In 1869, Victor Schemmel considered the multiplicative function which counts the number of sets of $m$ consecutive integers which are relatively prime to $n$ in the interval $[1,n)$. Denoting this function by $\varphi_m(n)$, we can see that $\varphi_m(n)$ generalizes the Euler's totient function $\varphi=\varphi_1$. This function was used by Lehmer in the context of magic squares and more recently, it also appeared in formulas enumerating cliques in direct product graphs. In this talk we consider $\varphi_m(n)$ for any integer $m$ and introduce certain asymptotic formulae for sums of Schemmel's function $\varphi_m(n)$ over $n$ and $m$. Regardless of the original definition we also call $\varphi_m$ with a non-positive $m$ Schemmel's function. This is a joint work with Jörn Steuding from the University of Würzburg.

In various joint works with Sary Drappeau, Adam Harper, Kaisa Matomaki, Maksym Radiziwill and Fernando Shao we better understand what is known about such estimates (as well as the corresponding estimates for prime numbers), and prove best possible results in some aspects.

For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof of the fact that if $G$ is a finitely generated multiplicative subgroup of a number field $K$ having rank $r$, then the ratio between $n^r$ and the Kummer degree $[K(\sqrt[n]{G}):K(\zeta_n)]$ is bounded independently of $n$. Moreover we present a concrete algorithm to compute those degrees if $K$ is either the field of rational numbers or a quadratic field. This is joint work with Pietro Sgobba and Sebastiano Tronto.

A singular modulus is the $j$-invariant of an elliptic curve with complex multiplication. Given a singular modulus~$x$ we denote by $\Delta_x$ the discriminant of the associated imaginary quadratic order. In this paper We show that the field $\mathbb{Q}(x,y)$, generated by two singular moduli $x$ and $y$, is generated by their sum ${x+y}$, unless~$x$ and$y$ are conjugate over $\mathbb{Q}$, in which case ${x+y}$ generates a subfield of degree at most $2$. We obtain a similar result for the product of two singular moduli.

Let $G$ be a finite abelian group. For any integer $a\geq 1$, we define the constant $s_{\leq a}(G)$ as the least positive integer $t$ such that any sequence $S$ over $G$ of length at least $t$ has a zero-sum subsequence of length $\leq a$ in it. In this talk, we prove this constant for many classes of abelian $p$-groups. In particular, it proves a conjecture of Schmid and Zhuang. This is a joint work with Dr. R. Thangadurai.

In this talk, we discuss the equation $F_{n+1}^{x} - $F_{n-1}^{x} = F_{m}, where $F_{n}$ and $F_{m}$ are respectively the $n$-th and $m$-th Fibonacci numbers. We find all the solutions in positive integer values $(m, n, x)$.

Given a real number $\tau$, we study the approximation of $\tau$ by signed harmonic sums $\sigma_N(\tau) := \sum_{n \leq N}{s_n(\tau)}/n$, where the sequence of signs $(s_N(\tau))_{N \geq 1}$ is defined ``greedily'' by setting $s_{N+1}(\tau) := +1$ if $\sigma_N(\tau) \leq \tau$, and $s_{N+1}(\tau) := -1$ otherwise. We give an accurate description of the behavior of the sequence of signs $(s_N(\tau))_{N \geq 1}$, highlighting a surprising connection with the Thue--Morse sequence. This is a joint work with Sandro Bettin and Giuseppe Molteni.

In the function field setting, automorphic forms of Drinfeld type can be viewed as analogue of classical weight 2 modular forms. The Hecke algebra associated to these automorphic forms is related to various topics in function field arithmetic, such as elliptic curves over function fields. I will present Sturm-type bounds for the generators of the Hecke algebra (joint work with Fu-Tsun Wei).

Given a totally real number field $K$, Galois over the rationals and satisfying certain conditions, we present a family of number fields $\{K(p)\}_p$ depending on a fixed $K$ and a varying rational prime $p$ such that $p$ is always ramified in $K(p)$ and the density of rational primes with residue degree $1$ in $K(p)/\mathbb{Q}$ can be given by a computable formula, bounded away from zero and bounded above by $(1/2)^{((n+1)/2)}$ where $n=[K:\mathbf{Q}]$. The formula depends on $n=[K:\mathbf{Q}]$ and $m_K$, an invariant of the number field $K$ which can be computed for a fixed $K$.

In this talk, We will discuss about an upper bound for the least prime square and least prime $p$ of the form $p \equiv f \pmod d$ where $(f,d)=1$ in an irrational non homogeneous Beatty sequence $\lbrace \lfloor \alpha n+\beta \rfloor : n=1,2,3 \dots \rbrace$, where $\alpha, \beta \in \mathbb{R}$ and $\alpha >1.$

Given an algebraic variety over the rationals, I will discuss the joint distribution of the number of prime factors of each of the coordinates as we vary an integral point. Under suitable assumptions and with the appropriate normalisation, we establish a multivariate central limit theorem, thus generalising the celebrated Erd\H{o}s--Kac theorem. This is based on joint work with Daniel Loughran and Efthymios Sofos.

After the work of Zhang--Maynard--Tao on bounded gaps between primes, we know that the quantity $$D_{\lambda,m}(x):= \frac{\#\lbrace n\leq x: \#[n,n+\lambda\log n]\cap \mathbb{P}\geq m\rbrace}{x},$$ satisfies $D_{\lambda,m}(x)\gg_{m} 1,$ when $0<\lambda<\varepsilon(m)$, for a certain $\varepsilon(m)>0$. Here $m\in\mathbb{N}$ and $\mathbb{P}$ is the set of prime numbers. However, this does not preclude the possibility that there are choices of $\lambda$ and $m$ for which the intervals $[n,n+\lambda\log n]$ contain exactly $m$ primes, for at most finitely many $n$. This was proven to be not the case by Freiberg, who showed that indeed for any positive real number $\lambda$ and any non-negative integer $m$, we have $$d_{\lambda,m}(x):= \frac{\#\lbrace n\leq x: \#[n,n+\lambda\log n]\cap \mathbb{P}= m\rbrace}{x}\geq x^{-\delta(x)},$$ where $\delta(x)=(\log\log\log\log x)^{2}/\log\log\log x$. I will explain how to refine such estimate proving that $d_{\lambda,m}(x)\gg_{m}1,$ whenever $0<\lambda<\varepsilon(m),$ for a certain $\varepsilon(m)>0$, thus showing the existence of a positive proportion of short intervals containing a given number of primes.

Supersingular isogeny graphs have been used in the Charles--Goren--Lauter cryptographic hash function and the supersingular isogeny Diffie--Hellman (SIDH) protocol of De\,Feo and Jao. A recently proposed alternative to SIDH is the commutative supersingular isogeny Diffie--Hellman (CSIDH) protocol, in which the isogeny graph is first restricted to $\mathbb{F}_p$-rational curves $E$ and $\mathbb{F}_p$-rational isogenies then oriented by the quadratic subring $\mathbb{Z}[\pi] \subset \mathrm{End}(E)$ generated by the Frobenius endomorphism $\pi: E \rightarrow E$. We introduce a general notion of orienting supersingular elliptic curves and their isogenies, and use this as the basis to construct a general oriented supersingular isogeny Diffie-Hellman (OSIDH) protocol. We describe the structure of this oriented isogeny graph and its navigation using isogeny chains and modular curve equations.

This is joint work with Leonardo Colò.

?Since Thue's breakthrough in 1909, various versions of Siegel's Lemma play an important role in diophantine approximation and transcendence theory. We discuss some of these together with some limitation results, finishing with a brief mention of recent work of Roger Baker and the speaker about the original version first explicitly formulated by Siegel in 1929.

Assume that $B$ is a large real number and let $c_1,c_2,c_3$ be three randomly chosen integers in the box $[-B,B]^3$. Consider the probability that the ``random" curve $$ c_1 X^2+c_2 Y^2+c_3 Z^2=0$$ has a non-zero solution $(X,Y,Z)$ in the integers. Serre showed in the 90's that this probability is $\ll (\log B)^{-3/2}$ and Hooley later showed that it is $\gg (\log B)^{-3/2}$. In joint work with Nick Rome we prove an asymptotic $\sim c (\log B)^{-3/2}$, where $c$ is a positive absolute constant.

Understanding solutions of Diophantine equations over rationals or more generally over any number field is one of the main problems of number theory. One of the most spectacular recent achievement in this area is the proof of Fermat's last theorem by Wiles. By the help of the modular techniques used in this proof and its generalizations it is possible to solve other Diophantine equations too. Understanding quadratic points on the classical modular curve or rational points on its twists play a central role in this approach. In this talk, I will summarize the modular method and mention some recent results about points on modular curves. This is joint work with Samir Siksek.

We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that the abelian surface does not contain curves of arithmetic genus less or equal than a fixed integer $\ell$. This approach naturally leads us to consider Weil restrictions of elliptic curves and non principally polarizable abelian surfaces.

In this article we prove level raising for cuspidal eigenforms modulo prime powers (for odd primes) of weight $k\geq 2$ and arbitrary character, extending the result in weight two established by the work of Tsaknias and Wiese and generalizing (partially) Diamond-Ribet's celebrated level raising theorems.

The one level density measures the statistical distribution of low-lying zeros across a family of L-functions. The Katz-Sarnak Density Conjecture states that such a quantity may be modeled by an analogous statistic regarding the eigenvalue distribution of a family of random matrices. I present a conjecture for the one level density across a symplectic family of Hecke L-functions, and a result verifying this conjecture, down to lower order terms, for test functions supported on (-1,1). The results offer evidence for a spectral interpretation of the zeros of such L-functions, in line with a suggestion first made my Hilbert and Polya.

Let $d>1$ be a squarefree integer and $(X_n,Y_n)$ be the $n$th solution of the Pell equation $X^2-dY^2=\pm 1$. Given your favourite set of positive integers $U$, one can ask what can we say about those $d$ such that $X_n\in U$ for some $n$? Formulated in this way, the question has many solutions $d$ since one can always pick $u\in U$ and write $u^2\pm 1=dv^2$ with integers $d$ and $v$ such that $d$ is squarefree obtaining in this way that $(u,v)$ is a solution of the Pell equation corresponding to $d$. What about if we ask that $X_n\in U$ for at least two different $n$'s? Then the answer is very different. For many interesting sets $U$ (like squares, Fibonacci numbers, rep-digits, factorials, etc.) the answer is that there are only finitely many such $d$ and they can all be effectively computed. We also look at the same problem for the sequence $(Y_n)_{n\ge 1}$. Here the problem is different. For example, if $U$ contains $1$ and infinitely many even positive integers, then there are infinitely many $d$ such that both $Y_1$ and $Y_2$ are in $U$. We show that if $U$ is the set of values of a binary recurrence, then there are only finitely many $d$ such that $Y_n\in U$ has three solutions $n$, and they are effectively computable. When $U=\{2^m-1:m\ge 1\}$, we show that there is no $d$ such that $Y_n\in U$ has three solutions $n$. The proofs use linear forms in logarithms and computations. These results have been obtained in joint work with various colleagues such as J. J. Bravo, C. A. G\'omez, S. Laishram, A. Montejano, L. Szalay and A. Togb\'e and recent Ph.D. students M. Ddamulira, B. Faye and M

The goal of this talk is to present an algorithmic procedure which allows to give a complete classification of number fields of degree 8 with minimum discriminant. The developed method can be used also to recover minimum discriminants for fields of degree 9. If time permits, there will be also some remarks about the similar problem of classifying number fields with low regulators.

Six years ago, M Ram Murty and I proved a growth result on non-Wieferich primes in arithmetic progressions. I will discuss the importance of non-Wieferich primes, present our result and techniques, and then talk about more recent, related work

In this talk, we shall prove the existence of a non-principal Euclidean ideal class in two (possibly infinite) families of bi-quadratic fields. This is a joint work with Dr. M. Subramani.

The celebrated Lang-Vojta Conjecture predicts degeneracy of S-integral points on varieties of log general type defined over number fields. It admits a geometric analogue over function fields, where stronger results have been obtained applying a method developed by Corvaja and Zannier. In this talk, we present a recent result for non-isotrivial surfaces over function fields dominating a two-dimensional torus. This extends Corvaja and Zannier?s result in the isotrivial case and it is based on a refinement of gcd estimates for polynomials evaluated at S-units. This is a joint work with A. Turchet.

Let $k$ be a number field and let $A$ be a commutative algebraic group defined over $k$. Let p be a prime. We show some sufficient conditions to have a local-global principle for divisibility by $p$ for the points in A. Those conditions also imply the triviality of the Tate-Shafarevich group Ш$(k,A[p])$, where $A[p]$ is the $p$-torsion subgroup of $A$. If $A$ is a principally polarized abelian variety, the vanishing of Ш$(k,A[p])$ implies a local-global principle for divisibility by $p$ for the elements of $H^r(k,A)$, for all positive integers $r$, giving a partial answer to a question posed by Cassels.

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to generalize the classical continued fractions. We propose an introductive fundamental study about MCFs in the field of the $p$-adic numbers. Firstly, we discuss their convergence. Then, we focus on a specific algorithm that, starting from a $m$-tuple of numbers in $Q_p$, produces the partial quotients of the corresponding MCF. We show that this algorithm is derived from a generalized $p$-adic Euclidean algorithm. We provide some results concerning the finiteness and periodicity of the above expansion, and the quality of the approximation.

For an elliptic curve over $\mathbb{Q} $ and an extension $L $of $\mathbb{Q} $ with uniformly bounded local degrees, $L(E_tor)$ has the Bogomolov property. In the proof we follow the proof of Habegger?s 2013 result on $\mathbb{Q}(E_tor)$ having the Bogomolov property.

Let $S_{k,m}(t)$ be the space of Drinfeld cusp forms of weight $k$, type $m$ and of level $\Gamma_0(t)$ and let $U_t$ be the Atkin operator. We study the degeneracy maps $S_{k,m}(1)\to S_{k,m}(t)$ and trace maps (the other way around) and use them to define oldforms and newforms. Looking at distribution of slopes, i.e. the $t$-adic valuation of $U_t$ eigenvalues, we shall describe various results and conjectures on the structure of $S_{k,m}(t)s.

In this talk we will present some results on abelian varieties whose representation associated to Galois action on their torsion group has as large image as possible. In particular, in 2016, in collaboration with S. Arias de Reyna, C. Armana, V. Karemaker, L. Thomas and N. Vila, we obtained a result in the case of dimension 3 abelian varieties. If time allows it, we will give a sketch of the proof.

In the the proof of the irrationality of zeta(2) Apery introduced numbers $b_n = \sum_{k=0}^n {n \choose k}^2{n+k \choose k}$ that are integral solutions of certain recursive relation. Zagier found six more sequences with similar property (sporadic sequences). Stienstra and Beukers showed that there is mod p congruence between b_{(p-1)/2} and p-th Fourier coefficeint of certain modular form. Osburn and Straub proved similar congruences for all but one of six Zagier's sporadic sequences and conjectured congruence for sixth sequence. We prove that remaining congruence by studying Atkin and Swinnerton-Dyer congruences between Fourier coefficients of certain cusp form for non-congurence subgroup.

We shall present two recent results linked with the Parity Principle. First, with Aled Walker, we proved that given any $X\ge2$ each progression $a\mod q$, with $a$ prime to $q$, contains a product of \emph{exactly} three primes of size at most $X$ provided that $q\le X^{3/16}$; Bound improved to $q\le X^{1/3}/900$ with Priyamvad Srivastav recently. Our sole ingredients being a sieve bound and a weak lower bound for $L(1,\chi)$, this apparently contradicts the Parity Principle but no: the Parity Principle deals with \emph{quantitative} estimates while our proof does not produce enough products! Secondly, if we are to keep asymptotic estimates and sieve devices, primes are accompanied with products of two primes, but how can one be more specific? Recently N. Debouzy refined an estimate of E. Bombieri and obtained, under the Elliott-Hamberstam conjecture, an asymptotic for the number of primes $p$ such that $p+2$ is either a prime or a product of two primes, one of them being of size at most $p^\epsilon$, for any arbitrary $\epsilon>0$.

This is a report on the work of my master student Francesco Campagna. For an elliptic curve E over a number field K, the set of primes pp of good reduction for which the finite group (E mod pp) is cyclic can be described by Galois theoretic means reminiscent of the situation arising for multiplicative groups (Artin's conjecture for primitive roots). I will discuss the analogy, the earlier results by Serre and Ram Murty and what we added to this.

Kummer (1851) and, many years later, Ihara (2005) posed conjectures on invariants related to the cyclotomic field $\mathbb Q(\zeta_q)$ with $q$ a prime. Kummer's conjecture concerns the asymptotic behaviour of the first factor of the class number of $\mathbb Q(\zeta_q)$ and Ihara's the positivity of the Euler-Kronecker constant of $\mathbb Q(\zeta_q)$ (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $\zeta_{\mathbb Q(\zeta_q)}(s)$ at $s=1$). If certain standard conjectures in analytic number theory hold true, then one can show that both conjectures are true for a set of primes of natural density 1, but false in general. Responsible for this are irregularities in the distribution of the primes.\\ \indent With my talk I hope to convince the listener that the apparently dissimilar mathematical objects studied by Kummer and Ihara actually display a very similar behaviour (partly joint work with Kevin Ford and Florian Luca).

We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in very short intervals of the form $I(f) :=\{ f(x) + a : a \in F_{p} \}$ for $f(x) \in F_{p}[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the Möbius $\mu$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a t Morse polynomial , we show that error terms are roughly of size $O(\sqrt{p})$ (with typical main terms of order $p$). We also give examples of $f$ for which there is no cancellation at all, and intervals where the heuristic primes are independent fails very badly. Time permitting we will discuss the curios fact that (square root) cancellation in Möbius sums is equivalent to (square root) cancellation in Chowla type sums.

In this talk, we will show that there is at most one value of the positive integer $X$ participating in the Pell equation $X^2 - dY^2 = k$, where $k \in {±1,±4}$, which is a Padovan numbers, with a few exceptions that we completely characterize.

Let $X$ be a smooth projective curve of genus $g \geq 1$ over a finite field $\F_{q}$, such that the function field $\F_{q}(X)$ is a geometric Galois extention of $\F_q(x)$ with $N=\#Gal(\F_{q}(X)/\F_{q}(x)).$ Let $M_{L}(2,1)$ be the moduli space of rank $2$ stable vector bundles over $\bar{X} \,=\, X \times_{\mathbb F_q} \bar{\F_{q}}$ with fixed determinant $L$, a degree $1$ line bundle. Let $N_q (M_L(2,1))$ be the cardinality of the set of $\F_{q}$-rational points of $M_{L}(2,1)$. We give an estimate of $N_q (M_L(2,1))$ in terms of $N, q \,\text{and}\, g$.

In a joint work with Sary Drappeau, we obtain results on the value distribution of  quantum  modular  forms. As particular examples we consider the distribution of  modular  symbols, the Estermann function at the central point, and the Kashaev invariant of hyperbolic knots.

It is known that, if K is a number field and L/K is an   infinite Galois extension, then the local degrees of L are uniformly   bounded at all rational primes if and only if the group Gal(L/K) has   finite exponent. Also motivated by some problems concerning the   Bogomolov property (a height gap property for sets of algebraic numbers), one can ask whether the simple non-uniform boundedness of   the local degrees of L is still equivalent to some (weaker) group   theoretical property of Gal(L/K). We will show that this is not the   case in general, by exhibiting several groups that admit two different   realisations over a given number field, one with bounded local degrees   at a given set of primes and one with infinite local degrees at the   same primes.

I will talk about a joint work with Jeffrey Vaaler, in which we extent a recent result of Dobrowolski and Smyth to establish a sharp lower bound for the Mahler measures of polynomials in any number of variables. These bounds depend on the number of non-vanishing coefficients of the polynomials, and are independent of their degrees.

We prove that, for every pair of positive integers $r$ and $n$, there exists an integer $B=B(r)$ such that the Diophantine equation $$ \prod _{m=1}^{n}\left(\sum_{i=1}^{k_m} a_{m,i} x_{m,i} - \sum _{j=1}^{l_m}b_{m,j}y_{m,j}\right)= B \quad {\textrm{with}} \quad \sum_{i=1}^{k_m} a_{m,i} = \sum _{j=1}^{l_m}b_{m,j} \qquad \forall m = 1,\ldots , n$$ is $r$-regular. This is a joint work with Ms. Bidisha Roy.

A classical result of Schur says that the equation $x+y=z$ is regular. We discuss some early generalizations of this result, report some recent results and state some open questions related to these problems.

The Stieltjes constants of the Dirichlet L-functions are the coefficients of the Laurent expansion of the Dirichlet L-functions at s=1. In my early works on these constants, I gave an explicit upper bound of the Laurent-Stieltjes constants of Dirichlet L-functions. In this talk, we give some interesting applications of the Laurent-Stieltjes constants. We also extend our results to the Selberg class.

It is classically known that a singular modulus (a $j$-invariant of a CM elliptic curve) is an algebraic integer. Habegger (2015) proved that at most finitely many singular moduli are units, answering a question of Masser (2011). However, his argument, being non-effective, did not imply any bound for the size of these singular units. I will report on a recent work with Philipp Habegger and Lars Kühne, where we prove that singular units do not exist at all. First, we bound the discriminant of any singular unit by $10^15$. Next, we rule out the remaining singular units using computer-assisted arguments.