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 SkolemĀ“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.