Number Theory Day - Wednesday 5 February - VENUE: ICTP - Luigi Stasi Seminar Room

[ICTP Math Section]

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_*_NUMBER THEORY DAY_
*

(ICTP - February 05, 2025)

Venue: Stasi Lecture Room, ICTP Leonardo Da Vinci Building, ICTP - Trieste


Organizers: Emanuel Carneiro (ICTP), Pietro Corvaja (University of 
Udine) and Umberto Zannier (SNS - Pisa)


Sponsors:

•        University of Udine (Department of Mathematics, Computer 
Science and Physics)
•        Scuola Normale Superiore di Pisa
•        Centro interuniversitario per la teoria dei numeri e sue 
applicazioni informatiche (Primi)
•        ICTP

Program:

11:00 - 11:50: David Masser (University of Basel, Switzerland)

Title: Equidistribution and heights

Abstract: Since Yuri Bilu (1997) we know that the conjugates of an 
algebraic number α of small height tend to be equidistributed near the 
unit circle. With Roger Baker (2023) we refined some aspects of weak 
convergence, for example with the test function log |1 − z|. We gave 
several applications, for example to the number of conjugates in the 
upper half plane: this number differs from half the degree d by at most 
8000 h^{1/3} d for the absolute logarithmic height h (now assumed 
non-zero) of α. We will describe some of this and also more recent 
applications to heights themselves, such as those of rational functions 
evaluated at roots of unity.


13:30 - 14:20: Mithun Das (ICTP, Italy)

Title: Effective equidistribution of Galois orbits for mildly regular 
test functions

Abstract: Abstract: We provide a detailed study on effective versions of 
the celebrated Bilu's equidistribution theorem for Galois orbits of 
sequences of points of small height in the N-dimensional algebraic 
torus, identifying the qualitative dependence of the convergence on the 
regularity of the test functions considered. We develop a general 
Fourier analysis framework that extends previous results obtained by 
Petsche (2005) and D'Andrea, Narváez-Clauss, and Sombra (2017).  This is 
also related to previous works by Pritsker (2011), Baker and Masser 
(2023), and Amoroso and Pleissis (2024). This is a joint work with 
Emanuel Carneiro.


15:00 - 15:50: Carlo Pagano (Concordia University, Canada)


Title: Hilbert 10 via additive combinatorics

Abstract: In 1970 Matiyasevich, building on earlier work of 
Davis--Putnam--Robinson, proved that every enumerable subset of Z is 
Diophantine, thus showing that Hilbert's 10th problem is undecidable for 
Z. The problem of extending this result to the ring of integers of 
number fields (and more generally to finitely generated infinite rings) 
has attracted significant attention and, thanks to the efforts of many 
mathematicians, the task has been reduced to the problem of 
constructing, for certain quadratic extensions of number fields L/K, an 
elliptic curve E/K with rk(E(L))=rk(E(K))>0. This was done under BSD by 
Mazur and Rubin.  In this talk I will explain joint work with Peter 
Koymans, where we use Green--Tao to construct the desired elliptic 
curves unconditionally, thus settling Hilbert 10 for every finitely 
generated infinite ring.


16:30 - 17:20: Gregory Debruyne (Ghent University, Belgium)


Title: Recent advances surrounding the Ingham-Karamata Tauberian theorem
Abstract: The Ingham-Karamata theorem is a cornerstone in Tauberian 
theory: it retrieves asymptotic information of a function from a 
regularity hypothesis on the function and from information about its 
Laplace transform. I will discuss some of my recent work surrounding 
this theorem and explain some applications in number theory.


All are welcome to attend.