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Webinar in Number Theory

• Every Tuesday once a month: a 50 mn talk
• Summer French time: 10:00-11:00,   Korean time: 17:00-18:00
      Winter French time: 9:00-10:00,   Korean time: 17:00-18:00
• Zoom link: ask the organizers

• 2024/12/10: Christian Maire (Besançon)
  Title: Governing field and embedding problems
  
  Abstract: In this talk, we will show how a governing field constructed on the Selmer group of a number field naturally arises in various contexts: genus theory, inverse Galois problem for the Hilbert $p$-class field tower, Schmidt's result on tame $G_S$ groups of cohomological dimension 2, tamely ramified inverse Galois problem for congruence subgroups, strong Massey property, and more.
  
  

• 2024/11/12: (9:00 France, 17:00 Korea) Heejong Lee (Purdue University)
  Title: Recent advances on the weight part of Serre's conjecture
  
  Abstract: The Langlands reciprocity associates a Galois representation with a certain modular form/automorphic representation. One can ask how this association reflects structures on both sides. For example, the weights of modular forms/automorphic representations are matched by the Hodge-Tate weights of Galois representations. The weight part of Serre's conjecture is a mod p analogue of this structural correspondence.
  
  In this talk, we introduce the weight part of Serre's conjecture and its role in the Langlands program. Then, we discuss a novel idea proving the conjecture for the group $GSp_4$ under technical assumptions. This is based on a joint work with Daniel Le and Bao Viet Le Hung.
  
  
• 2024/10/08: (10:00 France, 17:00 Korea) Jean-Stefan Koskivirta (University of Saitama)
  Title: Trivial-type automorphic forms
  
  Abstract: The motivation of the talk is to understand the relation between automorphic forms in characteristic p and the stack of G-zips of Pink-Wedhorn-Ziegler. We introduce the notion of trivial-type automorphic forms for this purpose. As an application, we prove the Cone Conjecture for weights of automorphic forms in characteristic p for Siegel modular varieties.
  
  
• 2024/09/10: (10:00 France, 17:00 Korea) Jaesung Kwon (Institute for Data Innovation in Science, Seoul National University)
  Title: Generation of cyclotomic Hecke fields by L-values of Hecke characters over totally real fields
  
  Abstract: Jun, Lee, and Sun posed the question of whether the cyclotomic Hecke field can be generated by a single critical L-value of a cyclotomic Hecke character over a totally real field. They provided an answer to this question in the case where the tame Hecke character is trivial. In this presentation, we explain an extension of their work to address the case of non-trivial Hecke characters over solvable totally real number fields. Our approach builds upon the primary estimation obtained by Jun-Lee-Sun, supplemented with new inputs, including global class field theory, duality principles, the analytic behavior of partial Hecke L-functions, and the non-vanishing of twisted sum of Gauss sums and Hyper Kloosterman sums.
  
  This work is joint with Hae-Sang Sun.
  
  
• 2023/05/22: Dr. Seok Hyeong Lee (Seoul National University)
  Title: Explicit construction for Bhargava's "Higher Composition Laws"
  
  Abstract: Bhargava's "Higher Composition Laws" give explicit one-to-one correspondence between rings of low ranks and certain integral forms. Generalizing Wood's extension of cubic and quartic ring parametrizations, we give a general algorithm of constructing rings out of well-posed integral forms which can be applied to all cases of Higher Composition Laws.
  
  
• 2023/05/15: Laurent Berger (ENS de Lyon)
  Title: Super-Hölder functions and vectors
  
  Abstract: I will define super-Hölder functions, an analogue in characteristic p of locally analytic functions. I will give examples of super-Hölder functions in certain situations of arithmetic interest. Joint work with Sandra Rozensztajn.
  
  
• 2023/04/24: Dr Jaeho Haan (Yeonse University)
  PDF Title: The local converse theorem for quasi-split SO(2n)
  
  Abstract: Local converse theorem (LCT) has many interesting applications. For example, global rigidity theorem and injectivity of global factorial lift of global generic cuspidal representations of classical groups immediately follows from it. Starting from the Jiang and Soudry's pioneering work for SO(2n+1), the LCT has now proved for almost all classical groups but quasi-split SO(2n). In this talk, we discuss the proof of LCT for quasi-split SO(2n) by studying the relation of gamma factors between SO(2n) and Sp(2n). If time permits, we also discuss the positive characteristic cases as well as the characteristic zero cases. This is a joint work with Yeansu Kim and Sanghoon Kwon.
  
  
• 2023/04/03: Guillaume Ricotta (University of Bordeaux)
  Title: Kloosterman paths of prime powers moduli
  
  Abstract: Let $n$ be a large enough fixed integer and p be a prime number. We prove that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums of moduli $p^n$ converge in law, as p tends to infinity, to an explicit random Fourier series in the Banach space of complex-valued continuous function on $[0,1]$. This is a joint work with Emmanuel Royer and Igor Shparlinski
  
  
• 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 cube-free polynomial. (This is a joint work with Yoonjin Lee and Jinjoo Yoo.)
  
  
• 2022/12/05: Anthony Poëls (Université Claude Bernard Lyon 1, France )
  PDF 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 non-Galois 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 )
  PDF 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 arithmetico-geometric 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 algebro-geometric setting (nefness, bigness, pseudo-effectiveness...). 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)
  PDF Title: On the arithmetic-geometric 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 Grunwald-Wang theorem, which deals with the case of abelian groups. Following more general concepts of arithmetic-geometric 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 Clermont-Ferrand)
  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)
  PDF Title: Rankin-Selberg 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 well-understood 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 non-archimedean local fields $F$ of positive characteristic, but it is rarely studied. Following the spirit of Cogdell and Piatetski-Shapiro, 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)
  PDF 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 quasi-split and tamely ramified, the depth zero category over $\overline{\mathbb{Z}}[\frac{1}{p}]$ is indecomposable. In general, for a quasi-split 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 Fargues-Scholze and Genestier-Lafforgue semisimple local Langlands correspondences. This is joint work with Jean-François Dat.
  
  
• 2022/05/02: Prof. Seungki Kim (University of Cincinnati)
  PDF Title: Adelic Rogers integral formula
  
  Abstract: The Rogers integral formula, a natural generalization of the well-known 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 Paris-Sud, 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 p-adic modular forms, which form a fractal-like 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 (p-adic) question and solved the case of dimension 3. Recently Breuil-Hellmann-Schraen studied the local Infinite Fern at well behaved crystalline points, and Hellmann-Margerin-Schraen, under strong Taylor-Wiles 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 Taylor-Wiles 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)
  PDF Title: Decidable problems on integral SL2-characters
  
  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 SL2-characters of surface groups. First, we prove that the set of integral SL2-characters 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 SL2-character of a finitely generated group is a decidable subset of the integers.
  
  
• 2022/03/07: Silvain Rideau-Kikuchi ( Institut de Mathématiques de Jussieu-Paris Rive Gauche)
  Title: H-minimality (with R. Cluckers, I. Halupczok)
  
  Abstract: The development and numerous applications of strong minimality and later o-minimality 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. O-minimality (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 (h-minimality), 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 o-minimality, this notion requires that definable sets of of the affine line are controlled by a finite number of points. Contrary to o-minimality though, one has to take special care of how this finite set is defined, leading us to a whole family of notions of h-minimality. I will then describe consequences of h-minimality, 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 Jun-Yong Park (MPIM, Bonn)
  PDF Title: Arithmetic Moduli of Elliptic Surfaces
  
  Abstract: By considering the arithmetic geometry of rational orbi-curves on modular curve $\overline{\mathcal{M}}_{1,1}$ where $\overline{\mathcal{M}}_{1,1}$ is the Deligne--Mumford 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 Kodaira--N\'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 Kodaira--N\'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 non-constant 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)
  PDF Title: Universal norms of p-adic 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 Fargues-Fontaine curve to generalise their results to p-adic representations.
  
  
• 2021/12/20: Professor Kwangho Choiy ( Southern Illinois University)
  PDF 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)
  PDF 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 Chebyshev-like 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)
  PDF Title: Equivariant BSD conjecture over global function fields
  
  Abstract: Under a certain finiteness assumption of Tate-Shafarevich 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 semi-stable 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 Clermont-Auvergne)
  PDF 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 j-invariant of elliptic curves within an isogeny class. Our second main result is an ''isogeny estimate'' in the spirit of theorems by Masser-Wüstholz and by Gaudron-Rémond. 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)
  PDF Title: On the generalized Diophantine m-tuples
  
  Abstract: For non-zero 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 Paris-Saclay, France)
  PDF Title: Non-Archimedean analytic curves and the local-global principle
  
  Abstract: In 2009, a new technique, called algebraic patching, was introduced in the study of local-global 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 non-Archimedean analytic curves. As an application, we will see various local-global 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 non-Archimedean analytic curves and will conclude by a presentation of a first step towards such results in higher dimensions.
  
  
• 2021/06/07: Hae-Sang Sun (UNIST, Korea)
  PDF Title: Cyclotomic Hecke L-values 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 L-function 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 L-function 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)
  PDF 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 Campagna-Stevenhagen and Lombardo-Tronto. 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: Chan-Ho Kim (KIAS, Korea)
  PDF 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 Mazur-Tate 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.