Davide Lombardo: Rational points on modular curves & generalised Fermat equations
Abstract: In 1972, Serre proved that the Galois representations arising from the p-power torsion points of non-CM elliptic curves over \(\mathbb{Q}\) have open image in \(\operatorname{GL}_2(\mathbb{Z}_p)\), and Mazur later initiated a vast programme to determine all such possible images explicitly -- for fixed p, it is known that there are only finitely many possibilities. Much progress has been made for small primes p, but a complete classification remains open beyond \(p \in \{2,3,13,17\}\).
In this talk, I will describe recent progress on this problem for p = 7, based on a surprising correspondence between rational points on modular curves and primitive integer solutions to certain generalised Fermat equations of signature (2,3,7), such as \( a^2 + 28b^3 = 27c^7. \) We show that these Diophantine equations can be reduced to determining the rational points on a finite collection of genus-3 curves. As a consequence, we are able for example to determine the rational points on a modular curve of genus 69 and establish that the 7-adic Galois images of elliptic curves over \(\mathbb{Q}\) are determined by their reduction modulo \(7^2\).
Harry Schmidt: On the \(L^2\)-norm of exponential polynomials
Abstract: In join work with Manon Parent we have investigated how small the \(L^2\)-norm of an exponential polynomial can get if we fix the norm of the its coefficient vector. We prove lower and upper bounds of this minimum in terms of the degree of the polynomial and we apply our methods to a variant of a problem of Hilbert. We prove that on any given interval the infimum of the \(L^2\)-norm of exponential polynomials with integer coefficients on this interval is 0.
I seminari si svolgeranno in presenza presso il Dipartimento di Matematica e Fisica, Largo S.L. Murialdo 1 - aula M1.
Per ulteriori informazioni, si può contattare gli organizzatori all'email amos.turchet@uniroma3.it.
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