Home
Titles and abstracts
Schedule
Practical information
Registration
Contributed talks
11th Atelier PARI/GP (April 1617th) 
Wednesday, April 19th
Morning
Session 1
 Session 2 
Reductions of elliptic curves Antonella Perucca (University of Luxembourg)  10:0010:15 
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 welldefined 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).

Explicit formula for the average of Goldbach and prime tuples representations
Marco Cantarini (Università di Parma)  10:0010:15 
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,
$$\frac{1}{\Gamma\left(k+1\right)}\sum_{n\leq N}r_{G}\left(n\right)\left(Nn\right)^{k},\,k \in \mathbb{R},\,k>0.$$

Unlikely Intersections in families of abelian varieties
Fabrizio Barroero (Universität Basel)  10:2510:40 
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.
 Waring's theorem for binary powers Carlo Sanna (Università di Torino)
 10:2510:40 
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 $b > 2$.
This is a joint work with Daniel M. Kane and Jeffrey Shallit: https://arxiv.org/abs/1801.04483

Isomorphism classes of Abelian varieties over finite fields
Stefano Marseglia (Stockholms Universitet)  10:5011:05 
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 easytostate 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.

Non Wieferich primes and Euclidean algorithm
Subramani Muthukrishnan (Harish Chandra Research Institute, Allahabad)  10:5011:05 
We prove results on non Wieferich primes in number fields and also it's connection to Euclidean number fields.

New instances of the MumfordTate conjecture and further applications
Victoria Cantoral Farfan (International Center for Theoretical Physics, Trieste)  11:3511:50 
In this talk we will present new cases of the MumfordTate conjecture for abelian varieties defined over number fields.
Moreover we will discuss some further applications in the direction of the SatoTate conjecture.

Some inequalities associated with the multiple Gamma functions and the Zeta functions Praveen Agarwal (Anand International College of Engineering, Jaipur)
 11:3511:50 
In the middle of $1980$, the multiple Gamma functions $\Gamma_n$ have been revived according to the study of determinants of Laplacians and, recently,
found an applicationin giving closedform evaluations of a class of series involving zeta functions. A class of mathematical constants are naturally
connected with the theory of multipleGamma functions. Here we introduce certain interesting inequalities associated with the multiple
Gamma functions, their related constants, and the zeta functions.

The minimal euclidean algorithm on the Gaussian integers
Hester Graves(Institute for Defense Analyses)  12:0012:15 
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.

Multiple zeta values at the nonpositive integers
Sadaoui Boualem(Khemis Miliana Univezrsity)  12:0012:15 
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*}

On cyclic extensions
Bayarmagnai Gombodorj (National University of Mongolia)  12:2512:40 
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 $\zeta_n + \zeta_{n}^{1}$.

Mean values of multiplicative functions over the function fields
Oleksiy Klurman (KTH, Royal Institute of Technology, Stockholm)  12:2512:40 
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.

Afternoon
Session 1
 Session 2 
Geometric primality tests using curves of genus 1 and 2
Eduardo Ruiz Duarte (Rijksuniversiteit Groningen)  14:4014:55 
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.

Diophantine approximation with prime variables
Alessandro Gambini (Università di Parma)  14:4014:55 
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
$$ \psi (k) =
\begin{cases}
(32k)/6k, & \text{if $1 < k \le 6/5$} \\
1/12, & \text{if $6/5 < k \le 2$}\\
(3k)/6k & \text{if $2 < k < 3$}\\
1/24 & \text{if $k=3$}
\end{cases}
$$

Counting rational points on genus one curves Manh Hung Tran (Chalmers University)  15:0515:20 
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.

On the BaezDuarte criterion about the Riemann hypothesis
Goubi Mouloud (Université Mouloud Maameri de Tizi Ouzou))  15:0515:20 
This work is a tentative to prove that there exist a Beurling function which satisfies the BaezDuarte 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.

On some conjectures on the MordellWeil and the TateShafarevich groups of an Abelian variety
Andrea Surroca  15:3015:45 
Following Manin's approach, we propose conjectural upper bounds for 1) the NéronTate height of the elements of a
system of generators of the MordellWeil group of an Abelian variety, as well as for 2) the cardinality of its TateShafarevich group.
We extend Manin's approach, initially for elliptic curves over the rationals (as revisited by Lang and GolfeldSzpiro), 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 SwinnertonDyer conjecture, HasseWeil 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 GoldfeldSzpiro
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 exact measures of the Sierpinski ddimensional tetrahedron in connection with a Diophantine nonlinear system
Fabio Caldarola (Università della Calabria)  15:3015:45 
The Sierpinski ddimensional 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 kdimensional
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 ddimensional tetrahedrons.

Expansions of quadratic numbers in a $p$adic continued fraction
Laura Capuano(University of Oxford)  16:1516:30 
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 padic 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 padic continued fraction.

Correlation of Multiplicative Functions
Pranendu Darba (Institute of Mathematical Sciences, Chennai)  16:1516:30 
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 in the 3dimensional hyperbolic space
Dimitrios Chatzakos (CEMPI, Lille and Université de Lille)  16:4016:55 
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 3manifold 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.

Correlations of Ramanujan expansions
Giovanni Coppola (Università di Salerno)  16:4016:55 
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 .

