Number Theory Seminars 2019

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December 27 (Fri), 2019, 10:30-12:00, Room 8101
December 30 (Mon), 2019, 10:30-12:00 and 14:00-15:30, Room 8101
December 31 (Tue), 2019, 10:30-12:00, Room 8101

Title: An introduction to affine Deligne-Lusztig varieties

Speaker: Yihang Zhu (Columbia University)

Abstract: Affine Deligne-Lusztig varieties (ADLVs) are algebraic varieties which are important for many applications in number theory. Most significantly they enter the study of Shimura varieties. In this lecture series, we will focus on ADLVs themselves, and survey their basic properties and some recent results. We will see that the study of ADLVs often has a strong group-theoretic flavor. We will try to assume minimal prerequisite from the audience. Apart from common algebraic number theory and algebraic geometry, some familiarity with reductive groups over fields should be enough.

December 12 (Thu), 2019, 4:00pm-6:00pm, Room 1423

Title: Integral models for the unipotent completion of a finitely generated group

Speaker: Dohyeong Kim (Seoul National University)

Abstract: Using a fragment of Lazard's theory, we construct integral unipotent completions of a finitely generated group. We will discuss its application to arithmetic, especially in view of the Chabauty-Kim method, by considering fundamental groups of curves.

October 24 (Thu), 2019, 4:00pm-6:00pm, Room 1423

Title: A Homotopy Lie Formula for the $p$-adic Dwork Frobenius operator

Speaker: Junyeong Park (POSTECH)

Abstract: Given a smooth projective complete intersection X over a finite field, there is the notion of zeta function defined as a generating function. Dwork's theory recovers the zeta function as a characteristic polynomial of his ''Frobenius operator''. In this talk, we will give an interpretation of this via deformation theory of homotopy Lie algebra. More precisely, we will first explain how to construct a Differential Gerstenhaber-Batalin-Vilkovisky (DGBV) algebra associated to a smooth projective complete intersection defined over the complex number field. This construction also works for the objects defined over a finite field to give a DGBV algebra as well. However, this is not enough to get the zeta function so we have to lift the DGBV algebra to a p-adic field, where the Dwork Frobenius operator is defined. Then we arrive at a natural setting to apply the deformation theory of homotopy Lie algebras. As a consequence, we get a formula for the p-adic Dwork operator in terms of a morphism of homotopy Lie algebras. This is a joint work with Dohyeong Kim and Jeehoon Park.

October 17 (Thu), 2019, 4:00pm-5:00pm, Room 1424

Title: Uniqueness of automorphic representations of $\text{GL}_n$ with certain local representations

Speaker: Dohoon Choi (Korea University)

Abstract: Let $\mathbb{A}$ be the adele ring of a number field $K$. The strong multiplicity one theorem says that an cuspidal representation of $\text{GL}(n, \mathbb{A})$ is uniquely determined by its local representation at $v$ for all but finitely many places of $K$. Based on this theorem, it is a natural question to find a subset $S$ of the set of finite places of $K$ such that an cuspidal representation of $\text{GL}(n, \mathbb{A})$ is uniquely determined by its local representation at $v$ for $v \in S$. In this talk, I will discuss about this question.

October 10 (Thu), 2019, 4:00pm-5:00pm, Room 1423

Title: Fontaine--Laffaille modules and their mod $p$ local-global compatibility

Speaker: Chol Park (UNIST)

Abstract: For a given mod $p$ Galois representation $\overline{\rho}$, one can define a mod $p$ automoprhic representation $\overline{\Pi}$ by a certain space of mod $p$ algebraic automoprhic forms on a unitary group. We wish that $\overline{\Pi}$ corresponds to $\overline{\rho}$ for a mod $p$ Langlands correspondence, but the structure of $\overline{\Pi}$ is quite mysterious as a representation. In this talk, we will discuss our approach to the conjecture, mod $p$ local-global compatibility which states that $\overline{\Pi}$ determines $\overline{\rho}$, in the case that $\overline{\rho}$ is Fontaine--Laffaille. This is a joint work with Daniel Le, Bao Le Hung, Stefano Morra, and Zicheng Qian.

September 24 (Tue), 2019, 4:00pm-6:00pm, Room 1424
September 26 (Thu), 2019, 4:00pm-6:00pm, Room 1424

Title: Universal norms and Control theorems in Iwasawa theory

Speaker: Gautier Ponsinet (Max Planck Institut, Bonn)

Abstract: The goal of this talks is to explain a work in progress on universal norms of $p$-adic Galois representations. The problem of universal norms appears in Iwasawa theory, notably to establish ''control theorems'' for Selmer groups. It was first studied by Mazur in his foundational work on Iwasawa theory for abelian varieties. The problems for abelian varieties has been solved by Coates and Greenberg. Their result has many applications in Iwasawa theory, Greenberg used it to generalize Mazur's control theorems for Selmer groups of abelian varieties in a wide class of extensions. We will first do an brief introduction to Iwasawa theory. Then we will focus on the role of Selmer groups and control theorems in Iwasawa theory. This will naturally lead to the problem of universal norms. In the second part, we will see how to use the Fargues-Fontaine curve to study the problem.
I - Introduction: Bloch-Kato's conjecture and Iwasawa theory
II - Control theorems for Selmer groups
III - Universal norms of $p$-adic Galois representations
IV - The Fargues-Fontaine curve

September 20 (Fri), 2019, 3:00pm-5:30pm, Room 8309

Title: On Gross-Prasad conjecture for $(\mathrm{SO}(2n+1), \mathrm{SO}(2))$ and Böcherer's conjecture

Speaker: Kazuki Morimoto (Kobe University)

Abstract: Böcherer conjectured an explicit formula between Bessel periods of Siegel cuspforms of degree 2 and central values of certain $L$-functions. In particular, this conjecture predicts a relationship between non-vanishing of Bessel periods and non-vanishing of central values of certain $L$-functions. Nowadays this relation on non-vanishing and this explicit formula are regarded as a special case of Gross-Prasad conjecture and refined Gross-Prasad conjecture, respectively. In this talk, we consider Gross-Prasad conjecture in the case of $(\mathrm{SO}(2n+1), \mathrm{SO}(2))$ and its refinement. In particular, we prove Böcherer's conjecture.

September 05 (Thu), 2019, 4:00pm-5:00pm, Room 1423

Title: Rank of elliptic curves

Speaker: Jaehyun Cho (UNIST)

Abstract: Under GRH for elliptic $L$-functions, we show that there are not so many elliptic curves with rank $\geq 27$. This is a joint work with Keunyoung Jeong.

August 06 (Tue), 2019, 2:00pm-3:30pm, Room 8101
August 08 (Thu), 2019, 2:00pm-3:30pm, Room 8101
August 20 (Tue), 2019, 2:00pm-3:30pm, Room 8101
August 22 (Thu), 2019, 2:00pm-3:30pm, Room 8101

Title: Iwasawa theory and the exact Birch--Swinnerton-Dyer formula I, II, III, IV

Speaker: John Coates (Cambridge)

Abstract: The conjectural exact Birch-Swinnerton-Dyer formula for the order of the conjecturally finite Tate-Shafarevich group of any elliptic curve defined over a number field remains one of the great mysteries of number theory, and up until now has been proven in very few cases. In my lectures, I plan to discuss recent ongoing work with Yongxiong Li, Yukako Kezuka, and Ye Tian which we believe will prove both the finiteness of the Tate-Shafarevich group and the exact formula for its order for a wide class of elliptic curves with complex multiplication, whose complex L-series does not vanish at $s = 1$.
Let $q$ be any prime congruent to 7 mod 16, $K = \mathbb{Q}(\sqrt{q)})$, and let $H$ the Hilbert class field of $K$. Let $A=H$ be Gross' $\mathbb{Q}$-curve with complex multiplication by the maximal order of $K$, and whose associated Hecke character has conductor $(\sqrt{q})$. Let $k$ be any non-negative integer, and let $R = r_1, \cdots, r_k$, where the $r_i$ are distinct rational primes such that $r_i$ is congruent to 1 mod 4 and $r_i$ is inert in $K$ for $i = 1, \cdots, k$. Let $A^{(R)}/H$ be the quadratic twist of $A$ by $H(\sqrt{R})/H$, and write $L(A^{(R)}/H, s)$ for its complex $L$-series, which is entire by Deuring's theorem. Recently, Yongxiong Li and I found a proof by arguments from Iwasawa theory that always $L(A^{(R)}/H, 1)$ is not 0. When $k = 0$ this is an old result of D. Rohrlich, which he proved using complex methods, but there seems little hope of proving the more general statement by such complex methods.
In my lectures, I shall briefly discuss the proof of this non-vanishing theorem, and then go on to explain how ideas from Iwasawa theory enable one to show that the Tate-Shafarevich group of $A^{(R)}/H$ is indeed finite, and that its order is given by the exact Birch-Swinnerton-Dyer formula.

August 08 (Thu), 2019, 1:00pm-2:00pm, Room 8101

Title: Kato's Euler systems and congruences of modular forms

Speaker: Chan-Ho Kim (KIAS)

Abstract: We discuss how Kato's zeta elements behave under congruences. This is joint work in progress with Jaehoon Lee and Gautier Ponsinet.

August 01 (Thu), 2019, 1:30pm-2:30pm, Room 1424

Title: Combinatorial proofs between some classes of partition functions

Speaker: Jaebum Sohn (Yonsei University)

Abstract: In this talk, we introduce some basic concepts about partition functions which include the meaning of hook length and $t$-core. After that, I will give combinatorial proofs between some classes of partition functions.

August 01 (Thu), 2019, 2:30pm-3:30pm, Room 1424

Title: Quotients of numerical semigroups and counting numerical semigroups using polytopes

Speaker: Hayan Nam (Iowa State University)

Abstract: A numerical semigroup is an additive monoid that has a finite complement in the set of non-negative integers. For a numerical semigroup $S$, the genus of $S$ is the number of elements in $\mathbb{N} \setminus S$, and the multiplicity is the smallest nonzero element in $S$. In the first half of the talk, we discuss a formula of the genus of a quotient of a numerical semigroup. In 2008, Bras-Amorós conjectured that the number of numerical semigroups with genus $g$ is increasing as $g$ increases. Later, Kaplan posed a conjecture that implies Bras-Amorós conjecture. In the second half, we prove Kaplan's conjecture when the multiplicity is 4 or 6 by counting the number of integer points in a polytope. Moreover, we find a formula for the number of numerical semigroups with multiplicity 4 and genus $g$.

August 01 (Thu), 2019, 4:00pm-5:00pm, Room 1424

Title: Construction of Galois representations for $\text{GSO}_{2n}$

Speaker: Arno Kret (University of Amsterdam)

Abstract: In this talk we report on some work in progress together with Sug Woo Shin, on the construction of Galois representations for certain forms of the group $\text{GSO}_{2n}$ over a totally real field $F$. More precisely, we consider a quasi-split form $G$ of $\text{GSO}_{2n}$ that is split if $n$ is even and non-split if $n$ is odd. Then we construct, for cohomological, cuspidal automorphic representations $p_i$ of $G$, a corresponding Galois representation, in case $p_i$ is at a finite place isomorphic (up to twist) to the Steinberg representation.

July 31 (Wed), 2019, 4:00pm-6:00pm, Room 1424

Title: Nearly overconvergent automorphic forms (expository talk)

Speaker: Gyujin Oh (Princeton University)

Abstract: We review the notion of nearly holomorphic modular forms, their original use in arithmetic and how they can be defined geometrically. We then discuss the notion of nearly overconvergent modular form and Coleman theory of it, following the development of Urban, Harron-Xiao, Z. Liu and Andreatta-Iovita.

July 29 (Mon), 2019, 1:00pm-3:00pm, Room 1424

Title: Depth 0 local Langlands correspondence for $\text{GL}_n$ (expository talk)

Speaker: Shin Eui Song (University of Maryland)

Abstract: We give an account of the methods of Yoshida to prove that the local Langlands correspondence for $\text{GL}_n$ is realized in the vanishing cycle cohomology of the deformation spaces of formal modules of height $n$, in the case of depth $0$ or level $p$. We first review the theory of formal modules with Drinfeld level structures, compute the explicit equation for some deformation space, realize them as a complete local ring of some unitary Shimura variety, compute its vanishing cycle cohomology, and apply the results of Deligne-Lusztig theory.

July 25 (Thu), 2019, 1:30pm-3:00pm, Room 1423

Title: The rational cuspidal divisor class group of X_0(N)

Speaker: Hwajong Yoo (Seoul National University)

Abstract: In this talk, we introduce a large subgroup $C(N)$ of the rational torsion subgroup of $J_0(N)$, which is called the rational cuspidal divisor class group of $X_0(N)$. It is conjecturally equal to the whole group itself. We then construct special rational cuspidal divisors which generate $C(N)$. We will show that they are indeed a "basis'' of the group $C(N)$ for any positive integer $N$.

July 22 (Mon), 2019, 4:00pm-6:00pm, Room 1424

Title: $p$-adic analogue of Riemann-Hilbert correspondence

Speaker: Yong Suk Moon (Purdue University)

Abstract: We will first explain the recent result of Diao-Lan-Liu-Zhu on the $p$-adic analogue of Riemann-Hilbert correspondence. Then we will talk about our joint work with Tong Liu proving that every relative crystalline representation with Hodge-Tate weights in $[0, 1]$ arises from a $p$-divisible group if the ramification is small, and explain its application to studying the correspondence.

June 27 (Thu), 2019, 4:00pm-5:00pm, Room 1209

Title: Irreducible components of Affine Deligne-Lusztig varieties

Speaker: Yihang Zhu (Columbia University)

Abstract: The set of irreducible components of an affine Deligne-Lusztig variety is interesting for many applications related to Shimura varieties. A natural symmetry group J acts on this set, and it is desirable to determine the orbits and the stabilizers of this action. In joint work with Rong Zhou, we prove a formula for the number of orbits, earlier conjectured by Miaofen Chen and Xinwen Zhu. In joint work in progress with Xuhua He and Rong Zhou, we show that all the stabilizers are "very special parahorics". In many cases this already characterizes the stabilizers.

March 28 (Thu), 2019, 4:00pm-6:00pm, Room 1424

Title: Big Heegner points and Heegner cycles

Speaker: Kazuto Ota (Keio University)

Abstract: The Big Heegner point, constructed by Howard, is a deformation of the Heegner point for the Galois representation attached to a Hida family. Castella showed that it interpolates Heegner cycles (higher weight-analogues of the Heegner point) attached to newforms arising from the Hida family, by using $p$-adic local methods such as Bloch-Kato logarithms. In this talk, we explain a slight refinement of his result by a different approach, which is global and more geometric.

March 18 (Mon), 2019, 4:00pm-5:30pm, Room 1423

Title: Heisenberg group and the Lerch zeta function

Speaker: Mehmet Kral (Sophia University)

Abstract: I want to talk about a recent paper of Jeffrey Lagarias with the above title. Representations of the Heisenberg group give rise to Lerch zeta and Lerch $L$-functions, and Fourier duality, which is intrinsic to the Heisenberg group itself, gives rise to the Functional equation of these Lerch zeta or $L$-functions.

January 31 (Thu), 2019, 5:00pm-6:00pm, Room 1423

Title: Equidistribution theorems for holomorphic Siegel cusp forms

Speaker: Henry Kim (University of Toronto/KIAS)

Abstract: We explain equidistribution theorems for a family of holomorphic Siegel cusp forms of $\text{GSp}_4$ in the level and weight aspects. A main tool is Arthur's invariant trace formula. While Shin-Templier used Euler-Poincare functions at the infinity in the formula, we use pseudo-coefficients of holomorphic discrete series to extract only holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms which have not been studied, and which correspond to endoscopic cuspidal representations with large discrete series at the infinity. We give several applications, including the vertical Sato-Tate theorem and low-lying zeros for degree $4$ spinor $L$-functions and degree $5$ standard L-functions of holomorphic Siegel cusp forms. This is a joint work with Satoshi Wakatsuki and Takuya Yamauchi. If time permits, we explain work in progress to the generalization to $\text{Sp}_{2n}$.

January 14 (Mon), 2019, 3:00pm-5:00pm, Room 8101
January 16 (Wed), 2019, 3:00pm-5:00pm, Room 8101
January 18 (Fri), 2019, 3:00pm-5:00pm, Room 8101

Title: $L$-invariants and $p$-adic Langlands program

Speaker: Yiwen Ding (BICMR)

Abstract: To a semi-stable non-crystalline $p$-adic Galois representation, one can associate the so-called Fontaine-Mazur $L$-invariants. Such invariants are however invisible in the classical local Langlands correspondence. A task in $p$-adic Langlands program is therefore to understand their counterpart in $p$-adic automorphic representations. In these lectures, we will first review some of Breuil's work on $L$-invariants in $\text{GL}_2(\mathbb{Q}_p)$-case (which somehow initialized the $p$-adic Langlands program). Then we discuss some joint work with Breuil on $L$-invariants in $\text{GL}_3(\mathbb{Q}_p)$-case. We will explain the construction, based on $p$-adic Langlands correspondence for $\text{GL}_2(\mathbb{Q}_p)$, of a locally analytic representation of $\text{GL}_3(\mathbb{Q}_p)$ which carries the information of all the $L$-invariants. We will also show a local-global compatiblity result on $L$-invariants.