Entanglement of Modular Forms
Samuele Anni, Aix-Marseille Université

This talk addresses the compelling question of whether two residual Galois representations attached to modular forms, possibly in different residual characteristics, can correspond to the same number field or, more generally, may have common non-trivial subfields. The study of entanglement for torsion representations attached to elliptic curves over the rationals has been already addressed in several papers. This question, linked to the inverse Galois and modular forms, highlights the importance of the study of congruences. Our ongoing research, in collaboration with Luis Dieulefait (Universitat de Barcelona) and Gabor Wiese (Université du Luxembourg), aims to uncover the conditions under which entanglement arises, offering new insights into the intricate relationships between modular forms, arithmetic geometry and representation theory.

Rank and non-vanishing in the family of elliptic curves y² = x³ − d x
Chantal David, Concordia University

The elliptic curves E_d : y² = x³ − d x, where d is a 4th power-free integer, form a family of quartic twists. We study in this talk the average analytic rank r(d) over the family. Under the GRH, we show that the average analytic rank is bounded by 13/6, and by 3/2 assuming a conjecture of Heath-Brown and Patterson about the distribution of quartic Gauss sums. Since the same result holds when we restricts to the subfamilies of curves E_d where the root number is fixed (i.e. W(E_d) = ±1), this shows that there is a positive proportion of curves with r(E_d) = 0 among the curves with even analytic rank, and a positive proportions of curves with r(E_d) = 1 among the curves with odd analytic rank. Our results are similar to the results obtained by Heath-Brown for the analytic rank of the quadratic twists d y² = x³ + a x + b under the GRH. For the quadratic twists, it was shown in the recent ground-breaking work of Smith that half of the quadratic twists have algebraic rank 0 and half of the quadratic twists have algebraic rank 1, under the assumption that the Tate-Shafarevic group is finite. For the case of the quartic twists E_d : y² = x³ − d x, no bound for the average algebraic rank is known. This is joint work with L. Devin, A. Fazzari and E. Waxman.

On the geometry of period lattices for elliptic curves
Lenny Fukshansky, Claremont McKenna College

Isomorphism classes of elliptic curves correspond to similarity classes of planar lattices, represented by points in the fundamental strip of the upper halfplane. The position of such a point determines certain geometric properties of the corresponding lattice and arithmetic properties of the corresponding elliptic curve. The purpose of this talk is to describe some such properties. We will focus on well-rounded, semistable, virtually rectangular and arithmetic lattices and their corresponding curves. We will also discuss the finite sequence of the so-called deep-hole lattices resulting from any point in the fundamental strip and the corresponding family of elliptic curves and their isogenies, which become especially interesting in the CM/arithmetic case. The talk is based on joint works with Pavel Guerzhoy, Stefan Kuehnlein, Florian Luca and Tanis Nielsen.

Local-global principles for abelian schemes
Wojciech Gajda, Adam Mickiewicz University

I will tell a story which originated almost 30 years ago with a paper by Corrales-Rodriganes and Schoof on an elliptic analogue of the support problem of Erdos. The paper led to investigations of many authors of certain local-global principles for detecting relations in Mordell-Weil groups of abelian varieties. In the final part of my talk I will discuss recent results on related local-global principle for homomorphisms between abelian schemes (joint work with Petersen) which are follow-ups of the paper by Corrales-Rodriganes and Schoof.

Class groups as Galois modules
Cornelius Greither, Universität der Bundeswehr München

We consider a G-Galois extension K/F of number fields, and the class group Cl_K of K as a module over the group ring Z[G]. It is a very classical fact that special values of zeta functions and L-functions "know a lot" about Cl_K. In particular if K/F is a CM extension and if we take minus parts (this will be explained), then the special value boils down to something rational and we may essentially predict the order of Cl_K^-. But very much more can be done. Indeed one can say a lot about Cl_K^- as a Z[G]-module. It is asking too much to predict that module up to isomorphism. As a more realistic goal, so-called Fitting ideals have been studied. In this talk the approach is different: we describe an interesting equivalence relation on finite Z[G]-modules, and we explain results determining the class group up to equivalence. Very roughly, only a small proportion of modules up to equivalence may be "realized" in this way. The latest progress is going beyond the case of abelian Galois groups, admitting certain smallish nonabelian groups G. In doing so, we have occasion to use fairly old results on the classification of certain torsionfree Z[G]-modules, which are interesting per se, e.g. in the case where G is dihedral of order 2p . This talk reports on joint work with T. Kataoka, based on previous work of Kataoka, Atsuta, Kurihara, the speaker, and many others.

Zeta functions and arithmetic geometry
Marc Hindry, Université Paris-Diderot

We will describe several situations where one can identify interesting arithmetic quantities (regulators, heights, cardinality of arithmetic groups, etc.) inside special values of zeta functions and infer asymptotic estimates for the latter via complex analysis. The model for this pattern is the Brauer-Siegel theorem, I will show that there are many other applications of this principle.

Challenges in generating random supersingular elliptic curves over finite fields of large characteristic
Annamaria Iezzi, Université Grenoble Alpes

Constructing supersingular elliptic curves with unknown endomorphism ring over finite fields of large characteristic is an open problem with significant implications for isogeny-based cryptography. In this talk, we will provide an overview of recent attempts to tackle this problem, we will examine their limitations and we will possibly discuss new ideas.

Constructing Class Groups of Imaginary Quadratic Fields with Large n-Rank
Mike Jacobson, University of Calgary

Constructing imaginary quadratic fields whose ideal class groups have large n-rank has proved to be a challenging practical problem, due in part to fact that we believe such examples to be very rare. One of the most successful methods for producing many fields of relatively small discriminant with large 3-rank is due to Diaz y Diaz; this was part of the method used by Quer to find 3 fields with 3-rank equal to 6 in 1987, which still stands as the current record. We describe generalizations to this method for constructing fields with large n-rank for an arbitrary integer n >= 3, and practical enhancements to improve the efficiency of the search procedure. We present some preliminary numerical results, including the first known example of an imaginary quadratic field whose class group has 7-rank equal to 4.

Abelian varieties over F_2 of prescribed order and dimension
Kiran Kedlaya, University of California San Diego

For A a simple abelian variety of dimension g over a finite field F_q, when q > 2 one can give a lower bound on the order of A(F_q) which is exponential in g. By contrast, for q = 2 it has been known since the 1970s that there are infinitely many simple abelian varieties of order 1. We show that for every fixed m > 1, there exist infinitely many simple abelian varieties over F_2 of order m; for certain values of m, we also show that these occur in every sufficiently large dimension.

Modular entanglements of elliptic curves
David Kohel, Aix-Marseille Université

Let N₁ and N₂ be two levels in {3,4,5}. We describe explicit 1-dimensional families parametrizing pairs of elliptic curves with projective A₄-entanglements between the Galois representations on the respective N₁-torsion and N₂-torsion subgroups. Following the work of Silverberg, Rubin and Silverberg, and Fisher, we derive explicit models for the associated modular surfaces parametrizing all such pairs of elliptic curves over a suitable cyclotomic extension of ℚ. This is joint work with Samuele Anni and Zoé Yvon.

Automorphisms of modular curves
Pietro Mercuri, Sapienza Università di Roma

First we recall the definition of modular curve and of modular automorphism of a modular curve. After this, we try to and (partially) successfully answer the question: Is every automorphism of a modular curve a modular automorphism?

Computing supersingular traces
Travis Morrison, Virginia Tech

Let p > 3 be a prime, let E be a supersingular elliptic curve defined over F_{p²} with j(E) ≠ 0, 1728, and let α be an endomorphism of E represented as a sequence of L isogenies of degree at most d. We prove that the trace of α may be computed in time O(n⁴(log n)² + dLn³) bit operations using a generalization of the SEA algorithm for computing the trace of the Frobenius endomorphism of an ordinary curve. When L = O(log p) and d = O(1), this complexity is quasi-quadratic in the size of the input and matches the heuristic complexity of the SEA algorithm. Our theorem is unconditional, unlike the SEA algorithm, since all isogenies of a supersingular elliptic curve are defined over an extension whose degree is bounded by a constant, independent of p. We also give practical speedups, including a fast algorithm to compute the trace of α modulo p, that provide a substantial speedup to the computation of the trace of α in practice.

Structure of Supersingular Elliptic Curve Isogeny Graphs
Renate Scheidler, University of Calgary

Supersingular elliptic curve isogeny graphs have isomorphism classes of supersingular elliptic curves over a finite field as their vertices and isogenies of some fixed degree between them as their edges. Due to their apparent "random" nature, supersingular isogeny graphs – which are optimal expander graphs – have been used as a setting for certain cryptographic schemes that are resistant to attacks by quantum computers. However, recent research, including the work presented here, reveals hidden structures in these graphs that may have implications for the security of supersingular isogeny based cryptosystems. This is joint work with Sarah Arpin (Virginia Tech) and our co-supervised undergraduate student Taha Hedayat (University of Calgary).

The kernel of the Gysin homomorphism for positive characteristic - From algebraic varieties to smooth schemes
Claudia Schoemann, HAWK University Göttingen

Let S be a smooth projective connected surface over an algebraically closed field k embedded into a projective space P^d and let C be a smooth projective curve embedded into S. Let CH0(S)_{deg=0} and CH0(C)_{deg=0} be the Chow groups of zero-cycles of degree 0 on S and C, respectively. The kernel of the Gysin homomorphism from CH0(C)_{deg=0} to CH0(S)_{deg=0} is a countable union of translates of an abelian subvariety A inside the Jacobian J of the curve C. Further there is a countable open subset U₀ contained in the subset U of (P^d)^* parametrizing the smooth curves such that A=B or A=0 for all curves parametrized by U₀, where B is the abelian subvariety of J corresponding to the vanishing cohomology H¹(C, k')_{van} of C. Hence the kernel of the Gysin homomorphism either is a countable union of shifts of B or, if A=0, it is countable. Passing to finite type smooth schemes as a generalisation of algebraic varieties we extend this result to the subset U of (P^d)^*, i.e. to all smooth curves C. Using the language of algebraic stacks this is done by constructing an increasing filtration of Zariski countable open substacks U_i of U for i in a countable set I, that respects the required isomorphism between geometric generic and special point and by applying a convergence argument. This is joint work with Rina Paucar (IMCA/UNI, Lima in Peru).

Planar self-avoiding walks in a rectangle
László Szalay, University of Sopron

Assume that there is given a connected graph. A self-avoiding walk on the graph is a walk that never visits the same vertex more than once. One of the basic problems in this area is to find the number of distinct self-avoiding walks from the origin on the integer lattice Z^d to a fixed vertex. This problem seems to be intractable in general. We investigated some specific cases on the plane (i.e. d=2), and discovered interesting properties related to the enumerative statements. The results are joint work with H. Belbachir, L. Major, L. Németh, and A. Pahikkala.

Speeding up the computation of group order of genus 2 curves over finite fields
Nicolas Thériault, Universidad de Santiago de Chile

In 2012, Gaudry and Schost obtained the first known randomly selected cryptographically interesting hyperelliptic curve of genus 2 over a field of 127 bits. Producing this curve required a very significant computation, in the order of one million CPU-hours, in which computing the order of a specific curve (after a comparatively fast selection process) would take on average close to 1000 CPU-hours. Furthermore, the curves considered had full 2-torsion groups, i.e. the curve had group order 16*prime. We present new developments in the computation of the group order of specific curves which allow us to significantly reduce the cost of several of the sub-algorithms used by Gaudry and Schost, specifically: improved division-by-L algorithms, specialized factorization algorithms and the use of special towers of field extensions, which allow us to compute the group order of each hyperelliptic curve in less than 100 CPU-hour. Furthermore, our algorithm applies to more general curves and we are able to obtain examples of curves with prime order (giving slightly stronger curves) over fields of 127 bits.

Lagrange’s theorem for certain finite flat group schemes
Emiliano Torti, University of French Polynesia

Around 1960s, Grothendieck asked if every finite flat group scheme over any base is killed by its order, i.e. the morphism multiplication-by-#G factors via the identity section. In this talk, we discuss the state of the art of the question as well as giving a positive answer in a new case when the special fiber of G belongs to a certain family of non-commutative finite flat group scheme. This extends partially the current best known result due to Schoof.

The zeta functions of Weierstrass and Riemann
Michel Waldschmidt, Université Pierre-et-Marie-Curie

In his book Auxiliary polynomials in Number Theory published in 2016 (Cambridge University Press), David Masser states a lemma: "The function ζ(z) has infinitely many non-real zeroes with real part different from 1/2" with the following comment: "With the powerful method of auxiliary polynomials we have now developed so much machinery that we can dispose of a well known conjecture in the negative." At the end of this chapter he proposes some exercises: "Show that the Riemann zeta function and the Weierstrass zeta function are algebraically independent over ℂ" with the hint: "the first is a function of s and the second a function of z." The next exercise is "No, seriously, show that the Riemann zeta function ζ(z) and the Weierstrass zeta function ζ(z) are algebraically independent over ℂ. Well, you know what I mean..." We develop this point of view in connection with Schanuel's Conjecture.

The first level of ℤₚ-extensions and compatibility of heuristics
Lawrence C. Washington, University of Maryland

Let K be an imaginary quadratic field in which the odd prime p does not split. When the p-part of the class group of K is cyclic, we describe the possible structures for the p-part of the class group of the first level of the cyclotomic ℤₚ-extension of K. This allows us to show the compatibility of the heuristics of Cohen--Lenstra--Martinet for class groups with the heuristics of Ellenberg--Jain--Venkatesh for how often the cyclotomic Iwasawa invariant lambda equals 1. This is joint work with Debanjana Kundu.

Rigorous methods in computational algebraic number theory
Benjamin Wesolowski, École normale supérieure de Lyon

Many number theoretic algorithms resort to heuristic assumptions for their analysis. This issue concerns some of the most fundamental problems of the field, such as the computation of class groups in subexponential time. In this talk, we will present new methods for the design of algorithms which, instead of ad hoc heuristics, only rely on the Extended Riemann Hypothesis.

Conway-Coxeter friezes as positive integral points
Robin Zhang, Massachusetts Institute of Technology

In the 1970s, Coxeter and Conway introduced infinite arrays of positive integers called frieze patterns. In the 2000s, it was discovered that these friezes could be extended to general Dynkin types with deep connections to representation theory. Most types of positive integral friezes have been completely enumerated using methods from discrete geometry, algebraic combinatorics, and cluster algebras. We describe how to view these enumeration problems as a question of bounding the number of positive integral points on certain n-dimensional affine varieties associated to Cartan matrices.