Consider an ordinary isogeny class of elliptic curves over a finite,
prime field. Inspired by a random matrix heuristic (which is so strong
it’s false), Gekeler defines a local factor for each rational prime.
Using the analytic class number formula, he shows that the associated
infinite product computes the size of the isogeny class.
I’ll explain a transparent proof of this formula; it turns out that this
product actually computes an adelic orbital integral which visibly counts
the desired cardinality. Moreover, the new perspective allows a natural
generalization to higher-dimensional abelian varieties.
----- This is joint work with Julia Gordon and S. Ali Altug.
Let E/ℚ be a semistable elliptic curve of analytic rank one, and let p>3 be
a prime for which the Galois group of the extension ℚ(E[p])/ℚ is
isomorphic to GL_{2}(F_{p}). In this note, following a slight
modification of the methods of Jetchev–Skinner–Wan, we use Iwasawa theory to
establish the p-part of the Birch–Swinnerton-Dyer formula for E. In particular,
we extend the main result of loc.cit. to primes of multiplicative reduction.
In 2000, Paul Vojta solved the n-squares problem under Bombieri's conjecture,
by explicitly finding all the curves of geometric genus 0 or 1 on certain surfaces
of general type related to this problem. In this talk I will sketch a refined
and generalized version of the geometric method implicit in Vojta's work.
I will also discuss new arithmetic applications conditional to Bombieri's conjecture
in the case of number fields, and unconditional for function fields.
Let A be an abelian surface over a number field k that is isogenous over
the algebraic closure of k to the square of an elliptic curve E. If E does
not have CM, by results of Ribet and Elkies concerning fields of
definition of elliptic k-curves, E is isogenous to a curve defined over a
polyquadratic extension of k. We show that one can adapt Ribet's methods
to study the field of definition of E up to isogeny also in the CM case,
as long as k contains the field of CM. As an application of this analysis,
we provide a number field over which abelian surfaces can be found
realizing each of the 52 possible Sato--Tate groups of abelian surfaces.
----- This is joint work with Francesc Fité.
Let K be a number field such that K/ℚ is galois and
χ the caharacter of a linear representation of Gal(K/ℚ).
The Artin conductor asociated to χ (denoted by f_{χ})
is a positive integer involved in the functional equation of the Artin
L-function attached to χ.
In 2011, we improved the lower bounds to the conductor obtained by Odlyzko
in 1976. Now, we will see that the growth of the Artin conductor is at most
exponential in the degree of the character.
We introduce and study a new way to catagorize supersingular
abelian varieties defined over a finite field by classifying them as fully
maximal, mixed or fully minimal. The type of A depends on the normalized
Weil numbers of A and its twists over its minimal field of definition. We
analyze these types for supersingular abelian varieties and curves under
conditions on the automorphism group. In particular, we present a
complete analysis of these properties for supersingular elliptic curves
and supersingular abelian surfaces in arbitrary characteristic. For
supersingular curves of genus 3 in characteristic 2, we use a
parametrization of a moduli space of such curves by Viana and Rodriguez to
determine the L-polynomial and the type of each.
----- This is joint work with Valentijn Karemaker.
Numerical semigroups are easy to define: they are just additive semigroups,
contained in the non-negative integers, such that they only miss a finite
amount of such integers. However, the computational problems that arise in
such structures are far from easy, specially from a computational viewpoint.
A rather mysterious set that seems to have the key to solve many of these
problems is the so-called Apéry set, an object whose origins lie in singularity
theory.
We will dedicate this talk to celebrate the Mathematics of Javier Cilleruelo.
He had a special intuition for additive problems with a combinatorial flavour
and his taste for Mathematics had a big impact on many of us.
He wrote more than 80 research articles with more than 44 coauthors from all
around the globe and he made.e profound contributions in a wide variety of problems.
We will present a selection of his results to give a glimpse of the impact of his work.