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John Cremona

Publications and source records attributed to John Cremona.

18 recordsLinked to original sources
arXiv2025

Bianchi Modular Forms over Imaginary Quadratic Fields with arbitrary class group

Let $K$ be an imaginary quadratic field and let $\mathcal{O}_K$ be its ring of integers. For an integral ideal $\mathfrak{n}$ of $\mathcal{O}_K$, let $\Gamma_0({\mathfrak{n}})$ be the congruence subgroup of level ${\mathfrak{n}}$ consisting of matrices in $\operatorname{GL}_2{\mathcal{O}_K}$ that are upper triangular mod ${\mathfrak{n}}$. In this paper, we discuss techniques to compute the space of Bianchi modular forms of level $\Gamma_0({\mathfrak{n}})$ as a Hecke module in the case where $K$ has arbitrary class group. Our algorithms and computations extend and complement those carried out for fields of class number $1$, $2$, and $3$ by the first author, and by his students Bygott and Lingham in unpublished theses. We give details and several examples for $K=\mathbb{Q}(\sqrt{-17})$, whose class group is cyclic of order $4$, including a proof of modularity of an elliptic curve over this field. We also give an overview of the results obtained for a wide range of imaginary quadratic fields, which are tabulated in the L-functions and modular forms database (\href{https://www.lmfdb.org/}{LMFDB}).

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arXiv2021

$\mathbb Q$-curves over odd degree number fields

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~$\ell$ which an elliptic curve without CM may have are those degrees which are already possible over~$\mathbb Q$ itself (in particular, $\ell\le37$), and we show the existence of a bound on the degrees of cyclic isogenies between $\mathbb Q$-curves depending only on the degree of the field. We also prove that the only possible torsion groups of $\mathbb Q$-curves over number fields of degree not divisible by a prime $\ell\leq 7$ are the $15$ groups that appear as torsion groups of elliptic curves over $\mathbb Q$. Complementing these theoretical results we give an algorithm for establishing whether any given elliptic curve $E$ is a $\mathbb Q$-curve, which involves working only over $\mathbb Q(j(E))$.

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arXiv2020

On rational Bianchi newforms and abelian surfaces with quaternionic multiplication

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 newform with nebentypus character of order 4 and eight inner twists.

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arXiv2020

Global methods for the symplectic type of congruences between elliptic curves

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 global symplectic criteria that apply in situations where the available local methods may fail. We report on the results of applying our methods for all primes $p\ge7$ to the elliptic curves in the LMFDB database, which currently includes all elliptic curves of conductor less than ${500000}$. We also show that while such congruences exist for each $p\le17$, there are none for $p \geq 19$ in the database, in line with a strong form of the Frey-Mazur conjecture.

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arXiv2020

The proportion of genus one curves over $\mathbb{Q}$ defined by a binary quartic that everywhere locally have a point

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 these curves that are locally soluble, computed as a product of local densities, is approximately 75.96%. We prove that the local density at a prime $p$ is given by a fixed degree-$9$ rational function of $p$ for all odd $p$ (and for the generalised equation, the same rational function gives the local density at every prime). An additional analysis is carried out to estimate rigorously the local density at the real place.

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arXiv2020

Sorting and labelling integral ideals in a number field

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 total order on the set of prime ideals of $K$ and then extend this to a total order on the set of all nonzero integral ideals of $K$. This order allows us to give a unique label of the form $N.i$, where $N$ is its norm and $i$ is the index of the ideal in the ordered list of all ideals of norm $N$. Our ideal labelling scheme has several nice properties: for a given norm, prime ideals always appear first, and given the factorisation of the norm, the bijection between ideals of norm $N$ and labels is computable in polynomial time. Our motivation for this is to have a well-defined and concise way to sort and label ideals for use in databases such as the LMFDB. We have implemented algorithms which realise this scheme, in Sage, Magma and Pari.

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arXiv2020

Computing classical modular forms

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

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arXiv2018

On Elliptic Curves of prime power conductor over imaginary quadratic fields with class number one

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-Oesterlé (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 theorem holds with no change, while for the remaining five fields the discriminant of a curve with prime conductor is either prime or the square of a prime. The proof is conditional in two ways: first that the curves are modular, so are associated to suitable Bianchi newforms; and second that a certain level-lowering conjecture holds for Bianchi newforms. We also classify all elliptic curves of prime power conductor and non-trivial torsion over each of the nine fields: in the case of $2$-torsion, we find that such curves either have CM or with a small finite number of exceptions arise from a family analogous to the Setzer-Neumann family over $\Q$.

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arXiv2018

Black Box Galois Representations

We develop methods to study $2$-dimensional $2$-adic Galois representations $ρ$ 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 primes. Using suitable finite test sets of primes, depending only on $K$ and $S$, we show how to determine the determinant $\detρ$, whether or not $ρ$ is residually reducible, and further information about the size of the isogeny graph of $ρ$ whose vertices are homothety classes of stable lattices. The methods are illustrated with examples for $K=\mathbb{Q}$, and for $K$ imaginary quadratic, $ρ$ being the representation attached to a Bianchi modular form. These results form part of the first author's thesis.

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arXiv2016

The L-functions and modular forms database project

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-function is the Riemann zeta-function, and as well as being ubiquitous in number theory itself, L-functions have applications in mathematical physics and cryptography. Two of the seven Clay Mathematics Million Dollar Millennium Problems, the Riemann Hypothesis and the Birch and Swinnerton-Dyer Conjecture, deal with their properties. Many different mathematical objects are connected in various ways to L-functions, but the study of those objects is highly specialized, and most mathematicians have only a vague idea of the objects outside their specialty and how everything is related. Helping mathematicians to understand these connections was the motivation for the L-functions and Modular Forms Database (LMFDB) project. Its mission is to chart the landscape of L-functions and modular forms in a systematic, comprehensive and concrete fashion. This involves developing their theory, creating and improving algorithms for computing and classifying them, and hence discovering new properties of these functions, and testing fundamental conjectures. In the lecture I gave a very brief introduction to L-functions for non-experts, and explained and demonstrated how the large collection of data in the LMFDB is organized and displayed, showing the interrelations between linked objects, through our website www.lmfdb.org. I also showed how this has been created by a world-wide open source collaboration, which we hope may become a model for others.

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arXiv2015

What is the probability that a random integral quadratic form in $n$ variables is isotropic?

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 is isotropic. In particular, we show that the probability that a random integral quaternary quadratic form is isotropic is $\approx 97.0\%$, in the case where the coefficients of the quadratic form are independently and uniformly distributed in the range $[-X,X]$ with $X\to\infty$. When random integral quaternary quadratic forms are chosen with respect to the Gaussian Orthogonal Ensemble (GOE), the probability of isotropy increases to $\approx 98.3\%$.

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arXiv2014

Tetrahedral Elliptic Curves and the local-global principle for Isogenies

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 $\mathbb{Q}$ 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 field. In this paper we provide a classification of failures for number fields which do contain this quadratic field, and we find a new `exceptional' source of such failures arising from the exceptional subgroups of $\mbox{PGL}_2(\mathbb{F}_l)$. By constructing models of two modular curves, $X_{\text{s}}(5)$ and $X_{S_4}(13)$, we find two new families of elliptic curves for which the principle fails, and we show that, for quadratic fields, there can be no other exceptional failures.

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arXiv2013

The proportion of plane cubic curves over ${\mathbb Q}$ that everywhere locally have a point

We show that the proportion of plane cubic curves over ${\mathbb Q}_p$ that have a ${\mathbb Q}_p$-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 ${\mathbb Q}$ that have points everywhere locally; numerically, this density is shown to be $\approx 97.3\%$.

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arXiv2012

Explicit n-descent on elliptic curves. III. Algorithms

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 algorithm for trivialising central simple algebras. This is of independent interest: for example, it could be used for parametrising Brauer-Severi surfaces.

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arXiv2009

Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves

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 number fields. Finally, for genus one models defined over Q, we develop a theory of reduction and again give explicit algorithms for n = 2, 3 and 4.

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arXiv2006

Explicit n-descent on elliptic curves, II. Geometry

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 factors through the Segre embedding P^{n-1} x P^{n-1} --> P^{n^2-1}. The comparison relies on an explicit version of the local-to-global principle for the n-torsion of the Brauer group of the base field.

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arXiv2006

Explicit n-descent on elliptic curves, I. Algebra

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.

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