. Further set f0(x) = 1. For d= 0, . . . , g consider
the unique smooth projective curve Cdbirational to the affine curve
y2=fd(x)f2g+1−d(x).
For d= 1, . . . , g, we have the congruence
(t−1)PCd(t)≡(td−1)(t2g+1−d−1) ≡t2g+1 +t2g+1−d+td+ 1 (mod 2),
while for d=0 we have
(t−1)PC0(t)≡t2g+1 −1≡t2g+1 + 1 (mod 2).
Proof. This is a direct application of lemma 5.3, combined with the fact that by
corollary 5.1 we have PC(t)≡fC(t) (mod 2). 
Proof of theorem 1.4 for podd. The inequality dimQLg(Fq)≤g+ 1 follows imme-
diately from the symmetry relation ag+i=qiag−isatisfied by the coefficients of
the L-polynomials; it thus suffices to establish the lower bound dimQLg(Fq)≥
g+ 1. Consider the g+ 1 curves C0, . . . , Cgof corollary 5.4 (any choice of the
irreducible polynomials fd(x) will work) and the corresponding L-polynomials
PC0(t), . . . , PCg(t). Let M⊆Z[t] be the Z-module generated by these polynomials;
it is clear that in order to prove the theorem it suffices to show that rankZM≥g+1.
Notice that M⊗F2is in a natural way a vector subspace of F2[t], and that
rankZM≥dimF2(M⊗F2).
Let N⊂F2[t] be the image of the linear map
M⊗F2→F2[t]
q(t)7→ (t−1)q(t).
The F2-vector space Nis generated by the g+1 polynomials (t−1)PCi(t) for
i= 0, . . . , g, hence, by corollary 5.4, by the g+ 1 polynomials
t2g+1 + 1 and t2g+1 +t2g+1−i+ti+ 1for i= 1, . . . , g.
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40 F. Ballini, D. Lombardo and M. Verzobio
It is immediate to check that these g+1 polynomials are F2-linearly independent,
which implies
rankZM≥dimF2(M⊗F2) = dimF2N=g+ 1.

5.2. Proof of theorem 1.4:p=2
We now give the proof of theorem 1.4 in the case p= 2. As in the case of odd
characteristic, we will exhibit g+ 1 curves whose L-polynomials form a basis of
Lg(Fq). Recall from definition 1.2 the set Pg(Fq).
Proof of theorem 1.4 for p=2.Fix 0 ≤r≤g. Let h(x)∈Fq[x] be a separable
polynomial of degree rsuch that h(0) 6= 0. Such a polynomial exists: for r= 0,1
we may take h(x) = 1 or h(x) = x+ 1, respectively, and for r≥2 it suffices to take
as h(x) the minimal polynomial of any element that generates Fqrover Fq.
Consider the affine curve defined by the equation y2+yh(x) = x2g+1−rh(x). We
claim that this curve is smooth. Indeed, an Fq-point (x0, y0) on the curve is singular
if and only if







y2
0+y0h(x0) = x2g+1−r
0h(x0)
h(x0) = 0
y0h0(x0) = (2g+ 1 −r)x2g−r
0h(x0) + x2g+1−r
0h0(x0)
.
Here the second and third equations are given by the vanishing of the partial
derivatives in yand xof the defining equation, respectively. By the second equation,
x0is a root of h. So, by the first one, y0= 0. Hence, the third equation becomes
x2g+1−r
0h0(x0) = 0: but x06= 0 since h(0) 6= 0, and h0(x0)6= 0 since his separable,
so the above system has no solutions. Let C/Fqbe the smooth pro jective curve
given by the completion of the curve above. The curve Chas genus g, because the
degree of x2g+1−rh(x) is 2g+ 1 and the degree of h(x) is at most g. In particular,
PC(t) is an element of Pg(Fq). We will show that the reduction of PC(t) modulo 2
has degree r.
Let `be an odd prime and let T`Jbe the `-adic Tate module of the Jacobian J
of C. Let fC,`∞(t) := det(tId −ρ`∞(Frob) |T`J). If α∈Fqis a root of fC,`∞(t)
with multiplicity d, then q/α is a root of fC,`∞(t) with multiplicity d. Hence, we
can write fC,`∞(t) = tgQC(t+q/t) with QC(t)∈Z[t] of degree g. Let r2be the
2-rank of J, as defined in [24, Section 1]. By [24, proposition 3.1], r2is equal to the
sum of the multiplicities of the non-zero roots of QC(t) modulo 2. Hence,
QC(t)≡tg−r2˜
QC(t)(mod 2)
with ˜
QC(t)∈F2[t] a polynomial of degree r2such that ˜
QC(0) 6= 0 (in F2). In [14,
proof of theorem 23], the authors show that the 2-rank of Jis equal to one less than
the number of distinct projective points where H1(X, Z ) := h(X/Z)Zg+1 vanishes
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On the L-polynomials of curves over finite fields 41
(see also [21]). In our case, since h(x) is separable, this implies r2= deg h(x) = r.
Hence, we have
QC(t)≡tg−r˜
QC(t)(mod 2)
with ˜
QC(t) of degree r. As qis a power of 2, we obtain
fC,`∞(t)≡tgQCt+q
t≡tgQC(t)≡t2g−r˜
QC(t)(mod 2).
By theorem 1.7,
PC(t)≡t2gfC,`∞t−1≡t2gt−2g+r˜
QCt−1≡tr˜
QCt−1(mod 2).(24)
Since ˜
QC(0) 6≡ 0(mod 2), we see that the reduction of PC(t) modulo 2 has degree r.
So, for each 0 ≤r≤g, we can find a smooth hyperelliptic curve Crof genus gsuch
that PCr(t) modulo 2 has degree r. Therefore, the polynomials {PCr(t)|0≤r≤g}
are linearly independent modulo 2. The result follows as in the proof of theorem
1.4.
Remark 5.5. The polynomial fC,`∞(t) is monic by definition, which implies that
also QC(t) and ˜
QC(t) are monic. By (24), the constant term of PC(t) modulo 2 is
1. Hence,
PCr(t)≡tr+1+
r−1
X
i=1
ai,rti(mod 2).
In fact, one can show that PCr(t)≡tr+ 1(mod 2). To see this, recall from [18,
theorem 3.1] that, for a smooth projective curve C/Fq, with q= 2f, one has
PC(t)≡det 1−tϕ−1
qH1
´et CFq,Z/2Z (mod 2),
where ϕq:Fq→Fqis the Frobenius automorphism x7→ xq. Next, recall that
H1
´et CFq,Z/2Zis canonically dual to J(Fq)[2], so that we may compute PC(t)
as the inverse characteristic polynomial of Frobenius acting on J[2]. For the curve
Cr, the explicit description of J[2] given in [14, proof of theorem 23] shows that
the action of ϕqon J[2] is the natural Galois action on the roots of h(x), that
is, an r-cycle. It follows that the characteristic polynomial in question is PC(t)≡
tr−1 (mod 2), as claimed.
6. Algebraic independence
Theorem 1.4 asserts that lemma 1.1 captures all the linear relations among the
coefficients of the polynomials PC(t). In this section, we prove an analogous result
that deals with higher-order polynomial relations on the coefficients. Lemma 1.1
already gives a number of constraints: for PC(t) = P2g
i=0 aitiwe have a0= 1 and
ag+i=qiag−ifor every i= 0, . . . , g; it is, therefore, natural to restrict our analysis
to a1, . . . , ag. The following is the main result of this section:
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42 F. Ballini, D. Lombardo and M. Verzobio
Theorem 6.1. Let g, d be positive integers. There is a constant eg,d such that
for any prime power q > eg,d and for any non-zero polynomial f(x1, . . . , xg)∈
Z[x1, . . . , xg]of degree ≤din each variable there is a curve C∈ Mg(Fq)with
L-polynomial PC(t) = P2g
i=0 aitisuch that f(a1, . . . , ag)6= 0.
Notice that, unlike theorem 1.4,eg,d cannot be equal to 0 for all gand d, since
for fixed qand gwe can always find a polynomial f(x1, . . . , xg) (that may depend
on q) which vanishes on all the finitely many values of (a1, . . . , ag).
As is the case for theorem 1.4, the proof of theorem 6.1 exploits the reduction of
f(x1, . . . , xg) modulo a positive integer N. In this case, instead of a direct compu-
tation of the action of the Frobenius on the N-torsion points, we use theorem 2.1,
which guarantees that, for qlarge enough, all the characteristic polynomials of the
matrices in GSpq
2g(Z/NZ) come from some element of Pg,N (Fq).
To be more precise, for a curve C∈ Mg(Fq) and PC(t)∈Z[t] its L-polynomial,
let fC(t) = t2gPC(1/t) be its reciprocal polynomial. By theorem 1.7,fC(t) is equal
to the characteristic polynomial of the action of the Frobenius of C(modulo every
`). Theorem 2.1 implies that, for qlarge enough (in terms of N) and for any
M∈GSpq
2g(Z/NZ), the characteristic polynomial of Mis equal to the reduction
of fC(t) modulo Nfor some C∈ Mg(Fq). We then prove that there are too many
characteristic polynomials of elements of GSpq
2g(Z/NZ) for their coefficients to lie
in the zero locus of some f(x1, . . . , xg) of fixed degree. We are free to choose N,
and we will always take it to be an odd prime number. We set N=rand use the
letter rto avoid confusion.
The following lemma is a version of the well-known Schwartz–Zippel bound.
Notice that a polynomial in gvariables having degree at most din each of them
has total degree at most dg.
Lemma 6.2. Let g, d be natural numbers with g≥1, let rbe a prime number and
let f(x1, . . . , xg)∈Fr[x1, . . . , xg]be a non-zero polynomial of degree ≤din each
variable. We have
#{(u1, . . . , ug)∈Fg
r|f(u1, . . . , ug) = 0} ≤ dg ·rg−1.
Next, we identify the set of characteristic polynomials of matrices in GSpq
2g(Fr).
We show the following more general result:
Proposition 6.3. Let nbe a positive integer, let Rbe a commutative ring with 1,
and let q∈R×. Let p(x) = a0+a1x+··· +a2nx2n∈R[x]be a monic polynomial
satisfying an−i=qian+ifor all i= 0, . . . , n. There exists M∈GSp2n(R)with
multiplier qand characteristic polynomial p(x).
Remark 6.4. The statement is a simple variant of [45, theorem A.1]. We give
a detailed argument since, unfortunately, the proof of [45, theorem A.1] seems to
contain some typos. For example, in op. cit., the matrix Bis declared to have deter-
minant 1, but the construction does not ensure this property; more importantly,
in some examples we tried, the given construction does not seem to yield matrices
with the claimed characteristic polynomials. Our construction is therefore slightly
different from that of [45, theorem A.1], which we could not fully understand.
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On the L-polynomials of curves over finite fields 43
Proof. We work with the symplectic form given by the matrix J= 0 Idn
−Idn0!.
We construct the desired Mas a block-matrix M= 0B
C D!, where B,C,D
satisfy the following:
1. B, C, D are square n×nmatrices with Binvertible;
2. Bis the symmetric matrix
B=










0 0 0 ··· 0 1
0 0 0 ··· 1b2
0 0 0 ··· b2b3
···
0 1 b2··· bn−2bn−1
1b2b3··· bn−1bn










,
or, in symbols,
Bij =bi+j−nδi+j≥n+1 =






0,if i+j≤n
1,if i+j=n+ 1
bi+j−n,if i+j > n + 1,
where we have set b1= 1 and δi+j≥n+1 =


1,if i+j≥n+ 1
0,otherwise. . Note that
any matrix Bof this form is invertible for any choice of the bi;
3. C=−q(tB)−1=−qB−1;
4. Dis the companion matrix given by D=










000··· 0 0 d1
100··· 0 0 d2
...
000··· 0 0 dn−2
000··· 1 0 dn−1
000··· 0 1 dn










. In
symbols,
Dij =






1,if i=j+ 1
di,if j=n
0,otherwise.
Here b2, . . . , bn∈Rand d1, . . . , dn∈Rare coefficients to be chosen later. We
check the conditions for the matrix Mto be symplectic with multiplier q. We
compute
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44 F. Ballini, D. Lombardo and M. Verzobio
tMJM = 0−tC B
tBC tBD −tDB!,
which is equal to qJ if and only if







−tCB =qId
tBC =−qId
tBD −tDB = 0.
The first two equations are equivalent to one another and automatically satisfied
by our choice of C. The third equation is equivalent to the matrix tBD =BD
being symmetric. We claim that this is achieved by taking (b1= 1 and) bk+1 =
Pk
i=1 bidn+i−kfor k= 1, . . . , n −1 (notice that d1does not occur). Indeed, the first
n−1 columns of the product BD are given by the second, third, . . .,nth column
of B, while the last one is the linear combination d1B1+d2B2+···+dnBn, where
we denote by Bithe ith column of B. From this, it is immediate to check that the
top-left block of BD of size (n−1) ×(n−1) is symmetric (independently of the
values of b2, . . . , bn, d1, . . . , dn), and we only need to impose that the last line of BD
is equal to (the transpose of) its last column. We can also ignore the coefficient in
position (n,n), so we compare the first n−1 coefficients of the last line of BD with
the first n−1 coefficients of its last column. The kth coefficient on the last line is
the coefficient on the last line of the (k+ 1)th column of B, that is, bk+1 . The kth
coefficient on the last column is given by
d1Bk1+d2Bk2+· ·· +dnBkn =
n
X
i=1
diBki =
n
X
i=1
diδk+i≥n+1bk+i−n
=
k
X
i0=1
bi0di0+n−k.
Thus, the symmetry condition is satisfied if and only if for k= 1, . . . , n −1 we
have bk+1 =Pk
i=1 bidn+i−k, as claimed. Also note that a symplectic matrix with
invertible multiplier is itself invertible (because the determinant of a symplectic
matrix is a power of its multiplier), so Mis invertible and therefore an element
of GSp2n(R). In particular, for any choice of d1, . . . , dn, we have constructed a
corresponding matrix Mthat is symplectic of multiplier qand has Das its bottom-
right block of size n×n. We now compute the characteristic polynomial of this
matrix M. Consider the identity
xIdn−B
−C x Idn−D! B0
xIdnB−1!= 0−Idn
x2Idn−xD −CB xB−1−DB−1!
= 0−Idn
(x2+q) Idn−xD xB−1−DB−1!,
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On the L-polynomials of curves over finite fields 45
where we have used that—by definition—CB =−qId. Taking determinants on
both sides and using that the determinant of the block-matrix B0
xId B−1!is 1,
we obtain
det(xId2n−M) = det 0−Idn
(x2+q) Idn−xD xB−1−DB−1!
= det((x2+q) Idn−xD),
where the last equality uses basic properties of the determinant of block matrices.
Finally, we can rewrite this in the form
det(xId2n−M) = xndet x+q
xIdn−D,
so the characteristic polynomial of Mis equal to xnpDx+q
x, where pD(x) is
the characteristic polynomial of D. To conclude the proof, it suffices to show that
we can choose Din such a way that xnpDx+q
x=p(x), where p(x) is the
polynomial given in the statement. This is easy: Dis a companion matrix, so
any monic polynomial with coefficients in Rcan be realized as pD(x) for suitable
values of d1, . . . , dn. Finally, it is an easy exercise to show that a monic polynomial
p(x) = P2n
i=0 aixithat satisfies an−i=qian+ifor all i= 0, . . . , n can be written as
xnp1x+q
xfor some monic polynomial p1∈R[x] of degree n.
Remark 6.5. Inspection of the proof shows that the following slightly stronger
statement is true for the case of Rbeing the fraction field of a domain A: if the
characteristic polynomial p(x) has coefficients in Aand q∈A, then we may choose
Mto have coefficients in A,even if the multiplier q is not invertible in A. This
applies in particular when A=Z`and R=Q`.
Corollary 6.6. Let rbe a prime and let qbe an integer prime to r. The set
{charpol M:M∈GSpq
2g(Fr)}has cardinality rg.
Proof. By proposition 6.3, the set in question is the set of all monic polynomials
in Fr[x] of degree 2gwhose coefficients aisatisfy ag−i=qiag+ifor all i= 0, . . . , g.
Since any choice of the coefficients a1, . . . , agcorresponds to precisely one such
polynomial, the total number of polynomials is rg.
Finally, we connect characteristic polynomials of matrices in GSpq
2g(Fr) with
characteristic polynomials of Frobenius:
Lemma 6.7. Let g, r be positive integers. There is a constant hg,r such that for
any prime power q > hg,r with (q, r)=1and for any element Mof GSpq
2g(Z/rZ),
there is a curve C∈ Mg(Fq)such that the reduction of fC(t)modulo ris the
characteristic polynomial of M.
Proof. This is an immediate consequence of results of Katz–Sarnak [29]. We give a
proof in the language of this article.
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46 F. Ballini, D. Lombardo and M. Verzobio
For g= 1, the result follows from the fact that (writing q=pn) every polynomial
of the form t2+at +qwith p-aand |a| ≤ 2√qis the L-polynomial of an elliptic
curve over Fq(see [50, theorem 4.1]). Consider first the prime powers q=pnfor
which psatisfies p > 2√p > r. The integers a= 1, . . . , r realize all the residue
classes modulo r, are not divisible by p, and satisfy |a| ≤ 2√q, so the corresponding
polynomials t2+at +qare all realized by elliptic curves over Fqand give all the
characteristic polynomials of elements in GSpq
2g(Z/rZ). Consider now the prime
powers q=pnfor the finitely many primes pthat safisfy 2√p≤ror p≤2√p,
with (p, r) = 1. Suppose that b2√qc ≥ pr, which holds for all nlarge enough (with
respect to p). The integers 1, . . . , b2√qccover all residue classes modulo pr, hence in
particular for every residue class modulo rthere is a∈ {1, . . . , b2√qc} that realizes
the given class modulo rand is not divisible by p(recall that (p, r) = 1). As above,
t2+at +qis the L-polynomial of an elliptic curve over Fq, and we are done.
For g≥2, the result follows from theorem 2.1, as we now show. Let p(t) be
the characteristic polynomial of M. Notice that µq
rgives positive mass to the sin-
gleton {p(t)}, since GSpq
2g(Z/rZ) is a finite set. In fact, since the cardinality of
GSpq
2g(Z/rZ) is independent of q(it is equal to # Sp2g(Z/rZ), provided only that
(q, r) = 1), we have µq
r{p(t)} ≥ cg,r >0 for some absolute constant cg,r . By theo-
rem 2.1, this implies that (charpolr)∗Pnaive
g,q is positive at {p(t)}for qlarge enough.
Repeating the argument for the finitely many possible polynomials p(t) concludes
the proof. 
We can now combine our bounds to conclude the proof of theorem 6.1.
Proof of theorem 6.1.Let rbe an odd prime number, which will later be required to
be large enough. We prove the result for every qwhich is not a power of r; repeating
the argument with a different rwill prove the statement for every q. First, we can
assume that our polynomial f(x1, . . . , xg)∈Z[x1, . . . , xg] has a coefficient which
is non-zero modulo r(otherwise, divide by an appropriate power of r). Hence, its
reduction modulo ris non-zero. By lemma 6.7, the set of characteristic polynomials
of curves in Mg(Fq) modulo ris the same as the set of characteristic polynomials of
matrices of GSpq
2g(Fr) for qlarge enough and relatively prime with r. Suppose that
for every M∈GSpq
2g(Fr), writing charpol(M) = P2g
i=0 aiti, we have f(a1, . . . , ag) =
0. By combining lemma 6.2 and corollary 6.6 we obtain rg≤dg·rg−1,which implies
r≤dg. If ris chosen larger than this quantity, we obtain a contradiction. 
Acknowledgements
We thank Umberto Zannier for bringing the problem to our attention, for many
useful suggestions, and especially for pointing out the relevance of the equidistribu-
tion results of Katz–Sarnak, noting that they imply the case qg0 of theorem 1.4.
In addition, the first author would like to thank Umberto Zannier for his guidance
during his undergraduate studies, on a topic that ultimately inspired much of the
work in this article. We are grateful to J. Kaczorowski and A. Perelli for sharing
their work [28] before publication. We thank Christophe Ritzenthaler and Elisa
Lorenzo Garc´ıa for their interesting comments on the first version of this article,
Zhao Yu Ma for a comment about remark 3.12, and the anonymous referees for
their helpful suggestions.
https://doi.org/10.1017/prm.2025.7 Published online by Cambridge University Press

On the L-polynomials of curves over finite fields 47
Funding
The second and third authors have been partially supported by MIUR grant PRIN
2017 ‘Geometric, algebraic and analytic methods in arithmetic’ and MUR grant
PRIN-2022HPSNCR (funded by the European Union project Next Generation EU),
and by the University of Pisa through PRA 2018 and 2022 ‘Spazi di moduli, rapp-
resentazioni e strutture combinatorie’. The third author has received funding from
the European Union’s Horizon 2020 research and innovation program under the
Marie Sk lodowska-Curie Grant Agreement No. 101034413.
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