Let $E$ be an elliptic curve defined over a number field $K$. Fix some prime number $\ell$. If $\alpha \in E(K)$ is a point of infinite order, we consider the set of primes $\mathfrak p$ of $K$ such that the reduction $(\alpha \bmod \mathfrak p)$ is well-defined and has order coprime to $\ell$. This set admits a natural density. By refining the method of R. Jones and J. Rouse (2010), we can express the density as an $\ell$-adic integral. We also prove that the density is a rational number whose denominator (up to powers of $\ell$) is uniformly bounded in a very strong sense. Finally, we describe a strategy for computing the density which covers every possible case. This is joint work with Davide Lombardo (University of Pisa).

We present the explicit formula, similar to the classical explicit formula for $\psi\left(x\right)$, for the average of the Goldbach representations $$r_{G}\left(n\right)=\underset{{\scriptstyle m_{1}+m_{2}=n}}{\sum_{m_{1},m_{2}\leq n}}\Lambda\left(m_{1}\right)\Lambda\left(m_{2}\right),\,n\in\mathbb{N}$$ and for the prime tuples representation $$r_{PT}\left(N,h\right)=\sum_{n=0}^{N}\Lambda\left(n\right)\Lambda\left(n+h\right),\,h,N\in\mathbb{N}$$ where $\Lambda\left(n\right)$ is the Von Mangoldt function. In the case of $r_{G}\left(n\right)$ we show also a truncated version of the formula. Furthermore we analyze the possibility to extend the technique to the Cesàro average of $r_{G}\left(n\right)$, that is, $$\frac{1}{\Gamma\left(k+1\right)}\sum_{n\leq N}r_{G}\left(n\right)\left(N-n\right)^{k},\,k \in \mathbb{R},\,k>0.$$

Two varieties whose dimensions do not sum up to at least the dimension of the ambient space should not intersect. This is the guiding philosophy that led several authors to propose conjectures about subvarieties of commutative algebraic groups and of Shimura varieties. After a brief historic introduction we will talk about results for curves in families of abelian varieties, mostly obtained in collaboration with Laura Capuano.

A natural number is a binary $k^{\text th}$ power if its binary representation consists of $k$ consecutive identical blocks. We prove an analogue of Waring's theorem for sums of binary $k^{\text th}$ powers. More precisely, we show that for each integer $k \ge 2$, there exists a positive integer $W(k)$ such that every sufficiently large multiple of $E_k :=\operatorname{gcd}(2^k - 1, k)$ is the sum of at most $W(k)$ binary $k^{\text th}$ powers. (The hypothesis of being a multiple of $E_k$ cannot be omitted, since we show that the gcd of the binary $k^{\text th}$ powers is $E_k$). Also, we explain how our results can be extended to arbitrary integer bases $b > 2$.

Deligne proved that the category of ordinary abelian varieties over a finite field is equivalent to the category of free finitely generated abelian groups endowed with an endomorphism satisfying certain easy-to-state axioms. Centeleghe and Stix extended this equivalence to all isogeny classes of abelian varieties over ${\mathbb F}_p$ without real Weil numbers. Using these descriptions, under some extra assumption on the isogeny class, we obtain that in order to compute the isomorphism classes of abelian varieties we need to calculate the isomorphism classes of (non necessarily invertible) fractional ideals of some orders in certain étale algebras over ${\mathbb Q}$. We present a concrete algorithm to perform these tasks and, for the ordinary case, to compute the polarizations and also the automorphisms of the polarized abelian variety.

We prove results on non Wieferich primes in number fields and also it's connection to Euclidean number fields.

In this talk we will present new cases of the Mumford-Tate conjecture for abelian varieties defined over number fields. Moreover we will discuss some further applications in the direction of the Sato-Tate conjecture.

In 1949, Motzkin introduced a new tool to study Euclidean domains. As a side effect, it described the minimal Euclidean algorithm for agiven domain. He showed that the minimal Euclidean algorithm onthe integers is $\phi_Z (x) = \log_2 |x|$. Lenstra gave an elegant proof of the minimal Euclidean algorithm on Z[i]. This talk will give anew, elementary proof and an alternate description of said function that allows for fast computation.

In this talk, we relate the special value at a non positive integer $\underline{\textbf{s}}=(s_{1}, ..., s_{n})= -\underline{\textbf{N}}= (-N_{1},..., -N_{n})$ obtained by meromorphic continuation of the multiple zeta function \begin{equation*} Z(\underline{s})= \sum_{\underline{m} \in \mathbb{N}^{*n}}{\prod_{i=1}^{n}{\frac{1}{(m_{1}+\dots +m_{i})^{s_{i}}}}}\end{equation*}

Let $n\geq 3$ be an integer and $\zeta_n$ be a primitive $n$-th root of unity. We will present a description of cyclic extensions of degree n in terms of $k$-rational points of the tori when the base field $k$ contains $\zeta_n + \zeta_{n}^{-1}$.

Understanding mean values and correlations of multiplicative functions over number fields plays key role in analytic number theory. Motivated by the recent work of Granville, Harper and Soundararajan we discuss mean values of multiplicative functions over the function fields \mathbb{F}_q[x]. In particular, we prove stronger function field analogs of several classical results due to Wirsing, Halasz, Hall, Tenenbaum explaining some surprising features that are not present in the number field setting. Our main result describes spectrum of multiplicative functions over the function fields.

This is based on a joint work with K. Soundararajan and C. Pohoata.

We revisit and generalize some geometric techniques behind deterministic primality testing for some integer sequences using curves of genus 1 over finite rings. Subsequently we develop a similar primality test using the Jacobian of a genus 2 curve.

This talk concerns a Diophantine approximation problem with 3 prime variables: we will prove that the inequality $$ |\lambda_1p_1+\lambda_2p_2+\lambda_3p_3^k-\omega|\le (\max(p_1,p_2,p_3^k))^{\psi(k)+\varepsilon} $$ where $$ \psi (k) = \begin{cases} (3-2k)/6k, & \text{if $1 < k \le 6/5$} \\ 1/12, & \text{if $6/5 < k \le 2$}\\ (3-k)/6k & \text{if $2 < k < 3$}\\ 1/24 & \text{if $k=3$} \end{cases} $$

The density of integral solutions of Diophantine equations can also be viewed geometrically as the density of integral points on algebraic varieties which is one of the classical problems in Diophantine geometry. We are interested in the case of projective varieties defined by homogeneous polynomials and this gives rise to the study of rational points on such varieties. In this talk we focus on smooth genus one curves which are closely related to elliptic curves. We give uniform upper bounds for the number of rational points of bounded height on smooth genus one curves in two forms: plane cubic curves and complete intersections of two quadric surfaces. The main tools to study this problem are descent and determinant methods.

This work is a tentative to prove that there exist a Beurling function which satisfies the Baez-Duarte Criterion about the Riemann hypothesis. Furthermore we connect the distance of the indicatrice function to space generated by any Beurling function in means of the well known digamma function.

Following Manin's approach, we propose conjectural upper bounds for 1) the Néron-Tate height of the elements of a system of generators of the Mordell-Weil group of an Abelian variety, as well as for 2) the cardinality of its Tate-Shafarevich group. We extend Manin's approach, initially for elliptic curves over the rationals (as re-visited by Lang and Golfeld-Szpiro), to an Abelian variety of arbitrary dimension, over an arbitrary number field. The dependence of the bounds is explicit in all the parameters, and the bounds given here are not conjectured but are implied by strong but nowadays classical conjectures: Birch and Swinnerton-Dyer conjecture, Hasse-Weil conjecture. On once hand, point 1) extends in higher dimension and for arbitrary number fields a conjecture of Lang. On the other hand, assuming also Szpiro's conjectures, with point 2) we extends a theorem of Goldfeld-Szpiro to Abelian varieties of arbitrary dimension $g$ over any number field $K$, and improved it in the case $g=1$ and $K$ the field of rational numbers.

The Sierpinski d-dimensional tetrahedron $\Delta^d$ is the generalization of the most known Sierpinski gasket which appears in many fields of mathematics. Considering the sequences of polytopes $\Delta^d_{n}$ that generate $\Delta^d$, we find closed formulas for the sum $v^{d,k}_n$ of the measures of the k-dimensional elements of $\Delta^d_n$, deducing the behavior of the sequences $v^{d,k}_n$. It becomes quite clear that traditional analysis does not have the adequate language and notations to go further, in an easy and manageable way, in the study of the previous sequences and their limit values; contrariwise, by adopting the new computational system for infinities and innitesimals developed by Y.D. Sergeyev, we achieve precise evaluations for every $k$-dimensional measure related to each $\Delta^d$, obtaining a set $W$ of values expressed in the new system, which leads us to a Diophantine problem in terms of classical number theory. To solve it, we work with traditional tools from algebra and mathematical analysis. In particular, we define two kinds of equivalence relations on $W$ and we get a detailed description of the partition of various of its subsets together with the exact composition of the corresponding classes of equivalence. Finally, we also show as the unique Sierpinski tetrahedron for each dimension d, is replaced, if we adopt Sergeyev's framework, by a whole family of infinitely many Sierpinski d-dimensional tetrahedrons.

It goes back to Lagrange that a real quadratic irrational always has a periodic continued fraction. Starting from decades ago, several authors generalised proposed different definitions of a $p$-adic continued fraction, and the definition depends on the chosen system of residues mod $p$. It turns out that the theory of $p$-adic continued fractions has many differences with respect to the real case; in particular, no analogue of Lagrange's theorem holds, and the problem of deciding whether the continued fraction is periodic or not seemed to be not known. In recent work with F. Veneziano and U. Zannier we investigated the expansion of quadratic irrationals, for the p-adic continued fractions introduced by Ruban, giving an effective criterion to establish the possible periodicity of the expansion. This criterion, somewhat surprisingly, depends on the real value of the p-adic continued fraction.

In this talk, after recalling some earlier results I will present some recent progress on asymptotic formula for the following correlation: $$M_{x}(g_{1}, g_{2}, g_3)=\frac{1}{x}\sum_{n\le x}g_{1}(F_1(n))g_{2}(F_2(n))g_{3} (F_3(n)),$$ where $F_1(x), F_2(x)$ and $F_3(x)$ are polynomials with integer coefficients and $g_1,g_2,g_3$ are multiplicative functions with modulus less than or equal to $1.$

The Prime Geodesic Theorem studies the asymptotic behaviour of lengths of primitive closed geodesics on hyperbolic manifolds. For $2$-dimensional manifolds this problem was first studied by Huber and Selberg. It turns out that the lengths of these geodesics obey an asymptotic distribution analogous to the Prime Number Theorem, and the error term has been extensively studied by use of the Selberg and the Kuznetsov trace formulas. In this talk, we discuss the Prime Geodesic Theorem on $3$-dimensional hyperbolic manifolds. For the Picard manifold, we improve on the classical pointwise bound of Sarnak, using the Kuznetsov formula combined with a recent large sieve inequality of Watt. Further, for a 3-manifold of finite area, we study the second moment of the error term using the Selberg trace formula. This is a joint work in progress with Giacomo Cherubini and Niko Laaksonen.

After the series of papers, coauthored, Finite Ramanujan expansions and shifted convolution sums of arithmetical functions and two papers on the arxiv, about the same studies, we keep up to date our investigations about the Reef.