The house of an algebraic integer is the maximum absolute value of its algebraic conjugates. Lower bounds for the house may be easier to prove than lower bounds for the height, as V. Dimitrov's recent proof of the Schinzel-Zassenhaus conjecture suggests. We prove an analogue for the house of a recent conjecture by G. Rémond on lower bounds for the height in some radical extensions

I will review the notion of p-adic modular forms and show thorugh examples how they provide valuable tools to study arithmetic questions, such as congruences of special values of L-functions

Families of abelian varieties can often be parameterized by algebraic spaces; for instance, elliptic curves, up to isomorphism, are parameterized by the modular curve $Y(1)$, which is isomorphic to the affine line. We address the following problem: given two curves $C_1,C_2$ in $Y(1)^2$, can we determine whether every point $(p_1,q_1) \in C_1$ is isogenous to some point $(p_2,q_2) \in C_2$ (meaning that $p_1$ and $p_2$ correspond to isogenous elliptic curves - and the same for $q_1$ and $q_2$)? This kind of questions arose in the very first approaches to the André-Oort conjecture by André and Edixhoven. We present several viewpoints focusing on o-minimality vs. effectivity.

For a prime $\ell$, the connected components of an ordinary $\ell$-isogeny graph of elliptic curves over a finite field have a very regular shape: generically, each of them appears as a collection of isomorphic trees branching off a common central cycle, a structure which is reminiscent of a volcano. After recalling the reasons behind such a structure, we discuss whether it is true that every graph that looks like a genuine isogeny volcano can be in fact realized as a connected component of an ordinary isogeny graph over some finite field. If we restrict to fields of prime order, the answer to this question is affirmative and we show this in a joint work with Henry Bambury and Fabien Pazuki.

Let $P(X)$ be a monic, quartic polynomial with integer coefficients, irreducible and with cyclic or dihedral Galois group. We prove that there exists $c_P >0$ such that for a positive proportion of integers $n$, $P(n)$ has a prime factor bigger than $n^{1+c_P}$.
This is a joint work with James Maynard.

In this talk we will discuss the inertial Langlands correspondence for ${\rm GL}_2$ from an algorithmic point of view, and give some applications. (Joint with Nuno Freitas and John Voight.)

Solving some Diophantine equations can be a very difficult problem. In this talk, we will consider an opposite problem - can we construct Diophantine equations of a certain type with a prescribed set of solutions? More precisely, we consider equations of a shape $f(x) = y^n$. For a given finite set $S$ of integral powers, we look for a polynomial $f$ with integral coefficients such that only integral powers that appear when $f$ is evaluated at integers are precisely those in $S$.

In this presentation we will discuss results of recent work on two dimensional Kummer theory. In particular, a family of cyclic polynomials in the optimal number of parameters will be presented. Here the base field is assumed to be an index 2 subfield of a cyclotomic extension of an arbitrary field

Kummer varieties are an essential tool for performing explicit computations with the Jacobians of hyperelliptic curves. While the equations of the Kummer varieties are well-known for curves of low genus defined over number fields, these models do not behave when we reduce them modulo a prime lying above 2. In this talk, I will review the basic theory of Kummer surfaces defined over fields of characteristic 2 and explain how to obtain an explicit model for the desingularisation of the Kummer surface associated with a genus 2 curve.

In 1853 Chebyshev conjectured, based on numerical experiments, that for any given x there are at least as many primes congruent to 3 modulo 4 than primes congruent to 1 modulo 4 in the interval [2,x]. This observation, known as Chebyshev's bias, has since been vastly generalized and investigated. Notably Rubinstein and Sarnak ('94) proved conditonally on GRH and a linear independence hypothesis on zeros of Dirichlet L-functions that Chebyshev's conjecture is correct approximately for 99.6% of the values of x. I will report on joint work with Daniel Fiorilli in which we obtain unconditional instances of Chebyshev's bias in the setting of Galois extensions of the rationals.

Is there any distinguished elliptic curve over $\mathbf{Q}$ in its isogeny class?
First, Mazur and Swinnerton-Dyer proposed the so-called {\it strong} curve which is an optimal quotient of the Jacobian of the modular curve $X_0(M)$, where $M$ is the conductor of the isogeny class. Later, Stevens suggested that it is better to consider the elliptic curve which is an optimal quotient of the Jacobian of the modular curve $X_1(M)$. In both cases the Manin constant plays a role, and the Stevens proposal seems to be more intrinsically arithmetic due to the intervention of Néron models, étale isogenies, and Parshin-Faltings heights. We define the Faltings curve as the one with minimal height in the isogeny class. Let $G$ be the natural graph attached to an isogeny class: a vertex for every elliptic curve in the class, and edges correspond to isogenies of prime degree among them. For every square-free integer $d$, we can consider the graph $G^d$ attached to the twisted elliptic curves in $G$ by the quadratic character of $\mathbf{Q}(\sqrt{d})$. It turns out that $G$ and $G^d$ are canonically isomorphic as abstract graphs (the isomorphism identifies the vertices with equal $j$-invariant.) In this talk we shall discuss the probability distribution of a vertex in $G^d$ to be a Faltings elliptic curve as $|d|$ grows up to infinity.

The Edwards model of an elliptic curve $E$ is a certain embedding of $E$ into the product $\mathbb{P}^1 \times \mathbb{P}^1$ of two projective lines. One can often arrange for the points at infinity on E to not be rational over the ground field k, in which case the set $E(k)$ of rational points lies in the affine plane $\mathbb{A}^1 \times \mathbb{A}^1$ and the group law on $E$ can be given by a single formula that works for all of $E(k)$. My talk will discuss these issues for elliptic curves, and will then report on joint work with E. V. Flynn (arxiv:2211.01450) in which we generalize this construction to a general principally polarized abelian surface $J$. We obtain a model of $J$ as a subvariety of $\mathbb{P}^3 \times \mathbb{P}^3$, described by 15 equations of small bidegree in the 8 variables on $\mathbb{P}^3 \times \mathbb{P}^3$. This is considerably fewer than the usual 72 quadric equations in 16 variables for an abelian surface embedded in $\mathbb{P}^{15}$.

In Diophantine approximation, the quality of the approximation - studied typically via exponents - is determined by the sequence of best approximation vectors. In this talk, we study how properties of degeneracy of the dimension of the space of consecutive best approximation enable to refine some results.

Classical Iwasawa theory predicts that the growth of the class number in p-adic towers of number fields should be governed by three explicit invariants $\mu$, $\lambda$, $\nu$. Analogous results have been proven for towers of coverings of knot complements, and in this case Ueki has shown how to relate the invariant ? to the p-adic Mahler measure of the Alexander polynomial of the knot. In this talk, based on joint work in progress with Daniel Vallières, I will report on analogues of this theorem for the growth of the number of covering trees in towers of Galois coverings of finite graphs, where the analogue of the Alexander polynomial was defined by Ihara.

Continued fractions are a classical and very powerful tool of number theory. Therefore, it has been natural to introduce them in the field of $p$-adic numbers. Unlike continued fractions in $\mathbb R$, there is not a standard algorithm, due to the fact that there is not a unique canonical way to define the integer part of a $p$-adic number. In this talk, after a general introduction on the theory of $p$-adic continued fractions, we present a new algorithm, which is obtained as a slight modification of a Browkin's algorithm. The new algorithm improves the properties of periodicity of Browkin's algorithm both in theoretical and experimental results. In particular, we present results about the pure periodicity and the length of the pre-periods and the periods for periodic continued fractions, together with some computational observations.

Since the fundamental work of Faltings on Mordell's conjecture, many conjectures have been made concerning the problems of when rational points of a variety over a number field are (potentially) Zariski dense. Varieties whose rational points are (potentially) Zariski dense are called special, and Campana characterised these varieties as the ones that (loosely speaking) don?t admit fibrations to varieties of general type. Conjecturally, this is equivalent to the fact that complex analytification of the variety is Brody-special; that is, it admits a dense entire curve. Inspired by the notion of Brody-special, in a joint work with Jackson Morrow, we introduced the notion of $K$-analytically special varieties over an algebraically closed non archimedean field $K$. In this presentation, I shall explain this definition and prove several results ($K$-analytically special sub-varieties of semi-abelian varieties are translate of semi-abelian varieties; $K$-analytically special varieties don't dominate pseudo-$K$-analytically Brody hyperbolic variety) that support the fact that our notion is the right one to test specialness in $p$-adic analytic geometry.

Let $(F_n)_{n \geq 1}$ be the sequence of Fibonacci numbers. It is well known that every positive integer $n$ can be written as a sum of distinct non-consecutive Fibonacci numbers. This is called the Zeckendorf expansion of $n$ and, apart from the equivalent use of $F_1$ instead of $F_2$ or vice versa, it is unique. Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf expansion of $(2^{-1} \bmod F_n)$, for every positive integer $n$ that is not divisible by~$3$. We extend their result by determining the Zeckendorf expansion of the multiplicative inverse of $a$ modulo $F_n$, for every fixed integer $a \geq 3$ and every positive integer $n$ with $\gcd(a, F_n) = 1$. Furthermore, we consider the problem of finding quadruples $(F_a, F_b, F_c, F_d)$ of Fibonacci numbers such that $F_a \equiv F_c^{-1} \pmod {F_b}$ and $F_b \equiv F_d^{-1} \pmod {F_a}$. Finally, we raise some open questions.

For any prime p which is at least 5, Mazur proved that the rational torsion subgroup of the modular jacobian J_0(p), the Jacobian of the modular curve $X_0(p)$, is a cyclic subgroup of order the numerator of $p-\frac{1}{12}$, and it is generated by the linear equivalence class of the two cusps 0 and infinity on $X_0(p)$. The Manin and Drinfeld theorem tells us that for any modular curve, the difference of two cusps is a torsion point on the Jacobian, and thus the subgroup of the Jacobian generated by difference of cusps which are fixed by the Galois group is a subgroup of the rational torsion subgroup of its Jacobian. The natural question, following from Mazur's theorem, is whether these groups are equal. In this talk, I will begin with a short introduction to modular curves, their moduli interpretation and their Jacobians. I will then give an overview of variations of the above question, and some interesting results.

We will present some recently discovered properties of the factors of Fermat numbers and other interesting results of geometric nature, which were obtained by applying techniques of extractive proof theory.

In joint work with Tim Browning and Joni Teräväinen we prove asymptotics for averages of arithmetic functions over values of random polynomials (in the style of Manjul Bhargava). We give applications to the Bateman?Horn conjecture for prime values of polynomials, the Chowla conjecture for values of polynomials and we prove the Hasse principle for almost all Châtelet hypersurfaces.

In 2013 P. Habegger proved the Bogomolov property for the field generated over $\mathbb{Q}$ by the torsion points of a rational elliptic curve. We explore the possibility of applying the same strategy of proof to the case of field extensions fixed by the kernel of some modular Galois representations. (Joint work with F. Amoroso)

The task of solving equations in singular moduli is motivated by the André-Oort property for modular varieties. Among others, a theorem of Pila and Tsimerman gives finiteness of $n$-tuples of multiplicatively dependent singular moduli, which is a problem of mixed type. We give a fully effective version of their theorem for $n=3$, thus bounding all triples of multiplicatively dependent j-invariants. The arguments involve a detailed study of the Galois actions of various fields generated by singular moduli, as well as fine properties of the q-expansion of the j-invariant. Joint work with Yuri Bilu (Bordeaux) and Sanoli Gun (Chennai).

The now-classical method of Montgomery and Odlyzko shows that there are infinitely many pairs of zeroes of the zeta-function closer than the average spacing. It has been known for almost 40 years that this method cannot show an infinitude of gaps less than half of the average spacing. But could it show infinitely many gaps of exactly half the average spacing? This is connected to the question of existence of Siegel zeroes. I shall outline the history of this problem, and show that even getting very close to one half is not possible by this method. This is joint work with Dan Goldston (San Jose State University) and Caroline Turnage-Butterbaugh (Carleton College).

A long-standing open problem in additive number theory is the following: how large does a set of integers have to be before it is guaranteed to contain a non-trivial arithmetic progression of length 3? In the first half of this talk we shall survey recent progress on this problem, and the techniques used to solve it and related questions about additive structure in finite abelian groups and the primes. In particular, we shall explain the idea behind the so-called ?arithmetic regularity lemma? pioneered by Green, which is a group-theoretic analogue of Szemerédi?s celebrated regularity lemma for graphs. In the second half of the talk we shall describe recent joint work with Caroline Terry (University of Chicago), which shows that under some natural combinatorial assumptions, the conclusions of the arithmetic regularity lemma can be significantly strengthened.

The exploitation of Motohashi's formula has led Conrey-Iwaniec and Petrow-Young to the uniform Weyl subconvex bound for all Dirichlet $L$-functions, a celebrated recent result in analytic number theory. Based on an adelic version of Motohashi's formula, we establish the uniform Weyl-type subconvex bound for some $L$-functions for ${{\rm PGL}}_2$. For the estimation of the dual weight functions, we use Katz's works on the hypergeometric sums. Our method also shows the limitation of the cubic moment by exhibiting bad bound for certain $L$-functions even if very short families are selected. This is joint work with Ping Xi.

I will briefly survey on the issue of effectivity in number theory, especially for integral points in diophantine analysis. I shall then briefly present some results recently obtained for some curves of genus 2 in work in progress with P. Corvaja.