**
Samuele Anni
**

The inverse Galois problem: abelian varieties, modular forms and Goldbach's
conjecture

*
The inverse Galois problem is one of the most interesting open problems in
group theory and also one of the easiest to state: is each finite group a
Galois group? Hilbert was the first to study this problem: Hilbert's
irreducibility theorem established a connection between Galois groups over the
field of rational numbers Q and Galois groups on Q[x], and this led him to show
that symmetric groups are realisable as Galois groups over Q.
My interest in this topic is related to the realization of linear and
symplectic groups as Galois groups over Q and over number fields, with a
particular emphasis on effective and explicit results.
In this talk I will describe "uniform" realizations for symplectic groups
(joint work with Vladimir Dokchitser) and generalisations to number fields,
using modular forms. If time allows, I will also speak about some results about
degrees of isogenies between semistable elliptic curves over totally real
fields, obtained using similar techniques.
*

**
Barry Fagin
**

Idempotent Integers: The complete class of numbers n with bipartite factorizations that work correctly with RSA

*
Semiprimes (n=p*q with p and q prime) are used as the basis of the RSA
cryptosystem due to the difficulty of factoring large integers. Public and
private keys (e,d) are chosen such that they are congruent to 1 mod phi(n) =
(p-1)(q-1), with encryption and decryption achieved through modular
exponentiation mod e and d.
What is often misunderstood in presenting public key cryptography is that the
use of semiprimes is required not because of correctness, but because of
security. The class of positive integers n with at least one factorization n=pq
such that p and q can correctly encrypt and decrypt was originally thought to
include only the semiprimes, and then later some Carmichael numbers. Recent
work has shown that characterization to also be incomplete. The complete class
consists exactly of those square-free numbers n=pq such that lambda(n) |
(p-1)(q-1), where lambda is the Carmichael function. Note that one or both of
(p,q) may be composite. We refer to such n as idempotent integers, and the
multiplicative partition n=pq as an idempotent factorization. Even though only
the semiprimes are useful cryptographically, idempotent integers are worthy of
study on their own as their discovery and construction lie at the intersection
of hard problems in computer science and number theory.
We discuss what it known about these integers, including:
Smallest examples with 2,3,4,5,6 and 7 factors;
Fully composite and semi-composite factorizations;
Less efficient and more efficient search techniques;
Constructive techniques;
Theorems.
In the process of identifying idempotent integers, we discovered that certain
idempotent integers are maximal in that all their bipartite multiplicative
partitions are idempotent. We show how to construct maximally idempotent
integers of arbitrary size using specially constructed number-theoretic graphs.
The largest maximally idempotent integer built by the author has 2081 digits,
with approximately 10^43 smaller maximally idempotent integers as divisors. We
will present numerous examples of idempotent and maximally idempotent integers
and go through examples of how to find and construct them.
*

**
Daniel Fiorilli
**

Omega results for cubic field counts via the Katz-Sarnak philosophy

*
I will discuss recent joint work with P. Cho, Y. Lee and A. Södergren. Since
the work of Davenport-Heilbronn, much work has been done to obtain a precise
estimate for the number of cubic fields of discriminant at most X. This
includes work of Belabas-Bhargava-Pomerance, Bhargava-Shankar-Tsimerman and
Taniguchi-Thorne. In this talk I will present a negative result, which states
that the GRH implies that the error term in this estimate cannot be too small.
Our approach involves low-lying zeros of Dedekind zeta functions of cubic
fields (first studied by Yang), and is strongly related to the Katz-Sarnak
conjectures and the ratios conjecture of Conrey, Farmer and Zirnbauer.
*

**
Olivier Fouquet
**

The Birch and Swinnerton-Dyer Conjecture for elliptic curves of rank 0 for almost all primes

*
The Birch and Swinnerton-Dyer Conjecture predicts the p-adic valuation of the
first non-zero term in the Taylor expansion of the L-function of an elliptic
curve about s=0 divided by a suitable period and regulator. In joint work with
Xin Wan, we prove it holds for all rational elliptic curves of rank 0 when
p≥167.
*

**
Pip Goodman
**

Superelliptic curves with large Galois images

*
In 1972, Serre proved his famous 'Open Image Theorem' for non-CM elliptic
curves. Since then, work has focused on analogues for higher dimensional
abelian varieties with trivial endomorphism ring. The results varying in
explicitness. In this talk we present an explicit large image result for
superelliptic jacobians. As part of this, we examine the somewhat surprising
structure of these images.
*

**
Richard Griffon
**

New cases of the generalised Brauer-Siegel theorem

*
Tsfasman and Vladuts made a conjecture concerning families of number fields
which, if true, would be a vast generalisation of the classical Brauer—Siegel
theorem. Their conjecture, call it BS-TV, is known to hold conditionally to
GRH, but also unconditionally in a number of cases. For instance, Lebacque and
Zykin have shown that BS-TV holds for asymptotically exact families of stepwise
Galois number fields. In this talk, I will report on recent joint work with
Philippe Lebacque, where we exhibit new families of number fields where the
BS-TV is true. Among other examples, we prove that any infinite global field
contained in a p-class field tower satisfies the generalised form of the
Brauer—Siegel theorem.
*

**
Marc Hindry
**

Arithmetic of algebraic surfaces over finite fields

*
Consider a smooth projective surface. Its Néron-Severi group is finitely
generated; if C1, … Cr are curves on the surface that form a basis of the
Néron-Severi group modulo torsion, the regulator is the absolute value of the
determinant of the intersection numbers of the Ci and Cj. We will discuss
reasonable conditions under which this regulator is bounded in terms of the
geometric genus of the surface (beware: no such universal bound can exist!).
When the curve is defined over a finite field of cardinality q, the problem is
more arithmetic and one can bring in arithmetic tools like zeta functions and
we show that, under appropriate conditions, the regulator is bounded by q power
the geometric genus (up to epsilon).
In fact this result is conditional to the finiteness of the Brauer group of the
surface and one obtains actually a bound for the product of the cardinality of
the Brauer group and the regulator. This is a partial generalisation of a
result of Richard Griffon (2018) concerning Fermat surfaces, but less precise,
namely one obtains only «half» of the analogue of Brauer-Siegel theorem.
*

**
Elisa Lorenzo García
**

Decomposing Jacobians via Galois covers

*
Let ϕ:X→Y be a cover between two algebraic curves of positive genus. We develop
tools that may identify the Prym variety of ϕ, up to isogeny, as the Jacobian
of a quotient curve C in the Galois closure of the composition of ϕ with a
well-chosen map Y→ℙ¹. This method allows us to recover all previously obtained
descriptions of a Prym variety in terms of a Jacobian that are known to us. We
also find algebraic equations for some of these new cases, including one where
X has genus 3, Y has genus 1 and ϕ is a degree 3 map totally ramified over 2
points. This is joint work with Davide Lombardo, Christophe Ritzenthaler and
Jeroen Sijsling.
*

**
Michelle Manes
**

Some Current Trends in Arithmetic Dynamics

*
Arithmetic dynamics is a relatively new field in which classical problems from
number theory and algebraic geometry are reformulated in the setting of
iterated (discrete) dynamical systems. For example, rational points on
algebraic varieties become rational points in orbits, and torsion points on
abelian varieties become points having finite orbits. Moduli problems also
appear, where for example the complex multiplication points in the moduli space
of abelian varieties correspond (sort of) to the postcritically finite points
in the moduli space of rational maps. I'll describe some background in
dynamics, a motivating analogy that drives much current research, and I will
highlight some recent progress.
*

**
Fabien Pazuki
**

Isogeny estimates - the tryptic mirror

*
We will state and compare three results concerning j-invariants in isogeny
classes. We begin by studying the case of elliptic curves over number fields.
In joint work with Griffon, we turn to elliptic curves over function fields.
The third part will focus on Drinfeld modules of rank 2 and is joint work with
Breuer and Razafinjatovo.
*

**
Samir Perlaza
**

Shannon's Mathematical Theory of Data Transmission — Case of Low Latency and High Reliability

*
In this talk, the mathematical formulation of the problem of data transmission
introduced by Claude Shannon is presented in the aim of building common grounds
for the analysis of low latency and high reliability communications systems.
Using this formulation, the fundamental limits on the decoding error
probability (reliability) and transmission duration (latency) are briefly
discussed to highlight the main limitations of the existing results. In
particular, the focus is on the difficulty of approximating the cumulative
distribution function of the sum of a ﬁnite number of i.i.d random variables
that arise in this problem. The main result is a new method to approximate
such cumulative distribution functions; and an upper bound on the approximation
error. Existing results such as the Gaussian (Berry-Esseen) and Saddlepoint
approximations are shown to be special cases of the new approximation. The new
approximation is shown to lead to more accurate calculations of the fundamental
limits of low latency and high reliability communications systems.
*

**
Renate Scheidler
**

Computing modular polynomials of rank-2 Drinfeld modules

*
Drinfeld modules represent the function field analogue of the theory of complex
multiplication. They were introduced as "elliptic modules" by Vladimir Drinfeld
in the 1970s in the course of proving the Langlands conjectures for GL(2) over
global function fields. Drinfeld modules of rank 2 exhibit very similar
behaviour to elliptic curves: they are classified as ordinary or supersingular,
support isogenies and their duals, and their endomorphism rings have an
analogous structure. Their isomorphism classes are parameterized by
j-invariants, and the n-th Drinfeld modular polynomial parameterizes pairs of
n-isogenous rank-2 Drinfeld modules, visualized by their isogeny graph whose
ordinary connected components take the shape of volcanos when n is prime. While
the rich analytic and algebraic theory of Drinfeld modules has undergone
extensive investigation, very little has been explored from a computational
perspective. This research represents a first foray in this direction,
introducing an algorithm for computing Drinfeld modular polynomials. This is
joint work with Perlas Caranay and Matthew Greenberg, as well as ongoing
research with Edgar Pacheco Castan. Some familiarity with elliptic curves is
expected for this talk, but no prior knowledge of Drinfeld modules is assumed.
*

**
Claudia Schoemann
**

The twisted forms of a semisimple group over the integral domain of a global function field

*
Let K=F_q(C) be the global field of rational functions on a smooth and
projective curve C defined over a finite field F_q. Any finite but non-empty
set S of closed points on C gives rise to an integral domain O_S=F_q[C-S] of K.
Given a semisimple and almost-simple group scheme G defined over Spec O_s with
a smooth fundamental group F(G), we describe the set of (O_S-classes of)
twisted forms of G in terms of the geometric invariants of F(G) and the
absolute type of the Dynkin diagram of G. This finite set is given by
H_fl^1(O_S,Aut(G)) seen as the disjoint union over P of H_fl^1(O_S,{}^P G^{ad})
modulo the Out(G)-action, where P are the G^{ad}-torsors, and turns out in most
cases to biject to a disjoint union of finite abelian groups. This is joint
work with Rony A. Bitan and Ralf Köhl.
*

**
René Schoof
**

Finite flat group schemes over Z

*
Just like finite abelian groups, commutative finite flat group schemes over Z
are successive extensions of simple finite flat group schemes.
Simple finite flat group schemes have order a power of a prime p.
In his 1966 “Driebergen” paper Tate asked whether for every prime p the
only simple finite flat group schemes over Z of p-power order are
the constant group scheme Z/pZ and its dual µₚ.
In joint work with Lassina Dembele we show that this true for p≤19.
*

**
Robin Zhang
**

A Galois–dynamics correspondence for unicritical polynomials

*
A universal fact about abelian varieties that was used in the proof of the
Mordell–Lang conjecture is that their torsion points always satisfy the Galois
homothety property. We describe a dynamical analogue with connections to the
irreducibility of dynatomic polynomials and to quadratic points on dynatomic
curves. Using existing results on explicit Hilbert irreducibility, we deduce
the non-existence (outside of a specified finite set) of quadratic polynomials
with periodic points of exact period 5 or 6 in quadratic number fields.
*