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

2022/02/07: Gautier Ponsinet (Post doctoral at the Università degli Studi di Genova)
Title: ??
Abstract: T??

2022/02/21: Dr JunYong Park (MPIM, Bonn)
Title: Arithmetic Moduli of Elliptic Surfaces
Abstract: By considering the arithmetic geometry of rational orbicurves on modular
curve $\overline{\mathcal{M}}_{1,1}$ where $\overline{\mathcal{M}}_{1,1}$ is the
DeligneMumford stack of stable elliptic curves, we formulate the moduli stack of
minimal elliptic fibrations over $\mathbb{P}^{1}$, also known as minimal elliptic
surfaces with section over any base field $K$ with $\mathrm{char}(K) \neq 2,3$.
Inspired by the classical work of [Tate] which allows us to determine the
KodairaN\'eron type of fibers over global fields, we establish Tate's correspondence
between the moduli stacks $\mathrm{Rat}_{n}^{\gamma}(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})$
of quasimaps with vanishing constraints
$\gamma$ and $\mathrm{Hom}^{\Gamma}_n(\mathcal{C}, \overline{\mathcal{M}}_{1,1})$
of twisted maps with cyclotomic twistings $\Gamma$. Afterward, we acquire the exact
arithmetic invariants of the moduli for each KodairaN\'eron types which naturally
renders new sharp enumerations with a main leading term of order
$\mathcal{B}^{\frac{5}{6}}$ and secondary & tertiary order terms
$\mathcal{B}^{\frac{1}{2}} ~\&~ \mathcal{B}^{\frac{1}{3}}$ on $\mathcal{Z}_{\mathbb{F}_q(t)}(\mathcal{B})$
for counting elliptic curves over $\mathbb{P}_{\mathbb{F}_q}^{1}$ with additive
reductions ordered by bounded height of discriminant $\Delta$. The emergence of
nonconstant lower order terms are in stark contrast with counting the semistable
(i.e., strictly multiplicative reductions) elliptic curves. In the end, we
formulate an analogous heuristic on $\mathcal{Z}_{\mathbb{Q}}(\mathcal{B})$ for
counting elliptic curves over $\mathbb{Q}$ through the global fields analogy.
This is a joint work with Dori Bejleri (Harvard) and Matthew Satriano (Waterloo).
Past speakers

2021/12/20: Professor Kwangho Choiy ( Southern Illinois University)
Title: Invariants in restriction of admissible representations of $p$adic groups
Abstract: The local Langlands correspondence, LLC, of a $p$adic group over complex
vector spaces has been proved for several cases over decades. One of interesting
approaches to them is the restriction method which was initiated for $SL(2)$ and its
inner form. It proposes in line with the functoriality principle that the LLC of
one group can be achieved from the LLC of the other group sharing the same derived
group. In this context, we shall explain how the method is extended to some other
cases of LLC's, the multiplicity formula in restriction, and the transfer of the
reducibility of parabolic induction.

2021/11/22: Lucile Devin (Université du Littoral)
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Title: Chebyshev's bias and sums of two squares
Abstract: Studying the secondary terms of the Prime Number Theorem in Arithmetic Progressions,
Chebyshev claimed that there are more prime numbers congruent to 3 modulo 4 than to 1 modulo 4.
We will explain and qualify this claim following the framework of Rubinstein and Sarnak.
Then we will see how this framework can be adapted to other questions on the distribution of
prime numbers. This will be illustrated by a new Chebyshevlike claim : "for more than half
of the prime numbers that can be written as a sum of two squares, the odd square is the square
sof a positive integer congruent to 1 mod 4.

2021/11/08: Wansu Kim (KAIST)
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Title: Equivariant BSD conjecture over global function fields
Abstract: Under a certain finiteness assumption of TateShafarevich groups, Kato and Trihan
showed the BSD conjecture for abelian varieties over global function fields of positive
characteristic. We explain how to generalise this to semistable abelian varieties ``twisted
by Artin character'' over global function field (under some additional technical assumptions),
and discuss further speculations for generalisations if time permits. This is a joint work
with David Burns and Mahesh Kakde.

2021/10/18: Richard Griffon (University ClermontAuvergne)
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Title: Isogenies of elliptic curves over function fields
Abstract: I will report on a recent work, joint with Fabien Pazuki, in which we study elliptic
curves over function fields and the isogenies between them. More specifically, we prove
analogues in the function field setting of two famous theorems about isogenous elliptic
curves over number fields. The first of these describes the variation of the Weil height of
the jinvariant of elliptic curves within an isogeny class. Our second main result is an
''isogeny estimate'' in the spirit of theorems by MasserWüstholz and by GaudronRémond.
After stating our results and giving quick sketches of their proof, I will, time permitting,
mention a few Diophantine applications.

2021/10/04: Dr. Seoyoung Kim (Queen's University)
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Title: On the generalized Diophantine mtuples
Abstract: For nonzero integers $n$ and $k\geq2$, a generalized Diophantine $m$tuple
with property $D_k(n)$ is a set of $m$ positive integers $\{a_1,a_2,\ldots, a_m\}$
such that $a_ia_j + n$ is a $k$th power for any distinct i and j. Define by $M_k(n)$
the supremum of the size of the set which has property $D_k(n)$. In this paper, we
study upper bounds on $M_k(n)$, as we vary $n$ over positive integers. In particular,
we show that for $k\geq 3$, $M_k(n)$ is $O(\log n)$ and further assuming the Paley
graph conjecture, $M_k(n)$ is $O((\log n)^{\epsilon})$. The problem for $k=2$ was
studied by a long list of authors that goes back to Diophantus who studied the
quadruple $\{1,33,68,105\}$ with property $D(256)$. This is a joint work with A.
Dixit and M. R. Murty.

2021/06/21: Vlerë Mehmeti (Université ParisSaclay, France)
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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.

2021/06/07: HaeSang Sun (UNIST, Korea)
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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.