Webinar in Number Theory
YoungJu Choie (POSTECH), BoHae Im (KAIST),
Laurent Berger (ENS de Lyon)
Link:
Forthcoming speakers

2021/06/21: Vlerë Mehmeti (Université ParisSaclay, France) (confirmed)
Title: NonArchimedean analytic curves and the localglobal principle
Abstract: In 2009, a new technique, called algebraic patching, was introduced in the study of localglobal
principles. Under different forms, patching had in the
past been used for the study of the inverse Galois problem. In this talk, I will present an extension
of this technique to nonArchimedean analytic curves.
As an application, we will see various localglobal principles for function fields of curves, ranging
from geometric to more classical forms. These results
generalize those of the previous literature and are applicable to quadratic forms. We will start by a
brief introduction of the framework of nonArchimedean
analytic curves and will conclude by a presentation of a first step towards such results in higher
dimensions.
Past speakers

2021/06/07: HaeSang Sun (UNIST, Korea)
PDF
Title: Cyclotomic Hecke Lvalues of a totally real field
Abstract: It is known that any Fourier coefficient of a newform of weight 2 can be expressed as a
polynomial with rational coefficients, of a single algebraic critical value of the corresponding
Lfunction twisted by a Dirichlet character of $p$power conductor for a rational prime $p$. In the talk,
I will discuss a version of this result in terms of Hecke Lfunction over a totally real field, twisted
by Hecke characters of $p$power conductors. The discussion involves new technical challenges that arise
from the presence of the unit group, which are (1) counting lattice points in a cone that $p$adically
close to units and (2) estimating an exponential sum over the unit group. This is joint work with Byungheup
Jun and Jungyun Lee.

2021/05/17: Riccardo Pengo (ENS Lyon, France)
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Title: Entanglement in the family of division fields of a CM elliptic curve
Abstract: Division fields associated to an algebraic group defined over a number field, which are
the extensions generated by its torsion points, have been the subject of a great amount of research,
at least since the times of Kronecker and Weber. For elliptic curves without complex multiplication,
Serre's open image theorem shows that the division fields associated to torsion points whose order
is a prime power are "as big as possible" and pairwise linearly disjoint, if one removes a finite
set of primes. Explicit versions of this result have recently been featured in the work of
CampagnaStevenhagen and LombardoTronto. In this talk, based on joint work with Francesco Campagna
(arXiv:2006.00883), I will present an analogue of these results for elliptic curves with complex
multiplication. Moreover, I will present a necessary condition to have entanglement in the family
of division fields, which is always satisfied for elliptic curves defined over the rationals.
In this last case, I will describe in detail the entanglement in the family of division fields.

2021/05/03: ChanHo Kim (KIAS, Korea)
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Title: On the Fitting ideals of Selmer groups of modular forms
Abstract: In 1980's, Mazur and Tate studied ``Iwasawa theory for elliptic curves over finite abelian extensions"
and formulated various related conjectures. One of their conjectures says that the analytically defined
MazurTate element lies in the Fitting ideal of the dual Selmer group of an elliptic curve. We discuss some
cases of the conjecture for modular forms of higher weight.