Number Theory Seminars 2016

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January 7, 2016 Sun, Hae-Sang (UNIST)

Title: Infinite examples of vanishing $\mu$-invariant of Mazur-Swinnerton-Dyer $p$-adic $L$-functions

Abstract: In the talk, I will present a conjecture on the modular symbols, which implies the vanishing ¥ì-invariant. I will also discuss a partial result on the conjecture and as a consequence, infinitely many examples that the ¥ì-invariant of Mazur-Swinnerton-Dyer p-adic L-function vanishes. This is a joint work with Kim, Myungil (UNIST).

January 11-13, 2016 Ashay Burungale (University of Arizona)

Title: On the non-triviality of Hecke $L$-values modulo $p$

Abstract: Let $\lambda$ be a Hecke character over a CM field. We give an overview of results on the non-triviality of central $L$-values of anticyclotomic twists of $\lambda$ modulo $p$. We also describe the case of Katz $p$-adic $L$-functions. These results are based on a strategy initiated by Hida and fundamentally rely on the Zariski density of mod $p$ reduction of a family of CM points on several copies of a Hilbert modular Shimura variety.

January 18-20, 2016 Seoyoung Kim (Brown University)

Title: The Main Theorem of Complex Multiplication

Abstract: The main theorem of complex multiplication allows us to regard the multiplication by $a^{-1}$ map as a map between two elliptic curves, which are defined by the Artin reciprocity, and the other way around. In this talk, I would formalize the theorem by giving reasonable preliminaries and present the brief proof of the main theorem. If time permits, I wish to discuss some applications of the main theorem or a slight generalization of the theorem when $\mathrm{End}(E) \neq \mathcal{O}_K$.

January 18-20, 2016 Yong Suk Moon (Harvard University)

Title: $p$-divisible Groups and Galois Deformation Rings in the Relative Case

Abstract: Fontaine-Mazur conjecture predicts that any global irreducible p-adic Galois representation which is potentially semi-stable at primes dividing p and unramified outside finitely many places is a subquotient of the p-adic cohomology of algebraic variety, up to Tate twist. To understand which Galois representations come from algebraic geometry, it is important to study the sub-moduli spaces which parametrize those with certain p-adic Hodge theoretic conditions. In the first talk, I will explain Kisin's result on the loci of potentially semi-stable Galois representations, and his application of this result to modularity lifting and Fontaine-Mazur c onjecture. In the second, I will explain the generalization of the above result on potentially semi-stable deformation ring to the case when the residue field of the associated p-adic field is not necessarily finite. In the third, I will talk about the generalization of Raynaud's theorem on extending p-divisible groups in the relative case, and its application to the study of the loci of Barsotti-Tate representations in the relative case.

January 21, 2016 Lucio Guerberoff (University College London)

Title: Periods relations for certain automorphic motives

Abstract: In this talk I will explain some results relating critical values of L-functions of cohomological automorphic representations of unitary groups over CM fields and periods. Roughly speaking we express the critical values as Petersson norms of holomorphic forms, and we explain the link with Deligne's conjecture, which predicts that these have a factorization in terms of quadratic periods, depending on the signature of the unitary group; these period relations would follow from Tate's conjecture.

February 2-5, 2016 Michael Griffin (Princeton University)

Title: Moonshine

Abstract: In the 1970¡¯s, as mathematicians worked to classify the finite simple groups, Ogg, McKay and others observed several striking apparent coincidences connecting the then-conjectural Monster group (the largest of the sporadic simple groups) to the theory of modular functions. These ¡®coincidences¡¯ became known as ¡°Monstrous Moonshine¡± and were made into a precise conjecture by Conway and Norton. They conjectured the existence of a naturally occurring graded infinite dimensional Monster module whose graded traces at elements of the monster group give the Fourier coefficients of distinguished modular functions. Borcherds proved the conjecture in 1992, embedding Moonshine in a deeper theory of vertex operator algebras. For this work Borcherds was awarded a Fields Medal. Fifteen years after Borcherds¡¯ proof, Witten conjectured important connections between Monstrous Moonshine and pure quantum gravity in three dimensions. Under Witten¡¯s theory, the irreducible components of the Monster module represent black hole states. Witten asked how these states are distributed. In joint work with Ken Ono and John Duncan, we answer Witten¡¯s question giving exact formulas for these distributions.

February 2-5, 2016 Michael Griffin (Princeton University)

Title: Moonshine and mock modular forms

Abstract: Monstrous Moonshine creates a bridge between the fields of representation theory and modular forms. Here we will review the theory of harmonic Maass forms and mock modular forms which extends the classical theory and provides a useful framework with which to study the McKay-Thompson series. Although the McKay-Thompson series are true modular functions, we may construct them as harmonic Maass forms by way of the Maass Poincare series. This construction yields exact formulas and asymptotics for the coefficients. These formulas in turn allow us to answer Witten's question concerning distribution of the irreducible representations in Monstrous Moonshine. We will also consider the construction of weight 1/2 vector valued Maass--Poincare series. These forms have appeared in recently observed variations of the moonshine phenomena; notably the Umbral Moonshine conjectures of Cheng, Duncan, and Harvey, however convergence in this case is particularly delicate and relies on the convergence of the Selberg-Kloosterman zeta function.

February 2-5, 2016 Michael Griffin (Princeton University)

Title: Umbral Moonshine

Abstract: In recent years, Moonshine type-phenomena have been observed for other groups besides the Monster. The Umbral Moonshine conjectures of Cheng, Duncan, and Harvey connects the automorphism groups of the 24 Niemeier lattices to the Fourier coefficients of certain mock modular forms. Many mathematical physicists anticipate physical interpretations for Umbral Moonshine similar to Witten's application of Monstrous Moonshine. The first case of Umbral Moonshine, connected to the Leech Lattice, is covered by Monstrous Moonshine, while the second is covered by Gannon's proof in 2013 of Moonshine for the Mathieu group M24. In joint work with Ken Ono and John Duncan, we verify the remaining 22 cases.

March 24, 2016 Bo-Hae Im (Chung-Ang University)

Title: Zeros of Quasi-Modular Forms and Weakly Holomorphic Modular Forms for the Hecke Groups of Low Levels

Abstract: We will talk about the locations of zeros of quasi-modular forms and weakly holomorphic modular forms for the Hecke groups of certain levels, especially level 2 and 3. Also we will give some numerical data for them. This is a joint work with SoYoung Choi.

April 7, 8, 2016 Junyeong Park (POSTECH)

Title: Categories for working number theorists 1,2,3

Abstract: In this talk, I will mainly cover the role of sheaf theory in number theory and its generalization. First, I will introduce Grothendieck topology on a category and sheaves with respect to it and describe sheaves on the category of schemes (with some topologies) as examples. Next, I will apply this theory to the functor of moduli of elliptic curves to explain the reason why the moduli of elliptic curve cannot be a scheme. There are many directions to 'fix' it, and the language of stacks is one of them, which is the direction I will introduce in this talk. In order to do this, the theory of 2-categories will be also covered.

April 11,12, and 14, 2016 Wansu Kim (King's College, London)

Title: p-adic geometry of certain Shimura varieties 1,2,3

Abstract: Recently, there have been many exciting new developments in the study of p-adic geometry and cohomology of a wide class of Shimura varieties. Underlying many of these developments is the new technique to study "infinite-level" Shimura varieties as p-adic analytic spaces, as well as new developments in p-adic Hodge theory (both of which are built upon P. Scholze's theory of perfectoid spaces). The aim of this series of talks is to illustrate the geometric results on Shimura varieties proved in the recent paper by Caraiani and Scholze, which "decomposes" certain Shimura varieties in a way that gives a meaningful decomposition of the cohomology, and discuss my work in progress to generalise this result for unramified Hodge-type Shimura varieties. In the first two hours, we will cover some backgrounds in p-adic Hodge theory and p-divisible groups. We will not necessarily strive to give a precise definition or construction of each main object, and may settle for giving some simple examples (even at the risk of over-simplifying).

April 28, 2016 Jae-Hyun Yang (Inha University)

Title: Differential operators on the Siegel-Jacobi space

Abstract: Jiong Yang and LinSheng Yin published the good paper, "Differential operators for Siegel-Jacobi forms" in the Science China Mathematics last year. In that paper, they obtained derivations of Jacobi forms, and then constructed a series of invariant differential operators on the Siegel-Jacobi space. In this talk, I explain the works of J. Yang and L. Yin, and then discuss subjects related to their works. Finally I give some open problems to be investigated in the future.

May 12, 2016 Henry Kim (University of Toronto)

Title: Universality of Artin L-functions in conductor aspect

Abstract: We establish the universality of Artin $L$-functions associated to a certain family of number fields in conductor aspect. When $s=1+it$ and $s=1$, our result is unconditional. In the critical strip, we assume a conjecture on large sieve inequality of Artin $L$-functions, or GRH. This is a joint work with Peter J. Cho.

May 26, 2016 Henry Kim (University of Toronto)

Title: Equidistribution theorems for holomorphic Siegel cusp forms

Abstract: We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms of $GSp_4$ 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.

May 27, 2016 Dohyeong Kim, (IBS-CGP)

Title: On the etale cohomology of some algebraic surfaces

Abstract: In this talk we study the etale cohomology of an algebraic surface that is defined over a local field. The main goal is to obtain geometric criteria which ensure that a naturally defined quotient of the second cohomology is unramified, or crystalline depending on the characteristic of the coefficient. An important issue is to deal with non-semistable reduction and we make a mild progress towards it. We also present some diophantine applications.

June 9, 2016 Byungchan Kim, (Seoul National University of Science and Technology)

Title: Partition cranks, a survey

Abstract: To explain Ramanujan's partition congruences, Andrews and Garvan introduced a crank function. Motivated from this work, several cranks for various partition functions have been introduced. We are going to survey some recent works on arithmetic of crank functions.

June 21, 2016 John Binder, (Massachusetts Institute of Technology)

Title: Fields of rationality of cusp forms and automorphic representations

Abstract: Given a classical cusp form f, the field of rationality Q(f) is the field obtained by adjoining all of its Fourier coefficients; if f is a member of the classical basis of cusp forms of weight k and level Gamma_1(N), the Q(f) is a number field and we call [Q(f):Q] the degree of f. Serre posited that, for a fixed weight k and an increasing sequence {N} of levels, that the proportion of cusp forms with degree bounded by some fixed A approaches 0; he proved the conjecture when there is an auxiliary prime p_0 that is coprime to each N_\lambda. In this talk, we will answer Serre's question by reformulating it in terms of cuspidal automorphic representations. The talk will have two parts. In the first part, we will discuss the connection between cusp forms and automorphic representations and compute the fields of rationality of some local representations. In the second part, we will discuss a Plancherel equistribution result for the local components of automorphic representations in families and use this result to answer Serre's question.

June 23, 2016 Jeongho Park, (Ulsan National Institute of Science and Technology)

Title: On the parity of ideal classes of number fields over a fixed prime

Abstract: In this talk we will consider real quadratic number fields and their prime ideals over a fixed rational prime, and raise a question about how often these ideals are principal. I will give a partial result concerning the parity of orders of ideal classes containing those prime ideals.

July 28, 2016 Dohyeong Kim, (IBS-CGP)

Title: Some computations in arithmetic Chern-Simons theory

Abstract: We investigate some computational aspects of arithmetic Chern-Simons theory. The theory has been recently proposed by Minhyong Kim, which is modelled on the Dijkgraaf-Witten theory. For any choice of a finite group and a homomorphism of a Galois group of a number field into it, the theory produces a canonical function, called Chern-Simons functional. The resulting functional can be viewed as a function on certain cohomology group. The present theory requires the base number field to contain certain roots of unity, or to be totally imaginary, depending on circumstances. In this talk, we aim to show that the Chern-Simons functional is non-trivial for the Klein 4-group, by giving an cohomology class and a representation of a Galois group, and evaluating the Chern-Simons functional.

August 11, 2016 Jayce Getz, (Duke University)

Title: Langlands' beyond endoscopy proposal and triple product L-functions

Abstract: The theory of (twisted) endoscopy is responsible for much of what we now know about the Langlands functoriality conjecture but it has intrinsic limitations. Langlands proposed a method for studying Langlands functoriality in general by building L functions into trace formulae and using analytic number theory to analyze the resulting expression. The proposal has only been carried out in very special cases corresponding more or less to the standard representation and symmetric square representation of GL(2) and the tensor product of GL(2) with itself. We explain how the analytic part of the proposal can be carried out in the case of the tensor product of three copies of GL(2).

August 11, 2016 Heekyoung Hahn, (Duke University)

Title: Langlands' beyond endoscopy proposal and the Littlewood-Richardson semigroup

Abstract: Langlands' beyond endoscopy proposal for establishing functoriality motivates the study of irreducible subgroups of GL(n) that stabilize a line in a given representation of GL(n). Such subgroups are said to be detected by the representation. In this talk we study the important special case where the representation of GL(n) is the triple tensor product representation. We prove a family of results describing when subgroups isomorphic to classical groups of type A_n, B_n, C_n, D_2n are detected.

August 25, 2016 Junguk Lee, (Yonsei University)

Title: On the structure of certain valued fields

Abstract: We show that for finitely ramified valued fields with perfect residue fields, for large enough n, any homomorphism of the n-th residue rings can be lifted to a homomorphism of valuation rings quite naturally. Here, the n-th residue ring is the quotient of the valuation ring by the n-th power of the maximal idea. Thus we have a lifting map for finitely ramified valued fields like as the lifting map of homomorphisms of residue fields to homomorphisms of valuation rings for the unramified case. Moreover this lifting map is compatible with composition of homomorphisms of n-th residue rings. This provides a functor from a category of certain principal Artinian local rings of length n to a category of certain valuation ring of fixed ramification index, which naturally generalizes the functorial property of Witt ring. The result also strengthens Basarab's result on the AKE-principle for finitely ramified henselian valued fields, which solve a question raised by Basarab, when residue fields are perfect. This is joint work with Wan Lee.

September 09, 2016 Chol Park, (KIAS)

Title: Semi-stable deformation rings in even Hodge--Tate weights

Abstract: We construct semi-stable deformation rings in Hodge--Tate weights (0,r) for r=even of absolutely irreducible 2-dimensional mod-p representations, by integral p-adic Hodge theory. This is a joint work with Lucio Guerberoff.

September 29, 2016 Zicheng Qian, (University of Paris-Sud)

Title: Some facts around the modular representations of GL_n(F_p)

Abstract: During the work on progress with Chol Park, I found the necessity to consider certain group operators of GLn(Fp) which are closely related to a special basis of principal series. The talk will start with recalling some elementary background of related representation theories. I will explain my idea through some explicit calculation.

October 13, 2016 Seunghwan Chang, (Ehwa Womans University)

Title: Computing shortest vectors in lattices of low dimension

Abstract: Finding vectors with minimal size in the Euclidean lattices is an interesting computational problem with variety of applications. The problem, usually called SVP (The Shortest Vector Problem), can be extremely difficult in high dimension. The LLL basis reduction algorithm by Lenstra, Lenstra and Lovsz efficiently computes lattice vectors with size "close" to minimal, i.e., it solves SVP, but only "approximately" in general. In this seminar, I will talk about results towards solving SVP "exactly" for lattices of low dimension.

October 25, 27, 28, 2016, 2pm-5pm, Gabriel Dospinescu, (E.N.S., Lyon)

Title: p-adic etale cohomology of the Drinfeld tower in dimension 1

Abstract: We will talk about work in progress with Pierre Colmez and Wieslawa Niziol, which describes the p-adic etale cohomology of the coverings of the p-adic upper half-plane (for GL_2(Q_p)) constructed by Drinfeld. The result is very similar to the l-adic case (due to Carayol, Faltings, Harris and Taylor), except that the classical Langlands correspondence is replaced by the p-adic one.

November 11, 2016 Yeansu Kim, (Chonnam National University)

Title: Classification of strongly positive descrete series representations of GSpin groups and its applications

Abstract: The classication of discrete series representations of connected reductive groups G over a local field F is one important step in local Langlands correspondence. More precisely, the admissible representation, one object in the Langlands program, has a ltration according to its growth properties of its matrix coefficients. $$supercuspidal \subset discrete series \subset tempered \subset admissible:$$ One strategy to study properties or theorems of admissible representations is first to prove those in the case of supercuspidal representations. If it works, we generalize those proofs to the case of discrete series, tempered and admissible representations following the above filtration. The natural question is how we generalize the theorems in the case of previous class to the case of next class. In this talk, I will explain the results on the first step (from supercuspidal representations to discrete series representations) which is called `classification of (strongly positive) discrete series representations' in the case of GSpin groups. One application of this classification results is to show the equality of L-functions from Langlands-Shahidi method and Artin L-functions through local Langlands correspondence. In the beginning of the talk, I will focus on the application of the results. Then, I will briefly explain the idea of the proof of the main theorems.

November 24, 2016 Yoonbok Lee, (Incheon National University)

Title: $a$-values of the Riemann zeta function

Abstract: Let $a$ be a nonzero complex number. The $a$-values of the Riemann zeta function are the complex numbers $s$ satisfying $\zeta(s)=a$. In this talk, we introduce known results and an ongoing project about the $a$-values of the Riemann zeta function.

December 1, 2016 Dong Uk Lee, (IBS)

Title: A full proof of Langlands-Rapoport conjecture and Kottwitz formula for local zeta functions of Shimura varieties

Abstract: We give a proof of the full statement of Langlands-Rapoport conjecture for Shimura varieties of abelian type with hyperspecial level, building on the recent work of Kisin on this conjecture. As a result, we establish the formula for the local zeta functions of the same kind of Shimura varieties, conjectured by Kottwitz. The latter is the first big step in the resolution of the conjecture of Langlands that the Hasse-Weil zeta functions of Shimura varieties are automorphic, and is also an important ingredient in some of the constructions of Galois representations using cohomology of Shimura varieties.

December 13, 2016 Elena Mentovan, (CalTech)

Title: Towards a geometric realization of p-adic automorphic forms on unitary Shimura varieties at non-split unramified primes

Abstract: A crucial construction in the study of p-adic automorphic forms on Shimura varieties is their realization as global functions on the ordinary Igusa tower. Following Hida's approach, we consider the cases when the ordinary locus is empty, and explore the relation between p-adic automorphic forms and global functions on the \mu-ordinary Igusa tower. This is joint work in progress with E. Eischen.

December 15, 2016, 4pm-5pm, Junghun Lee, (Nagoya University)

Title: An ergodic value distribution of certain meromorphic functions

Abstract: We will introduce a result on a certain mean-value of meromorphic functions by using specific ergodic transformations, which we call affine Boolean transformations. As examples, we introduce some applications to zeta functions and L-functions. We also introduce an equivalence of the Lindelof hypothesis of the Riemann zeta function in terms of its certain ergodic value distribution associated with affine Boolean transformations. This result can be obtained to use Birkhoff¡¯s ergodic theorem to transform the mean-value into a computable integral which allows us to completely determine the mean-value of this ergodic type. In the talk, we will briefly explain the proof of the results. This is a joint work with Ade Irma Suriajaya.

December 15, 2016, 5pm-6pm, Ade Irma Suriajaya, (Nagoya University)

Title: Distribution of zeros of the derivatives of the Riemann zeta function and Dirichlet L-functions

Abstract: Speiser in 1935 showed that the Riemann hypothesis is equivalent to the first derivative of the Riemann zeta function having no zeros on the left-half of the critical strip. This results shows that the distribution of zeros of the Riemann zeta function is related to that of its derivatives. The number of zeros and the distribution of the real part of non-real zeros of the derivatives of the Riemann zeta function have been investigated by Berndt, Levinson, Montgomery, and Akatsuka. Berndt, Levinson, and Montgomery investigated the general case, meanwhile Akatsuka gave sharper estimates under the truth of the Riemann hypothesis. This result is further improved by Ge. In the first half of this talk, we introduce these results and generalize the result of Akatsuka to higher-order derivatives of the Riemann zeta function. Analogous to the case of the Riemann zeta function, the number of zeros and many other properties of zeros of the derivatives of Dirichlet L-functions associated with primitive Dirichlet characters were studied by Yildirim. In the second-half of this talk, we improve some results shown by Yildirim for the first derivative and show some new results. We also introduce two improved estimates on the distribution of zeros obtained under the truth of the generalized Riemann hypothesis. We also extend the result of Ge to these Dirichlet L-functions. Finally, we introduce an equivalence condition analogous to that of Speiser¡¯s for the generalized Riemann hypothesis, stated in terms of the distribution of zeros of the first derivative of Dirichlet L-functions associated with primitive Dirichlet characters.