John E Cremona

• MA, DPhil (Oxon.)
• Professor Emeritus at University of Warwick

We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H ∈ Z [ X ] H\in \mathbb {Z}[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given algebraic integer is the j j -invariant of an elliptic curve with complex multiplicatio...

We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given algebraic integer is the j-invariant of an elliptic curve with complex multiplication (CM), and if so, the as...

We consider the question of determining whether two binary cubic forms over an arbitrary field $K$ whose characteristic is not $2$ or $3$ are equivalent under the actions of either GL$(2,K)$ or SL$(2,K)$, deriving two necessary and sufficient criteria for such equivalence in each case. One of these involves an algebraic invariant of binary cubic fo...

By reformulating and extending results of Elkies, we prove some results on \({{\mathbb {Q}}}\)-curves over number fields of odd degree. We show that, over such fields, the only prime isogeny degrees \(\ell \) that a \({{\mathbb {Q}}}\)-curve without CM may have are those degrees that are already possible over \({{\mathbb {Q}}}\) itself (in particul...

We study the rational Bianchi newforms (weight 2, trivial character, with rational Hecke eigenvalues) in the LMFDB that are not associated to elliptic curves, but instead to abelian surfaces with quaternionic multiplication. Two of these examples exhibit a rather special kind of behaviour: we show they arise from twisted base change of a classical...

We discuss practical and some theoretical aspects of computing a database of classical modular forms in the L-functions and modular forms database (LMFDB).

We describe a systematic investigation into the existence of congruences between the mod p torsion modules of elliptic curves defined over \mathbb{Q} , including methods to determine the symplectic type of such congruences. We classify the existence and symplectic type of mod p congruences between twisted elliptic curves over number fields, giving...

We determine the probability that a random polynomial of degree $n$ over $\mathbb{Z}_p$ has exactly $r$ roots in $\mathbb{Q}_p$, and show that it is given by a rational function of $p$ that is invariant under replacing $p$ by $1/p$.

We consider the proportion of genus one curves over ℚ of the form z2 = f(x,y) where f(x,y) ∈ ℤ[x,y] is a binary quartic form (or more generally of the form z2 + h(x,y)z = f(x,y) where also h(x,y) ∈ ℤ[x,y] is a binary quadratic form) that have points everywhere locally. We show that the proportion of these curves that are locally soluble, computed a...

We define a scheme for labelling and ordering integral ideals of number fields, including prime ideals as a special case. The order we define depends only on the choice of a monic irreducible integral defining polynomial for each field $K$, and we start by defining for each field its unique reduced defining polynomial, after Belabas. We define a to...

In this paper we consider the proportion of genus one curves over $\mathbb{Q}$ of the form $z^2=f(x,y)$ where $f(x,y)\in\mathbb{Z}[x,y]$ is a binary quartic form (or more generally of the form $z^2+h(x,y)z=f(x,y)$ where also $h(x,y)\in\mathbb{Z}[x,y]$ is a binary quadratic form) that have points everywhere locally. We show that the proportion of th...

We prove local results on the $p$-adic density of elliptic curves over $\mathbb{Q}_p$ with different reduction types, together with global results on densities of elliptic curves over $\mathbb{Q}$ with specified reduction types at one or more (including infinitely many) primes. These global results include: the density of integral Weierstrass equat...

We discuss practical and some theoretical aspects of computing a database of classical modular forms in the L-functions and Modular Forms Database (LMFDB).

We describe a systematic investigation into the existence of congruences between the mod $p$ torsion modules of elliptic curves defined over $\mathbb{Q}$, including methods to determine the symplectic type of such congruences. We classify the existence and symplectic type of mod $p$ congruences between twisted elliptic curves over number fields, gi...

We study the rational Bianchi newforms (weight 2, trivial character, with rational Hecke eigenvalues) in the LMFDB that are not associated to elliptic curves, but instead to abelian surfaces with quaternionic multiplication. Two of these examples exhibit a rather special kind of behaviour: we show they arise from twisted base change of a classical...

The main result of this paper is to generalize from $\Q$ to each of the nine imaginary quadratic fields of class number one a result of Serre and Mestre-Oesterl\'e of 1989, namely that if $E$ is an elliptic curve of prime conductor then either $E$ or a $2$-isogenous curve or a $3$-isogenous curve has prime discriminant. The proof is conditional in...

The main result of this paper is to extend from $\Q$ to each of the nine imaginary quadratic fields of class number one a result of Serre (1987) and Mestre-Oesterl\'e (1989), namely that if $E$ is an elliptic curve of prime conductor then either $E$ or a $2$-, $3$- or $5$-isogenous curve has prime discriminant. For four of the nine fields, the theo...

We develop methods to study $2$-dimensional $2$-adic Galois representations $\rho$ of the absolute Galois group of a number field $K$, unramified outside a known finite set of primes $S$ of $K$, which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of...

We develop methods to study $2$-dimensional $2$-adic Galois representations $\rho$ of the absolute Galois group of a number field $K$, unramified outside a known finite set of primes $S$ of $K$, which are presented as Black Box representations, where we only have access to the characteristic polynomials of Frobenius automorphisms at a finite set of...

The Langlands Programme, formulated by Robert Langlands in the 1960s and since much developed and refined, is a web of interrelated theory and conjectures concerning many objects in number theory, their interconnections, and connections to other fields. At the heart of the Langlands Programme is the concept of an L-function. The most famous L-funct...

We show that the density of quadratic forms in $n$ variables over $\mathbb {Z}_p$ that are isotropic is a rational function of $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. When real quadratic forms in $n$ variables are distributed according to the Gaussian Orthogonal Ensemble (GOE) of r...

We develop an explicit theory of congruence subgroups, their cusps, and Manin symbols for arbitrary number fields. While our motivation is in the application to the theory of modular symbols over imaginary quadratic fields, we give a general treatment which makes no special assumptions on the number field.

We show that the density of quadratic forms in $n$ variables over ${\mathbb Z}_p$ that are isotropic is a rational function in $p$, where the rational function is independent of $p$, and we determine this rational function explicitly. As a consequence, for each $n$, we determine the probability that a random integral quadratic form in $n$ variables...

We show that the proportion of plane cubic curves over ℚp that have a ℚp-rational point is a rational function in p, where the rational function is independent of p, and we determine this rational function explicitly. As a consequence, we obtain the density of plane cubic curves over ℚ that have points everywhere locally; numerically, this density...

We study the failure of a local-global principle for the existence of l-isogenies for elliptic curves over number fields K. Sutherland has shown that over Q there is just one failure, which occurs for l = 7 and a unique j-invariant, and has given a classification of such failures when K does not contain the quadratic subfield of the l-th cyclotomic...

We consider the family of irreducible cyclically presented groups on n generators whose generating word (in the standard rewrite) has length at most 15. Using the software packages KBMAG, quotpic and MAGMA, together with group and number theoretic methods, we show that if 6 ≤ n ≤ 100 then the group is non-trivial. In an appendix we list the 47 case...

We review a number of ways of "visualizing'' the elements of the Shafarevich-Tate group of an elliptic curve $E$ over a number field $K$. We are specifically interested in cases where the elliptic curves are defined over the rationals, and are subabelian varieties of the new part of the jacobian of a modular curve (specifically, of $X_0(N)$, where...

This is the third in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as curves of degree n in P^{n-1}. The methods we describe are practical in the case n=3 for elliptic curves over the rationals, and have been implemented in Magma. One important ingredient of our work is an al...

We give an account of the complex Arithmetic-Geometric Mean (AGM), as first studied by Gauss, together with details of its relationship with the theory of elliptic curves over $\C$, their period lattices and complex parametrisation. As an application, we present efficient methods for computing bases for the period lattices and elliptic logarithms o...

In this paper we consider models for genus one curves of degree n for n = 2, 3 and 4, which arise in explicit n-descent on elliptic curves. We prove theorems on the existence of minimal models with the same invariants as the minimal model of the Jacobian elliptic curve and provide simple algorithms for minimising a given model, valid over general n...

We give an extension and correction to a result stated in the first author's paper Classical In- variants and 2-descent on elliptic curves, J. Symb. Comp. 31 (2001), concerning the equivalence of binary quartics. In the earlier version the cases where I = 0 or J = 0 were not fully treated, and neither were the cases of reducible quartics or those w...

A well known theorem of Mestre and Schoof implies that the order of an elliptic curve E over a prime field F_q can be uniquely determined by computing the orders of a few points on E and its quadratic twist, provided that q > 229. We extend this result to all finite fields with q > 49, and all prime fields with q > 29.

We study families of integer circulant matrices and methods for determining which are unimodular. This problem arises in the study of cyclically presented groups, and leads to the following problem concerning polynomials with integer coefficients: given a polynomial f(x)inmathbb{Z}[x] , determine all those ninmathbb{N} such that operatorname{Res}(f...

Let K be a p-adic local field and E an elliptic curve defined over K. The component group of E is the group E(K)/E 0(K), where E 0(K) denotes the subgroup of points of good reduction; this is known to be finite, cyclic if E has multiplicative reduction, and of order at most 4 if E has additive reduction. We show how to compute explicitly an isomorp...

We describe an algorithm for determining elliptic curves defined over a given number field with a given set of primes of bad reduction. Examples are given over $\Q$ and over various quadratic fields.

This is the second in a series of papers in which we study the n-Selmer group of an elliptic curve. In this paper, we show how to realize elements of the n-Selmer group explicitly as curves of degree n embedded in P^{n-1}. The main tool we use is a comparison between an easily obtained embedding into P^{n^2-1} and another map into P^{n^2-1} that fa...

This is the first in a series of papers in which we study the n-Selmer group of an elliptic curve, with the aim of representing its elements as genus one normal curves of degree n. The methods we describe are practical in the case n=3 for elliptic curves over the rationals, and have been implemented in Magma.

Let E be an elliptic curve over the rationals. A crucial step in determining a Mordell-Weil basis for E is to exhibit some positive lower bound λ>0 for the canonical height ĥ on non-torsion points. We give a new method for determining such a lower bound, which does not involve any searching for points.

Tabulating elliptic curves has been carried out since the ear- liest days of machine computation in number theory. After some histor- ical remarks, we report on signican t recent progress in enlarging the database of elliptic curves dened over Q to include all those of conduc- tor N 130000. We also give various statistics, summarize the data, descr...

Let E be an elliptic curve over a number field K. Let h be the logarithmic (or Weil) height on E and be the canonical height on E. Bounds for the difference are of tremendous theoretical and practical importance. It is possible to decompose as a weighted sum of continuous bounded functions Ψυ:E(Kυ)→R over the set of places υ of K. A standard method...

Let F be a field whose characteristic is not 2 and K = F(t). We give a simple algorithm to find, given a,b,c 2 K , a nontrivial solution in K (if it exists) to the equation aX2+bY 2+ cZ2 = 0. The algorithm requires, in certain cases, the solution

This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank 0 abelian varieties A f that are optimal quo-tients of J 0 (N) attached to newforms. We prove theorems about the ratio L(A f , 1)/Ω A f , develop tools for computing with A f , and gather data about certain arithmetic invariants of the nearly 20, 000 abelian...

We present efficient algorithms for solving Legendre equations over Q (equivalently, for finding rational points on rational conics) and parametrizing all solutions. Unlike existing algorithms, no integer factorization is required, provided that the prime factors of the discriminant are known.

In this paper we study the equation x^2+7=y^m , in integers x, y, m with m> 3, using a Frey curve and Ribet's level lowering theorem. We adapt some ideas of Kraus to show that there are no solutions to the equation with m composite and m > 15, and none with m prime and 10 < m < 10^8 .

The second author developed in LMS J. Comput. Math. 2, 64-94 (1999; Zbl 0927.11020) a reduction theory for binary forms of degrees three and four with integer coefficients in detail, the motivation in the case of quartics being to improve 2-descent algorithms for elliptic curves over ℚ. We extend some of these results to forms of higher degree. One...

This paper concerns the existence and algorithmic determination of minimal models for curves of genus 1, given by equations of the form y = Q(x) where Q(x) has degree 4. These models are used in the method of 2-descent for computing the rank of an elliptic curve. Our results are complete for unramified extensions of Q 2 and Q 3 and for all p-adic f...

Introduction The method of descent has been used since classical times for studying the arithmetic of elliptic curves. More recently, explicit algorithms for determining the Mordell-Weil and Selmer groups of elliptic curves over the rational eld Q , general number elds, and other global elds, have been developed. One of the best such general algori...

. The classical theory of invariants of binary quartics is applied to the problem of determining the group of rational points of an elliptic curve defined over a field K by 2-descent. The results lead to some simplifications to the method first presented by Birch and Swinnerton-Dyer in [1], and can be applied to give a more efficient algorithm for...

Corrigendum to Proposition 14, on page 87 of the paper ‘Reduction of binary cubic and quartic forms’, LMS Journal of Computation and Mathematics, Volume 2, pp. 62–92.

Introduction Let E be an elliptic curve defined over Q. In this note we present related methods to do the following tasks: 1. Prove that a given finite set of points in the Mordell-Weil group E(Q) is independent; 2. Make the group law in the 2-Selmer group S 2 (E=Q) explicit, and hence show that a given finite set of elements in S 2 (E=Q) is indepe...

A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number fields, and 2-descent on elli...

In this paper we describe an algorithm for computing the rank of an elliptic curve dened over a real quadratic eld of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over Q. Several examples are included.

We review a number of ways of "visualizing" the elements of the Shafarevich-Tate group of an elliptic curve E over a number field K. We are specifically interested in cases where the elliptic curves are defined over the rationals, and are subabelian varieties of the new part of the jacobian of a modular curve (specifically, of X0(N), where N is the...

this article is in preparation [6]. 2 The First Descent

. The classical theory of invariants of binary quartics is applied to the problem of determining the group of rational points of an elliptic curve defined over a field K by 2-descent. The results lead to some simplifications to the method first presented by Birch and Swinnerton-Dyer in [1], and can be applied to give a more efficient algorithm for...

A geometrical construction is described that generates all fullerene isomers C n in which the twelve pentagonal faces are arranged in four separate fully fused triples. When such an isolated-pentagon-triple (IPT) fullerene achieves its maximal symmetry, it may belong to one of only five point groups (D 2 , D 2h , D 2d , T, T d ). At large n, the cl...

We present an improved method of computing the periods of a newform for {\mathversion{normal}$\Gamma$}$_0(N)$, which converges faster than the method used in [Cremona 1992] (and originally in [Tingley 1975]). We also present some shortcuts that speed up the process of computing all modular elliptic curves of a given conductor $N$. As an application...

The Weil-Taniyama conjecture states that every elliptic curve E/ℚ of conductor N can be parametrized by modular functions for the congruence subgroup Γ 0 (N) of the modular group Γ=PSL(2,ℤ). Equivalently, there is a nonconstant map φ from the modular curve X 0 (N) to E. We present here a method of computing the degree of such a map φ for arbitrary...

The full text is at http://homepages.warwick.ac.uk/staff/J.E.Cremona/book/fulltext/index.html

The Weil-Taniyama conjecture states that every elliptic curve E/Q of conductor N can be parametrized by modular functions for the congruence subgroup T0(N) of the modular group r = F5L(2, Z). Equivalently, there is a nonconstant map ip from the modular curve Xq(N) to E. We present here a method of computing the degree of such a map φ for arbitrary...

The Weil-Taniyama conjecture states that every elliptic curve E/ of conductor N can be parametrized by modular functions for the congruence subgroup 0(N) of the modular group = PSL(2, ). Equivalently, there is a non-constant map from the modular curve X 0 (N) to E. We present here a method of computing the degree of such a map for arbitrary N. Our...

In this paper we explore the arithmetic correspondence between, on the one hand, (isogeny classes of) elliptic curves E defined over an imaginary quadratic field K of class number one, and on the other hand, rational newforms F of weight two for the congruence subgroups Ѱ(n), where n is an ideal in the ring of integers R of K. This continues work o...

In this note we extend the computations described in [4] by computing the analytic order of the Tate-Shafarevich group III for all the curves in each isogeny class ; in [4] we considered the strong Weil curve only. While no new methods are involved here, the results have some interesting features suggesting ways in which strong Weil curves may be d...

This paper concerns certain two-dimensional abelian varieties A which are Q-simple factors of J0(N) and have extra twist by the character associated to a quadratic number field k. Results of Ribet [22] and Momose [21] are used to give a simple necessary and sufficient condition for A to split over k. The L-series of A over k is the square of the Me...

The modular symbols method developed by the author in [4] for the computation of cusp forms for Γ 0 (N) and related elliptic curves is here extended to Γ 1 (N). Two applications are given: the verification of a conjecture of Stevens [14] on modular curves parametrised by Γ 1 (N); and the study of certain elliptic curves with everywhere good reducti...

This paper treats the following simple geometric problem: Given a polyhedron in the integer lattice in k: dimensions, can it be %hrunk” to a similar one (i.e., one whose sides remain in the same ratio as the original polyhedron) while still remaining on the integer lattice? For 1 5 k < 4, we give necessary and sufficient conditions for this to be p...

The aim of this note is to prove Theorem 1. Let n ≥ 3, and let p 1 , p 2 ,…, P n be primes in ℕ: = {z ∈ ℤ:z > 0}, each congruent to 1 (mod 4), which satisfy both of the following conditions: (i) every unit in ℚ(√(p 1 p 2 )) has norm + 1; (ii) the graph γ = γ(p 1 , p 2 , …, p n ) associated with p 1 , p 2 , …, p n is odd (in the sense of [1]).

On obtient de nouveaux resultats sur la distribution des d∈N √d∈Q pour lesquels l'equation de Pell negative x 2 −dy 2 =−1 (x,y)∈Z est soluble. On relie la recherche des criteres de resolubilite a l'enumeration de certains graphes

General formulas are presented for the vertex numbers, , of pentagon+hexagon polyhedra of icosahedral, tetrahedral or dihedral symmetries. Criteria for uniqueness of representation, isomer counts and grouping of pentagons are established. All polyhedra with 256 vertices or less and belonging to T, D 5, D6or their supergroups are listed. With the ad...

By A. N. Andrianov: pp. 374. DM.184.-. (Springer-Verlag, 1987)

We show how the classical theory of reduction of real binary forms with respect to the action of SL(2, Z) may be extended to a reduction theory for binary forms with complex coefficients under the action of certain discrete groups. In particular, we give some explicit results concerning the reduction of binary cubics and quartics with coefficients...