Webinar in Number Theory
YoungJu Choie (POSTECH), BoHae Im (KAIST),
Laurent Berger (ENS Lyon)
Link:
The webinar will resume on Monday October 17 2022
Forthcoming speakers

2022/12/12: Professor Jungyun Lee (Kangwon University, Korea)
Title: The mean value of the class numbers of cubic function fields
Abstract: We compute the mean value of $L(s,\chi)^2$ evaluated at $s=1$ when chi goes through
the primitive cubic Dirichlet characters of $A:=F_q[T]$, where $F_q$ is a finite field with $q$
elements and $q \equiv 1 \ \text{mod 3}$. Furthermore, we find the mean value of the class numbers
for the cubic function fields $K_m=k(\sqrt[3]{m})$, where $k:= F_q(T)$ is the rational function
field and $m$ in $A$ is a cubefree polynomial.
(This is a joint work with Yoonjin Lee and Jinjoo Yoo.)
Past speakers

2022/12/05: Anthony Poëls (Université Claude Bernard Lyon 1, France )
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Title: Rational approximation to real points on quadratic hypersurfaces
Abstract: This is a joint work with Damien Roy. Let $\mathbb{Z}$ be a quadratic hypersurface
of $\mathbb{R}^n$ defined over $\mathbb{Q}$ (such as the unit sphere). We compute the largest exponent
of uniform rational approximation of the points belonging to $\mathbb{Z}$ whose coordinates
together with 1 are linearly independent over $\mathbb{Q}$. We show that it depends only on $n$
and on the Witt index (over $\mathbb{Q}$) of the quadratic form defining $\mathbb{Z}$. This completes
a recent work of Kleinbock and Moshchevitin.

2022/11/21: Professor Jaehyun Cho (UNIST, Korea)
Title: The average residue of the Dedekind zeta function
Abstract: We find the explicit formula for the average residue of the Dedekind
zeta functions over all nonGalois cubic fields. The main tool is a recent
result of Bhargava, Taniguchi, and Thorne's on improving the error term in
counting cubic fields.

2022/11/07: François Ballaÿ (Université de Caen Normandie )
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Title: Positivity in Arakelov geometry and arithmetic Okounkov bodies
Abstract: Arakelov theory is a powerful approach to Diophantine geometry that
develops arithmetic analogues of tools from algebraic geometry to tackle
problems in number theory. It permits to study the arithmeticogeometric
properties of a projective variety over a number field by looking at its
adelic line bundles, which are usual line bundles equipped with a suitable
collection of metrics. Since the seminal work of Zhang on arithmetic ampleness,
several notions of positivity for adelic line bundles have been introduced
and studied in analogy with the algebrogeometric setting (nefness, bigness,
pseudoeffectiveness...). In this talk, I will present these notions and
emphasize their connection with the study of height functions in Diophantine
geometry. I will then describe how these positivity properties can be studied
through convex analysis, thanks to the theory of arithmetic Okounkov bodies
introduced by Boucksom and Chen

2022/10/17: Professor Joachim Koenig (Korea National University of Education)
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Title: On the arithmeticgeometric complexity of the Grunwald problem
Abstract: The Grunwald problem for a group $G$ over a number field $k$ asks
whether, given Galois extensions of $k_p$ of Galois group embedding into
$G$ at finitely many completions $k_p$ of $k$ (possibly away from some
finite set of primes depending only on $G$ and $k$), there always exists a
$G$extension of $k$ approximating all these local extensions. This
question grew naturally out of the GrunwaldWang theorem, which deals with
the case of abelian groups. Following more general concepts of
arithmeticgeometric complexity in inverse Galois theory, we develop a
notion of complexity of Grunwald problems by looking for Galois covers of
varieties which encapsulate solutions to arbitrary Grunwald problems
(for a given group). In particular, we determine the groups $G$ for which
solutions to arbitrary Grunwald problems may be obtained via specialization
of a $G$cover of {\it curves}. Joint with D. Neftin.

2022/06/27: Baptiste Peaucelle (University of ClermontFerrand)
Title: Exceptional images of modular Galois representations
Abstract: Given a modular form $f$ and a prime ideal $\lambda$ in the coefficient
field of $f$, one can attach a residual Galois representation of dimension 2 with
values in the residue field of $\lambda$. A theorem of Ribet states that this
representation has small image for a finite number of prime ideals $\lambda$. In
this talk, I will explain how one can bound explicitly these exceptional ideals,
and how to compute them for some types of small image.

2022/06/13: Prof. Yeongseong Jo (The University of Maine)
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Title: RankinSelberg integrals in positive characteristic and its connection to Langlands functoriality
Abstract: The prominent Langlands functoriality conjecture predicts deep relationships
among representations on different groups. One of the wellunderstood cases is a local
functorial transfer of irreducible generic supercuspidal representations of
${\rm SO}_{2r+1}(F)$ to irreducible supercuspidal ones of ${\rm GL}_{2r}(F)$ over
$p$adic fields $F$. This functorial lift is defined by Lomel\'{\i} over
nonarchimedean local fields $F$ of positive characteristic, but it is rarely studied.
Following the spirit of Cogdell and PiatetskiShapiro, the purpose of this talk is to
take one more step further to investigate the transfer thoroughly. We first consider
the image of the map. Somewhat surprisingly, this is related to poles of local exterior
square $L$functions via integral representations due to Jacquet and Shalika. We then
discuss whether the map is injective. It turns out that the problem is relevant to what
is known as the local converse theorem for ${\rm SO}_{2r+1}(F)$.

2022/05/23: Thomas Lanard (University of Vienna)
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Title: Depth zero representations over $\overline{\mathbb{Z}}[\frac{1}{p}]$
Abstract: In this talk, I will talk about the category of depth zero representations
of a $p$adic group with coefficients in $\overline{\mathbb{Z}}[\frac{1}{p}]$. When
the group $\mathbf{G}$ is quasisplit and tamely ramified, the depth zero category
over $\overline{\mathbb{Z}}[\frac{1}{p}]$ is indecomposable. In general, for a
quasisplit group, we will see that the blocks (indecomposable summands) of this
category are in natural bijection with the connected components of the space of
tamely ramified Langlands parameters. In the last part, I will explain some potential
applications to the FarguesScholze and GenestierLafforgue semisimple local
Langlands correspondences. This is joint work with JeanFrançois Dat.

2022/05/02: Prof. Seungki Kim (University of Cincinnati)
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Title: Adelic Rogers integral formula
Abstract: The Rogers integral formula, a natural generalization of the wellknown Siegel
integral formula, first appeared in the 1950's as an essential tool in the geometry of
numbers. Very recently, there has been a surprising resurgence of interest in the
formula, thanks in much part to its usefulness in homogeneous dynamics, and a number
of variants and extensions have been proposed. I will introduce the audience to the
relevant literature, in particular the recently proved formula over an adele of a
number field.

2022/04/04: Valentin Hernandez (Université ParisSud, Orsay)
Title: The Infinite Fern in higher dimensions
Abstract: In general, deformations spaces of residual Galois representation
are quite mysterious objects. It is natural to ask if there is at least
enough modular points in their generic fiber X. A related question is the
density of the padic modular forms, which form a fractallike object
called the Infinite Fern. In dimension 2, in most cases Gouvea and Mazur
proved that this infinite fern is Zariski dense in X. In higher dimension
we look at \emph{polarized} Galois representation, and the analogous
question becomes much more complicated. Chenevier explained a strategy by
looking for \emph{good} (called generic) points in Eigenvarieties, studied
the analogous local (padic) question and solved the case of dimension 3.
Recently BreuilHellmannSchraen studied the local Infinite Fern at well
behaved crystalline points, and HellmannMargerinSchraen, under strong
TaylorWiles hypothesis, managed to prove the density of the (global)
Infinite Fern (in a union of connected components) in all dimensions using
the \emph{patched} Eigenvariety. In this talk I would like to explain how
to only use the local geometric input to deduce the analogous density result
without using the TaylorWiles hypothesis, but using another kind of
\emph{good} points as in Chenevier's strategy. This is a joint work with
Benjamin Schraen.

2022/03/21: Junho Peter Whang (Seoul National University)
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Title: Decidable problems on integral SL2characters
Abstract: Classical topics in the arithmetic study of quadratic forms include
their reduction theory and representation problem. In this talk, we discuss
their nonlinear analogues for SL2characters of surface groups. First, we
prove that the set of integral SL2characters of a surface group with
prescribed invariants can be effectively determined and finitely generated,
under mapping class group action and related dynamics. Second, we prove that
the set of values of an integral SL2character of a finitely generated group
is a decidable subset of the integers.

2022/03/07: Silvain RideauKikuchi ( Institut de Mathématiques de JussieuParis Rive Gauche)
Title: Hminimality (with R. Cluckers, I. Halupczok)
Abstract: The development and numerous applications of strong minimality and later
ominimality has given serious credit to the general model theoretic idea that
imposing strong restrictions on the complexity of arity one sets in a structure
can lead to a rich tame geometry in all dimensions. Ominimality (in an ordered
field), for example, requires that subsets of the affine line are finite unions
of points and intervals.
In this talk, I will present a new minimality notion (hminimality), geared
towards henselian valued fields of characteristic zero, generalising previously
considered notions of minimality for valued fields (C,V,P ...) that does not,
contrary to previously defined notions, restrict the possible residue fields
and value groups. By analogy with ominimality, this notion requires that definable
sets of of the affine line are controlled by a finite number of points. Contrary
to ominimality though, one has to take special care of how this finite set is
defined, leading us to a whole family of notions of hminimality. I will then
describe consequences of hminimality, among which the jacobian property that
plays a central role in the development of motivic integration, but also various
higher degree and arity analogs.

2022/02/21: Dr JunYong Park (MPIM, Bonn)
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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).

2022/02/07: Gautier Ponsinet (Post doctoral at the Università degli Studi di Genova)
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Title: Universal norms of padic Galois representations
Abstract: In 1996, Coates and Greenberg observed that perfectoid fields appear naturally in Iwasawa theory.
In particular, they have computed the module of universal norms associated with an abelian variety in a perfectoid field extension.
A precise description of this module is essential in Iwasawa theory, notably to study Selmer groups over infinite algebraic field extensions.
In this talk, I will explain how to use properties of the FarguesFontaine curve to generalise their results to padic representations.

2021/12/20: Professor Kwangho Choiy ( Southern Illinois University)
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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.