# Titles and Abstracts

### Explicit Methods in Number Theory: Conference in Honour of John Cremona's 60th Birthday

Barinder Banwait (Essen)

Traces of Frobenius and an inverse Galois Problem for cubic surfaces over finite fields

For a cubic surface over a finite field of $q$ elements, the trace of Frobenius acting on the Picard group must lie in the set $\{-2,-1,0,1,2,3,4,5,7\}$. Serre has asked which of these values actually arise for which $q$. Building on results of Swinnerton-Dyer we give an answer to this, and related, questions. This is work in progress joint with Francesc Fité and Daniel Loughran.

Manjul Bhargava (Princeton)

A positive proportion of plane cubics fail (respectively satisfy) the Hasse principle

Bryan Birch (Oxford)

Insoluble Curves

I will survey some questions about whether curves have soluble points, and the connection with an algebraic version of Hilbert's 13th problem.

Henri Cohen (Bordeaux)

L-functions and modular forms in Pari/GP

I will describe two important and new packages now available in Pari/GP: a package to work with L-functions, and a package for working with classical modular forms based on the use of several trace formulas. The theory behind these packages will be explained, and I will present a computer demo of the main functionalities. The L-function package is work joint with B. Allombert, K. Belabas, and P. Molin, and the modular forms package is work joint with K. Belabas and N. Mascot, with essential input from A. Booker and collaborators.

Lassina Dembele (Warwick)

On the existence of abelian surfaces with everywhere good reduction

A famous result of Fontaine (and Abrashkin) states that there is no abelian variety over the rationals with everywhere good reduction. Fontaine's proof of this result relies on the non-existence of certain finite flat group schemes. His technique has been refined by several people (including Schoof, Brumer and Calegari) to prove non-existence of semi-stable abelian varieties over various fields. But one has to expect that such non-existence results are the exception rather than the norm. Indeed, as the base field varies, we must hope to find more abelian varieties with everywhere good reduction.

In this talk, I will present a search method for finding abelian surfaces with everywhere good reduction over real quadratic fields. One main feature of this approach is that it allows for the determination of abelian surfaces with trivial endomorphism rings in some cases.

(This is a progress report on joint work with Abhinav Kumar.)

Tom Fisher (Cambridge)

Binary quartics and the Cassels-Tate pairing

Let $E$ be an elliptic curve over a number field $K$. We may improve the upper bound on the rank of $E(K)$ coming from $2$-descent to that coming from $4$-descent by computing the Cassels-Tate pairing on the $2$-Selmer group. Cassels described a method for computing this pairing that involves solving conics over the fields of definition of the $2$-torsion points of E. More recently, S. Donnelly found a method that only involves solving conics over $K$. I will describe a variant of his method phrased in terms of the invariant theory of binary quartics. One ingredient is the construction of certain $(2,2,2)$-forms that describe triples of $2$-Selmer elements that sum to zero.

Hendrik Lenstra (Leiden)

Abelian automorphisms towers

For each group one can form its automorphism group, and upon iteration one obtains the automorphism tower of the group. The lecture is about abelian groups for which all groups appearing in the automorphism tower are likewise abelian.

Ariel Pacetti (Leverhulme Trust Visiting Professor)

Computing tables of elliptic curves over $\mathbb{Q}(i)$

Elliptic curves over $\mathbb{Q}$ played a central role in our nowadays understanding of many number theoretical problems, including it various applications to diophantine problems. In this talk we will present an algorithm which follows the original ideas over $\mathbb{Q}$ to compute elliptic curves over $\mathbb{Q}(i)$, with special emphasis in the prime power conductor case. During the talk we will mention why this case is more easy and will mention some generalizations to other imaginary quadratic number fields.

This is a joint work with M. Bennett, J. Cremona and N. Vescovo.

Soma Purkait (Kyushu)

Hecke algebras, new vectors and characterization of the new space

Let $p$ be a prime. Let $K_0(p^n)$ be the subgroup of $\mathrm{GL}_2(\mathbb{Z}_p)$ consisting of matrices with lower left entry in $p^n \mathbb{Z}_p$. We shall consider the Hecke algebra of $\mathrm{GL}_2(\mathbb{Q}_p)$ with respect to $K_0(p^n)$ and its subalgebra that is supported on $\mathrm{GL}_2(\mathbb{Z}_p)$ and describe them using generators and relations. This will allow us to explicitly describe the representations of $\mathrm{GL}_2(\mathbb{Z}_p)$ having a $K_0(p^n)$ fixed vector. We will translate this information to the classical setting which will lead us to characterize the space of newforms for $\Gamma_0(M)$ as a common eigenspace of certain Hecke operators which depends on prime divisors of $M$.

Tony Scholl (Cambridge)

Special values of L-functions

We will discuss some particular cases of conjectures on the special values of L-functions of motives.

Haluk Sengun (Sheffield)

Mod $p$ automorphic forms

Lately, classes in the mod $p$ cohomology of locally symmetric spaces have been called 'mod $p$ automorphic forms' by some mathematicians. In this talk, we shall discuss why these classes deserve this name and what we know/don't know/speculate about them.

Samir Siksek (Warwick)

Semistability and Serre's Uniformity

Serre's uniformity conjecture asserts that for a prime $\ell>37$ and $E$ an elliptic curve over the rationals without complex multiplication, the mod $\ell$ representation of $E$ is surjective. Serre in fact proved his conjecture for semistable elliptic curves. We will consider the analogue of Serre's uniformity conjecture both for semistable elliptic curves over totally real fields and for semistable principally polarized abelian varieties, with the help of group theory, class field theory, modularity and Merel's uniform boundedness theorem. This talk is based on joint work with Samuele Anni and Pedro Lemos.

Denis Simon (Caen)

Squares represented by cubic forms

Michael Stoll (Bayreuth)

The Generalized Fermat Equation $x^2 + y^3 = z^{11}$ slides

Generalizing Fermat's original problem, equations of the form $x^p + y^q = z^r$, to be solved in coprime integers, have been quite intensively studied. It is conjectured that there are only finitely many solutions in total for all triples $(p,q,r)$ such that $1/p + 1/q + 1/r < 1$ (the 'hyperbolic case'). The case $(p,q) = (2,3)$ is of special interest, since several solutions are known. To solve it completely in the hyperbolic case, one can restrict to $r = 8,9,10,15,25$ or a prime $\ge 7$. The cases $r = 7,8,9,10,15$ have been dealt with by various authors. In joint work with Nuno Freitas and Bartosz Naskrecki, we are now able to solve the case $r = 11$ and prove that the only solutions (up to signs) are $(x,y,z) = (1,0,1), (0,1,1), (1,-1,0), (3,-2,1)$. We use Frey curves to reduce the problem to the determination of the sets of rational points satisfying certain conditions on certain twists of the modular curve $X(11)$. A study of local properties of mod-$11$ Galois representations cuts down the number of twists to be considered. The main new ingredient is the use of the 'Selmer group Chabauty' techniques developed recently by the speaker to finish the determination of the relevant rational points.

Andrew Sutherland (Massachusetts Institute of Technology)

Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

Let $E/\mathbb{Q}$ be an elliptic curve and let $\mathbb{Q}(3^\infty)$ denote the compositum of all cubic extensions of $\mathbb{Q}$. While the the group $E(\mathbb{Q}(3^\infty))$ is not finitely generated, one can show that its torsion subgroup is finite; this holds more generally for any Galois extension of $\mathbb{Q}$ that contains only finitely many roots of unity. I will describe joint work with Daniels, Lozano-Robledo, and Najman, in which we obtain a complete classification of the $20$ torsion subgroups that can occur, along with an explicit description of the elliptic curves $E/\mathbb{Q}$ that realize each possibility. This is achieved by determining the rational points on a corresponding set of modular curves using a variety of techniques, some of which I will discuss.

Lynne Walling (Bristol)

Explicit action of Hecke operators on half-integral weight Siegel Eisenstein series

While classical Eisenstein series of integral weight are well-understood (for any level and character), there are many gaps in our knowledge regarding Siegel Eisenstein series, especially those with half-integral weight.

In this talk I will focus primarily on half-integral weight Siegel Eisenstein series of degree $n$ and level $4\mathcal{N}$ where $\mathcal{N}$ is odd and square-free (allowing arbitrary character modulo $4\mathcal{N}$). I will begin by defining such Eisenstein series, determining criteria for when they are non-zero, then proceed to evaluate the action of the Hecke operators $T_j(p^2)$ ($1\le j\le n$, $p$ an odd prime) on those Eisenstein series attached to the $\Gamma_0(4\mathcal{N})$-orbits of elements in $\Gamma_0(4)$. I will show that the subspace spanned by these Eisenstein series can be simultaneously diagonalised, explicitly computing the eigenvalues, and yielding a multiplicity-one result. Finally, I compare these eigenvalues to those of integral weight Siegel Eisenstein series. During the course of this talk I will point out the obstacles preventing me from extending this work to half-integral weight Siegel Eisenstein series attached to $\Gamma_0(4\mathcal{N})$-orbits of elements of $Sp_n(\mathbb{Z})$ not in $\Gamma_0(4)$ and the Hecke operators $T_j(4)$.