Prym-Tyurin varieties via Hecke algebras

Abstract

Let $G$ denote a finite group and $\pi: Z \to Y$ a Galois covering of smooth projective curves with Galois group $G$. For every subgroup $H$ of $G$ there is a canonical action of the corresponding Hecke algebra $\mathbb{Q}[H \backslash G/H]$ on the Jacobian of the curve $X = Z/H$. To each rational irreducible representation $\mathcal{W}$ of $G$ we associate an idempotent in the Hecke algebra, which induces a correspondence of the curve $X$ and thus an abelian subvariety $P$ of the Jacobian $JX$. We give sufficient conditions on $\mathcal{W}$, $H$, and the action of $G$ on $Z$, which imply $P$ to be a Prym-Tyurin variety. We obtain many new families of Prym-Tyurin varieties of arbitrary exponent in this way.

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arXiv:0805.4563v2 [math.AG] 18 Aug 2008
PRYM-TYURIN VARIETIES VIA HECKE ALGEBRAS
ANGEL CAROCCA, HERBERT LANGE, RUB´I E. RODR´IGUEZ AND ANITA M. ROJAS
Abstract. Let G denote a finite group and π : Z → Y a Galois covering of smooth
projective curves with Galois group G. For every subgroup H of G there is a canonical
action of the corresponding Hecke algebra Q[H\G/H] on the Jacobian of the curve X =
Z/H. To each rational irreducible representation W of G we associate an idempotent in
the Hecke algebra, which induces a correspondence of the curve X and thus an abelian
subvariety P of the Jacobian JX. We give sufficient conditions on W, H, and the
action of G on Z for P to be a Prym-Tyurin variety. We obtain many new families of
Prym-Tyurin varieties of arbitrary exponent in this way.
1. Introduction
A Prym-Tyurin variety of exponent q is by definition a principally polarized abelian
variety (P,Ξ), for which there exists a smooth projective curve C and an embedding
P ֒→ JC into the Jacobian of C such that the restriction of the canonical polarization Θ
of JC is the q-fold of Ξ:
Θ|P = qΞ.
One point of interest in these varieties is that the structure of a Prym-Tyurin variety
allows to study the geometric properties of the underlying abelian varieties via curve
theory.
According to the Theorem of Matsusaka-Ran [BL, 11.8.1] Prym-Tyurin varieties of
exponent 1 are exactly the canonically polarized products of Jacobians. Welters showed
in [W] that Prym-Tyurin varieties of exponent 2 are exactly the classical Prym varieties
or some of their specializations. The Abel-Prym-Tyurin map of (P,Ξ) is the following
composition of maps
αP: C −→ JC =?
property of being a Prym-Tyurin variety can then be expressed by certain properties of
the curve αP(C) in P. One can deduce a criterion (see [BL, Welters’ criterion 12.2.2]) for
(P,Ξ) to be a Prym-Tyurin variety in term of a curve in P. Using this it is easy to see,
using complete intersection curves on the Kummer variety of P, that every principally
polarized abelian variety of dimension g is a Prym-Tyurin variety of exponent 2g−1(g−1)!.
The problem is to construct Prym-Tyurin varieties of small exponent ≥ 3. There are
in fact not many examples. A series of examples was given by Kanev in [K2] using Weyl
groups of type An,Dn,E6and E7. Generalizing the construction of [LRR], Salomon gave
αC
JC →?P = P,
where αC denotes the usual Abel map and? A the dual of an abelian variety A. The
1991 Mathematics Subject Classification. 14H40, 14K10,
Key words and phrases. Prym-Tyurin variety, Hecke algebra, correspondence.
The first, third and fourth author were supported by Fondecyt grants 1060743, 1060742 and 11060468
respectively.
1

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2 ANGEL CAROCCA, HERBERT LANGE, RUB´I E. RODR´IGUEZ AND ANITA M. ROJAS
in [S] examples of Prym-Tyurin varieties of arbitrary exponent using some special graph
constructions. There are a few other examples which shall not be mentioned here.
The construction of these examples are based on the following two facts: (i) The use of
correspondences: The set of abelian subvarieties of a polarized abelian variety is in canon-
ical bijection to the set of symmetric (with respect to the Rosati involution) idempotents
of its endomorphism algebra. On the other hand, it follows almost from the definitions
that the obvious map
DivQ(C × C) −→ EndQ(JC)
from the algebra of rational correspondences of C to the endomorphism algebra of its
Jacobian is surjective. Hence one can describe an abelian subvariety of JC by correspon-
dences. (ii) Kanev’s Criterion: In [K1] Kanev showed that an abelian subvariety of JC is
a Prym-Tyurin variety of exponent q if it is given by an integral effective fixed-point free
correspondence on C whose associated endomorphism of JC satisfies the equation
x2+ (q − 2)x − (q − 1) = 0. (1.1)
This is a direct generalization of Wirtinger-Mumford’s construction of classical Prym
varieties and, in fact, most Prym-Tyurin varieties are given by constructing integral sym-
metric fixed point free correspondences satisfying (1.1). There is an analogous result for
correspondences with some fixed points, due to Ortega [O], which also can be applied.
Our construction of such correspondences is, roughly speaking, as follows: Let G denote
a finite group and π : Z → Y be a Galois covering of smooth projective curves with Galois
group G. The group action induces a homomorphism of the group algebra Q[G] into the
endomorphism algebra EndQ(JZ). We identify the elements of Q[G] with their images
in EndQ(JZ). Hence any α ∈ Q[G] defines an abelian subvariety Im(α) of JZ. For the
details we refer to Section 3.1. On the other hand, every element α =?
Dα=αgΓg∈ DivQ(Z × Z).
g∈Gαgg ∈ Q[G]
defines a rational correspondence on Z, namely
?
g∈G
where Γgdenotes the graph of the automorphism g of Z. The problem is to find inte-
gral symmetric fixed-point free correspondences satisfying (1.1) in this way. For this we
proceed as follows:
For simplicity we first describe a special case. Let V be an irreducible complex represen-
tation of G and W the associated rational irreducible representation. For any subgroup
H of G, the covering π factorizes via a covering ϕH : X −→ Y where X denotes the
quotient Z/H. The Hecke algebra Q[H\G/H] is a subalgebra of Q[G] and the above
homomorphism induces a homomorphism
Q[H\G/H] −→ EndQ(JX).
which factorizes via DivQ(X × X). Then we associate to the pair (H,W) an integral
symmetric correspondence DH,Won X in a canonical way. It is defined via the projectors
of Q[G] given in [CaRo]. Our essential assumption is
H is a subgroup satisfying dimVH= 1 and maximal with this property,(1.2)
where VHdenotes the fixed subspace of V under the action of H.

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In fact, under this assumption we can show (Proposition 3.9) that a modified version
KXof DH,Wis an effective integral symmetric correspondence on X, which in the special
case Y = P1satisfies equation (1.1). Moreover we can compute its fixed points in terms
of the ramification of π : Z → P1.
In order to formulate the result, we use the following notation: We denote by KV the
field generated by the values of the character of V . Moreover, if C denotes a conjugacy
class of cyclic subgroups of the group G, a branch point y ∈ Y of the covering π is
called of type C, if the stabilizer of any point z in the fibre π−1(y) is a subgroup in the
class C. We define the geometric signature of the covering π : Z → Y to be the tuple
[g;(C1,m1),...,(Ct,mt)], where g is the genus of the quotient curve Y , the covering π has
a total of?t
j=1mjbranch points, and exactly mjof them are of type Cjfor j = 1,...,t
(see [R]). Using this we prove the following
Theorem. Let W denote a nontrivial rational irreducible representation of G, with as-
sociated complex irreducible representation V , and H a subgroup of G satisfying (1.2).
Suppose that the action of G has geometric signature [0;(C1,m1),...,(Ct,mt)] satisfying
(1.3)
t?
j=1
mj
?q[KV : Q](dimV − dimVGj) − ([G : H] − |H\G/Gj|)?= 0,
for an integer q given in terms of the algebraic data, where Gj is of class Cj, Then the
correspondence KXdefines a Prym-Tyurin variety of exponent q.
Using the theorem one obtains Kanev’s examples and several more, some of which are
included in Sections 5.1 and 5.2. However we could not find examples that exhibited the
full force of the theorem; we were interested in finding Prym-Tyurin varieties of small
exponent bigger than 2, using complex representations whose field of definition properly
contains Q, as these would provide entirely new examples, different from the existing ones.
In fact, we wrote a computer program which for many groups G and all of their subgroups
and irreducible complex representations such that Q ? KV verifies the hypothesis of the
theorem. In this way we found many examples of exponents 1 and 2, which however are
not interesting from our point of view.
The new idea was to start, instead of one irreducible representation, with several rep-
resentations satisfying some additional conditions (see Hypothesis 3.7) and generalize the
above result to Theorem 4.9. This gave in fact many examples, most of them new, and
which also include the above mentioned examples of Salomon. For details see Section 5.3
and the forthcoming paper [CLRR], in which more examples are given and these Prym-
Tyurin varieties will be investigated in detail.
The contents of the paper are as follows. Section 2 contains some algebraic preliminar-
ies. The essential result is Proposition 2.4, which describes the coefficients of a certain
idempotent of the group algebra in term of the character of V . In Section 3 we define the
correspondence KX and derive its properties. Section 4 contains the proof of our main
result Theorem 4.9. We also compare our construction with the construction of [K2].
Finally in Section 5 we give some examples.

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4 ANGEL CAROCCA, HERBERT LANGE, RUB´I E. RODR´IGUEZ AND ANITA M. ROJAS
We suppose throughout that the curves are defined over any algebraically closed field
of characteristic 0. Moreover the curves always will be smooth and projective.
2. Algebraic preliminaries
2.1. The group algebra. Let G be a finite group. In order to fix the notation, we
start by recalling some basic properties of representations of G (see [CR]).
field K of characteristic 0 we denote by K[G] the group algebra of G over K. It is a
semisimple algebra, whose simple components correspond one-to-one to the irreducible K-
representations of G. We may identify the elements of K[G] with the K-valued functions
on G. From this point of view, the multiplication in K[G] is convolution
?
for f1,f2∈ K[G] and g ∈ G. In this paper the field K will be either C or Q.
For any complex irreducible representation V of G, we denote by χV its character, by
LV its field of definition and by KV the subfield KV = Q(χV(g) | g ∈ G). LV and KV are
finite abelian extensions of Q. We denote by
For any
(f1f2)(g) =
g1g2=g
f1(g1)f2(g2)
mV = [LV : KV]
the Schur index of V . For any automorphism ϕ of LV/Q we denote by Vϕthe represen-
tation conjugate to V by ϕ.
If W is a rational irreducible representation of G, then there exists a complex irreducible
representation V of G, uniquely determined up to conjugacy in Gal(LV/Q), such that
?
We call V the complex irreducible representation associated to W.
The central idempotent eV of C[G] that generates the simple subalgebra of C[G] corre-
sponding to V and the central idempotent eWof Q[G] that generates the simple subalgebra
of Q[G] corresponding to W are given by
?
Lemma 2.1. If e is an idempotent of K[G], the map
W ⊗QC ≃
ϕ∈Gal(LV/Q)
Vϕ≃ mV
?
τ∈Gal(KV/Q)
Vτ
(2.1)eV =dimV
|G|
g∈G
χV(g−1)gandeW=dimV
|G|
?
g∈G
trKV/Q(χV(g−1))g.
EndK[G](K[G]e) → eK[G]e,ϕ ?→ ϕ(e)
is an anti-isomorphism of K-algebras.
Proof. Note first that ϕ(e) = ϕ(e · e) = e · ϕ(e) ∈ eK[G]e. For the proof that the map is
“anti”, suppose that ϕi(e) = fie for i = 1,2. Then
ϕ1ϕ2(e) = ϕ1(ϕ2(e)) = ϕ1(f2e) = f2ϕ1(e) = f2ef1e = ϕ2(e)ϕ1(e).
Finally, for any efe ∈ eK[G]e, the map ϕ ∈ EndK[G](K[G]e), defined by ϕ(f′e) = f′efe
for all f′∈ K[G], maps to efe. Hence the map is bijective.
?

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2.2. The Hecke algebra of a subgroup. Let H be a subgroup of G. The element
(2.2)pH=
1
|H|
?
h∈H
h
is the central idempotent of K[H] corresponding to the trivial representation of H. More-
over, the left ideal K[G]pH defines the K-representation of G induced by the trivial
representation of H for any field K as above. In the sequel we denote this representation
by ρH.
The K-algebra pHK[G]pH, considered as a subalgebra of K[G], consists of the K-valued
functions on G which are constant on each double coset HgH of H in G. It is called the
Hecke algebra over K of H in G, and it is usually denoted by K[H\G/H].
The idempotent eV of (2.1) is central in C[G]. This implies that the element
(2.3)
is an idempotent of C[G] (or zero) which satisfies
• hfH,V = fH,V = fH,Vh for all h ∈ H,
• the left ideal C[G]fH,V defines the representation V with multiplicity dimVH.
The last property follows from the fact that ρH≃ C[G]pHand V occurs in ρHwith mul-
tiplicity dimVH.
fH,V = pHeV = eVpH
Similarly, since the idempotent eWof (2.1) is central in Q[G], the element
(2.4)
is an idempotent of Q[G] (or zero) which satisfies
• hfH,W= fH,W= fH,Wh for all h ∈ H,
• the left ideal Q[G]fH,Wdefines the representation W with multiplicitydimVH
The last property and the fact thatdimVH
mV
is an integer follow from the equation dimVH=
?ρH,V ? and the fact that ρHis a rational representation. For a complete proof see [CaRo,
Theorem 4.4].
fH,W:= pHeW= eWpH
mV
.
According to [H, Theorem 5.1.7] there exists a set of representatives
{gij∈ G | i = 1,...,d and j = 1,...,ni}
for both the left cosets and right cosets of H in G, such that
G = ⊔d
i=1Hgi1HandHgi1H = ⊔ni
j=1gijH = ⊔ni
j=1Hgij
are the decompositions of G into double cosets, and of the double cosets into right and
left cosets of H in G. Moreover, we assume g11= 1G.
Clearly χV(g−1) = χV(g) is an algebraic integer for all g ∈ G ([CR, Corollary 30.11]).
This implies that trKV/Q(χV(g)) = trKV/Q(χV(g−1)) is an integer for every g ∈ G. We
conclude
?
and use this to prove the following lemma.
(2.5)
h∈H
trKV/Q(χV(hg−1
ij)) =
?
h∈H
trKV/Q(χV(gijh)),