Number Theory Seminars 2017

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January 5, 2017 Jung Kyu Canci (Univ. of Basel)

Title: Scarcity of periodic points for rational functions over a number field

Abstract: I will present a joint work with S. Vishkautsan where we provide an explicit bound on the number of periodic points of a rational function of degree at least 2 defined over a number field. The bound depends only on the number of primes of bad reduction and the degree of the function, and is linear in the degree. We show that under stronger assumptions (but not so strong) the dependence on the degree of the map in the bounds can be removed. Our results are consequences of some more general results about integral points on some varieties.

February 23, 2017 Joachim Koenig (Technion)

Title: On specializations of regular Galois extensions over number fields

Abstract: For a number field K, we investigate the set of specializations of a regular Galois extension E|K(t) with Galois group G, and its relation to some classical problems. The Beckmann-Black Problem asks whether every G-extension of K arises as a specialization of some such E|K(t). We use a notion of "parametric sets" of regular Galois extensions to provide negative answers to more restrictive versions of the Beckmann-Black Problem. The Grunwald Problem asks whether, for given local extensions (all with Galois group embedding into G) at finitely many primes of K, there always exists a G-extension of K which approximates all these local extensions. We investigate to what extent the specializations of some given regular Galois extensions can provide positive answers to this question. (The first part is joint work with Francois Legrand. The second part is joint work in progress with Francois Legrand and Danny Neftin.)

March 24, 2017 Jaehyun Cho (UNIST)

Title: Simple zeros of L-functions

Abstract: We show that every automorphic L-functions of $GL_2(A_\mathbb{Q})$ has infinitely many simple zeros. It is a joint work with A. Booker and M. Kim.

April 6-7 (4 hours), 2017 Christophe Breuil (University of Paris-Sud)

Title: Eigenvarieties and the locally analytic Langlands program for $GL_n$

Abstract: Let $p$ be a prime number and $n$ an integer $>=2$. The lectures will be a 4 hours (detailed) sketch of the proof by E. Hellmann, B. Schraen and the speaker that all expected companion constituents for $GL_n(Q_p)$ do indeed occur in the socle of the locally analytic Hecke eigenspaces of completed cohomology (for compact unitary groups) in the crystalline case and under standard Taylor-Wiles hypothesis.

April 12, 2017, 5:00pm-6:00pm, Room 1424 Chan-Ho Kim (KIAS)

Title: Bertolini-Darmon-Pollack-Stevens

Abstract: We discuss an overconvergent construction of anticyclotomic $p$-adic $L$-functions of modular forms of non-critical slope.

May 26, 2017, 4:00pm-5:00pm, Room 1424 Hae-Sang Sun (UNIST)

Title: Variance of modular symbols

Abstract: Based on numerical calculations, Mazur-Rubin established a conjecture on the distribution of modular symbols, namely period integrals on cusp forms for congruence subgroups. Regarding the modular symbol as a random variable for the set of rational numbers with a fixed denominator, they conjectured that it is asymptotically normal as the denominator goes to infinity. In the talk, I will discuss recent results of Petridis-Risager and present how to calculate the variance of the modular symbols.

May 26, 2017, 5:00pm-6:00pm, Room 1424 Myoungil Kim (UNIST)

Title: On the conjecture of Kurihara

Abstract: Let $E/\mathbb{Q}$ be an elliptic curve with supersingular reduction at an odd prime $p$. In his Inventiones paper, M. Kurihara obtained some explicit descriptions of the $p$-primary component of the Tate-Shafarevich groups of $E$ over each finite extension $K_n$ contained in the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ under technical assumptions. Based on this result he also proposed a conjecture that is finer than Iwasawa main conjecture in some sense . In this talk, we will briefly review Kurihara's result. If time permits, then we will also discuss whether one can remove some technical assumptions.

June 20-21, 2017, 4:00pm-6:00pm, Room 1424 Jeehoon Park (Postech)

Title: Arithmetic Chern-Simons theory

Abstract: The Chern-Simons theory is a gauge theory which is a version of (2+1)-dimensional TQFT (topological quantum field theory). It provided a useful framework and tools to understand the topology of knots in a 3-manifold, for example, the Jones polynomial of knots. The arithmetic Chern-Simons theory for Galois representations, initiated by Minhyong Kim, is an arithmetic analogue of the Chern-Simons theory, which is expected to attack the number theory problem (Galois theory problem, L-functions, Iwasawa theory, and etc) guided by physics (quantum field theory) and topology principles and techniques appearing in the Chern-Simons theory. In this talk, we provide a definition of the arithmetic Chern-Simons action. Then we will explain how to compute it using the boundary torsors and their trivializations, and its arithmetic application. This is a joint work with Hee-Joong Chung, Dohyeong Kim, Minhyong Kim, and Hwajong Yoo.

June 22, 2017, 4:00pm-5:00pm, Room 1424 Ashay Burungale (Paris 13/IAS)

Title: On the non-vanishing of special L-values modulo p

Abstract: We give a brief exposition of Namikawa's work on the mod p non-vanishing of special values of L-function of an automorphic representation on GL_2 over an imaginary quadratic field twisted by a finite order Hecke character over the imaginary quadratic field.

June 22, 2017, 5:00pm-6:00pm, Room 1424 Dong Quan Ngoc Nguyen (University of Notre Dame)

Title: Polynomial families, and Waring's problem

Abstract: Let A be a commutative ring. A subset X of A^n is a polynomial family with d parameters if it is the range of a polynomial map from A^d to A^n. It is an old question of Skolem (1938) whether the group SL_2(A) with A being the set of integers is a polynomial family. Only recently, Vaserstein (2010) answered Skolem´s question in the affirmative. For the first part of the talk, I will discuss my result of Skolem's conjecture in the function field case SL_2(A), where A is the polynomial ring over a finite field of q elements. The Waring problem asks whether for each positive integer n, every nonnegative integer is the sum of a bounded number of nth powers. Equivalently, it asks whether for each positive integer n, the set of nonnegative integers is a polynomial family with a bounded number of parameters, say d for the polynomial map from Z^d to Z defined by sending (x_1,...., x_d) to x_1^n +.......+ x_d^n. The second part of the talk is a joint work with Michael Larsen (Indiana University) on the Waring problem for unipotent algebraic groups.

June 23, 2017, 4:00pm-6:00pm, Room 1424 Ashay Burungale (Paris 13/IAS)

Title: Horizontal variation of the arithmetic of elliptic curves

Abstract: Let E be an elliptic curve over the rationals. Let K be an imaginary quadratic field and H_K the corresponding Hilbert class field. We discuss recent results on the arithmetic of E over H_K as K varies (joint with H. Hida and Y. Tian).

June 29, 2017, 4:00pm-5:00pm, Room 1423 Junho Lee (Mokpo University)

Title: A system of fundamental units for a family of some cubic fields

Abstract: In this talk, we talk about cubic fields having excetional units and their unit groups. Next, we introduce some Diophantine equation related to certain cubic field and share with you how to solve it. It is a joint work with St\'{e}phane R. Louboutin.

August 9, 2017, 4:00pm-5:00pm, Room 1424 Jaehoon Lee (UCLA)

Title: Controlling Selmer groups of elliptic curves

Abstract: We discuss various control theorems of Selmer groups of elliptic curves. (The most part of the talk will be expository.)

August 24, 2017, 4:00pm-5:00pm, Room 1424 Chan-Ho Kim (KIAS)

Title: On the Iwasawa main conjecture for modular forms

Abstract: We provide a simple numerical criterion to verify the Iwasawa main conjecture for all members of a Hida family once and for all under mild assumptions. Explicit examples will be given. This is joint work with Myoungil Kim and Hae-Sang Sun.

October 18 (Wed), 2017, 4:00pm-5:00pm, Room 1424 Kyoungmin Kim (SKKU AORC)

Title: Spinor representations of positive definite ternary quadratic forms

Abstract: For a positive definite integral ternary quadratic form $f$, let $r(k,f)$ be the number of representations of an integer $k$ by $f$. The famous Minkowski- Siegel formula implies that if the class number of $f$ is one, then $r(k,f)$ can be written as a constant multiple of a product of local densities which are easily computable. In this talk, we consider the case when the spinor genus of f contains only one class. In this case the above also holds if k is not contained in a set of finite number of square classes which are easily computable. By using this fact, we prove some extension of the results given on the representations of generalized Bell ternary forms and on the representations of ternary quadratic forms with some congruence conditions. This is a joint work with Jangwon Ju and Byeong-Kweon Oh.

October 18 (Wed), 2017, 5:00pm-6:00pm, Room 1424 Jangwon Ju (SNU)

Title: Universal sums of generalized octagonal numbers

Abstract: An integer of the form $P_8(x) = 3x2 - 2x$ for some integer $x$ is called a generalized octagonal number. A quaternary sum $\Phi_{a,b,c,d}(x, y, z, t) = aP_8(x) + bP_8(y) + cP_8(z) + dP_8(t)$ of generalized octagonal numbers is called universal if $\Phi_{a,b,c,d}(x, y, z, t) = n$ has an integer solution $x, y, z, t$ for any positive integer $n$. In this talk, we show that if $a = 1$ and $(b,c,d)$ = $(1,3,3)$, $(1, 3, 6)$, $(2, 3, 6)$, $(2, 3, 7)$ or $(2, 3, 9)$, then $\Phi_{a,b,c,d}(x, y, z, t)$ is universal. These were conjectured by Sun. We also give an effective criterion on the universal- ity of an arbitrary sum $a_1 P_8(x1) + a_2 P_8(x_2) + \cdots + a_k P_8(x_k)$ of generalized octagonal numbers, which is a generalization of "15-theorem" of Conway and Schneeberger. This is a joint work with Byeong-Kweon Oh.

November 14 (Tue), 2017, 4:00pm-6:00pm, Room 1424 Joachim Koenig (Technion/Wurzburg)
November 16 (Thu), 2017, 4:00pm-6:00pm, Room 1424

Title: On the inverse Galois problem with restricted ramification 1 and 2

Abstract: In this series of talks, I will consider various strong versions of the inverse Galois problem with added conditions on ramified primes. I will first discuss various versions of "minimal ramification problems" and review known results and conjectures. Here "minimal ramification" can be understood as the problem to minimize the number of primes ramifiying in an extension with a given Galois group, or also as the problem to minimize the ramification indices. I will also discuss relations to conjectures and heuristics about the distribution of Galois groups (such as the Malle-Bhargava heuristics) and the distribution of class groups (such as the Cohen-Lenstra heuristics). I will furthermore present new results on Galois realizations with restricted ramification indices, obtained in joined work with D.Neftin and J.Sonn. We use two essentially different approaches, one coming from Shavarevich's method for solvable group, and another one coming from geometric Galois theory and specialization of Galois coverings. Finally, I will present some results on so-called intersective polynomials. The notion of intersective polynomials, that is, integer polynomials with a root in every Q_p, but not in Q, was introduced by Sonn. Their construction is related to restricted-ramification problems. I will present conjectures and new results on the existence of such polynomials for prescribed Galois groups. This last talk in particular will be very easily accessible without a lot of theoretical background.

November 30 (Thu), 2017, 4:00pm-5:00pm, Room 1424 Paul Hamacher (Technische Universität München)

Title: On the irreducible components of affine Deligne-Lusztig varieties

Abstract: In our talk we discuss the geometric properties of affine Deligne-Lusztig varieties. They can be thought of as varieties inside the affine Grassmannian which are analogous to (classical) Deligne-Lusztig varieties, which were constructed by Deligne and Lusztig in order to study representations of finite groups of Lie type. They naturally appear when studying of the arithmetic of Shimura varieties and offer a "low tech" approach to the local Langlands correspondence.

December 14 (Thu), 2017, 4:00pm-6:00pm, Room 1424 Rei Otsuki (Keio University)

Title: The $p$-adic $L$-functions for higher weight modular forms in the supersingular case

Abstract: Let $f(z)= \sum a_n q^n$ be an eigen cusp form of weight $k \geq 2$. In the case when $f(z)$ is supersingular at $p$, we have two $p$-adic $L$-functions attached to $f(z)$, but those are power series with unbounded coefficients. In the case when $a_p = 0, k \geq 2$, Pollack constructed $p$-adic $L$-functions with bounded coefficients, and in the case when $k = 2$, Sprung constructed similar $p$-adic $L$-functions. In this talk, I will talk about generalization of Sprung's method in the case when $k > 2$ and $p \geq k-1$.

December 15 (Fri), 2017, 4:00pm-6:00pm, Room 1424 Takahiro Kitajima (Keio University)

Title: Non-existence of nontrivial finite $\Lambda$-submodules of +/- Selmer groups

Abstract: Let $E$ be an elliptic curve over $\mathbb Q$ which has supersingular reduction at some odd prime number $p$, $F$ a finite abelian number field and $F_{\infty}/F$ the cyclotomic $\mathbb Z_p$-extension. Assume that $a_p=0$. Then Kobayashi defined the plus and the minus Selmer groups of $E$ over $F_{\infty}$ and proved that their Pontryagin duals are $\Lambda$-torsion. It is a basic question whether the Pontryagin duals of the plus and the minus Selmer groups of $E$ over $F_{\infty}$ have nontrivial finite $\Lambda$-submodules or not. In this talk, we present a result on the non-existence of non-trivial finite $\Lambda$-submodules. Our result is a generalization of B.D. Kim's result in 2013. This is joint work with R. Otsuki.

December 18 (Mon), 2017, 4:00pm-6:00pm, Room 8406 Jaeho Haan (KIAS)

Title: Introduction to the Gan-Gross-Prasad conjecture

Abstract: In this talk, I will introduce the Gan-Gross-Prasad conjecture and local Langlands correspondence. This is a preliminary talk for the Hiraku Atobe's lecture series.

December 19 (Tue), 2017, 4:00pm-6:00pm, Room 1424 Hiraku Atobe (University of Tokyo)
December 21 (Thu), 2017, 3:00pm-5:00pm, Room 1423

Title: An explicit description of the local theta correspondence 1 and 2

Abstract: The theory of local theta correspondence is a major theme in the representation theory and the theory of automorphic forms. There are two basic problems. One is when a local theta lift is nonzero, and the other is how to describe it if it is nonzero. In this lecture, I address these two problems for tempered representations in the case of unitary dual pairs over p-adic fields in terms of the local Langlands correspondence. This lecture is based on a joint work with Wee Teck Gan.

December 26 (Wed), 2017, 4:00pm-5:00pm, Room 1424 Dohyeong Kim (University of Michigan, Ann Arbor)

Title: Discrete Hodge star operator and the conjecture of Prasanna-Venkatesh

Abstract: For an oriented triangulated manifold, Scott Wilson defined a discrete analogue of the Hodge star operator. We will show that this is independent of a triangulation if the manifold is 3-dimensional. Specializing to the case of arithmetic 3-manifolds, we will interpret the existence of such an operator as a modest evidence for a special case of the conjecture of Prasanna-Venkatesh.

December 27 (Wed), 2017, 4:00pm-6:00pm, Room 1424 Kazuto Ota (Keio University)

Title: On the Mazur-Tate refined conjecture for modular forms

Abstract: The Mazur-Tate refined conjecture relates arithmetic invariants with Mazur-Tate elements which are elements of certain group rings and are constructed from special values of L-function. In this talk, we discuss the rank part of this conjecture for elliptic modular forms. More precisely, under some assumptions we show that the order of vanishing of Mazur-Tate elements attached to modular forms is greater than or equal to the rank of Selmer groups.