The 11th(2024) NCTS–POSTECH-PMI Joint Workshop on Number Theory
Abstract
January 8th (Mon)
09:30-10:20
Title: Arithmetic invariants on supersingular EO strata
Abstract: In this talk we shall discuss the problem of variations of arithmetic invariants (for example endomorphism rings and autuomorphism groups) on the supersingular locus of the Siegel modular varieties. Oort’s conjecture asserts that every geometric generic su-persingular polarized abelian variety of dimension > 1 has automorphism group {±1}. We introduce a stratification on a smaller closed subvariety which is a union of supersingular EO strata and show that in open dense stratum every geometric generic point has automorphism group {±1} for g > 1 even and p > 3, confirming Oort’s conjecture in this case. This is joint work in progress with V. Karemaker.
10:50-11:40
Pei-Xin Liang (Academia Sinica)
Title: Cohomological Properties of Multinorm-One Tori
Abstract: In this talk we investigate the Tate–Shafarevich group X1(k, T) of a multinorm-one torus T over a global field k. We explore a few functo-rial maps among cohomology groups and trying to extend a few result of Bayer-Fluckiger–Lee–Parimala [Adv. in Math., 2019] to some more general multinorm-one tori. We also give a uniform proof of a result of Demarche–Wei for a criterion of the vanishing of X1(k, T), and of the main result of Pollio [Pure App. Math. Q., 2014] for the case where the étale k-algebra in question is a product of two abelian extensions. Moreover, we improve the explicit description of X1(k, T) in Lee [J. Pure Appl. Alg., 2022] by removing an intersection condition. This is a joint work with Chia-Fu Yu and Yasuhiro Oki.
13:00-13:50
Title: Hopf algebra structures of multiple zeta values
Abstract: Please Multiples zeta values in positive characteristic were introduced by Thakur as ana-logues of classical multiple zeta values of Euler. In this talk we present algebraic structures of MZV’s in positive characteristic by constructing both the stuffle algebra and the shuffle algebra of these MZV’s, which leads the Hopf algebra structures to them. In particular, we completely solve a problem suggested by Deligne and Thakur and establish Shi’s conjectures. This is a joint work with Hojin Kim, K. N. Le, T. Ngo Dac and L. H. Pham.
14:20-15:10
Title: On generalizations of elliptic sheaves
Abstract: Drinfeld modules were introduced as an analogue of elliptic curves in the func-tion field setting by Drinfeld in 1974. To establish a Langlands correspondence, Drinfeld studied moduli spaces of elliptic sheaves or equivalently, shtukas. In this presentation, following a brief introduction to the function field realm, I will talk on some well-known generalizations of elliptic sheaves and introduce a new one, called generalized D-elliptic sheaves. I will also present some basic results on their moduli spaces and state a uniformization theorem; building on the work of Laumon-Rapoport-Stuhler, of Hartl and of Rapoport-Zink. In the last part, I will delve into the connections between (generalized) shtukas and (generalized) elliptic sheaves.
15:40-16:30
Shih-Yu Chen (Kyoto University)
Title: ALGEBRAICITY OF RATIOS OF RANKIN–SELBERG L-FUNCTIONS
Abstract: For consecutive critical values of Rankin–Selberg L-functions for GLn × GLn′ , we have the celebrated result of G. Harder and A. Raghuram on the algebraicity of the ratios when nn′ is even. As a different aspect of ratios of critical values, we consider ratios of product of different Rankin–Selberg L-functions at a fixed critical point. In this talk, we introduce our result on the algebraicity of the ratios under a regularity condition. As applications, we prove new cases of Blasius’ and Deligne’s conjectures on critical values of tensor product L-functions and symmetric power L-functions of modular forms.
16:40-17:30
Title: Herglotz function and Number Theory
Abstract: The Herglotz function
named after Herglotz and Zagier used it to obtain a Kronecker limit formula for real quadratic fields. In this talk we further study the properties and application of the Herglotz function.
January 9th (Tue)
09:30-10:20
Dohoon Choi (Korea University)
Title: Finiteness for crystalline representations of the absolute Galois group of a totally real field
Abstract: Let K be a totally real field and GK := Gal(K/K) its absolute Galois group, where K is a fixed algebraic closure of K. Let ℓ be a prime and E a finite extension of ℓ. Let S be a finite set of finite places of K not dividing ℓ. Assume that K, S, Hodge-Tate type h and a positive integer n are fixed. In this paper, we prove that if ℓ is sufficiently large, then, for any fixed E, there are only finitely many isomorphism classes of crystalline representations r : GK → GLn(E) unramified outside S ∪ {v : v|ℓ}, with fixed Hodge-Tate type h, such that r|GK′ ≃ ⊕r′ for some finite totally real field extension K′ of K unramified at all places of K over ℓ, where each representation r′ over E is an 1-dimensional representation of GK′ or a totally odd irreducible 2-dimensional representation of GK′ with distinct Hodge-Tate numbers.
10:50-11:40
Cheng-Chiang Tsai (Academia Sinica)
Title: Wave-front sets for p-adic Lie algebras
Abstract: For an irreducible admissible representation of a p-adic reductive group there is the notion of its wave-front set, which is a set of nilpotent orbits that describes the asymptotic behavior of the character near the identity. By a theorem of Moeglin-Waldspurger, the set also describes the least degenerate Whittaker models, which is a double generalization of local components of Fourier expansions for modular forms. In this talk, we study the linearized version of the question, namely the wave-front sets for the Lie algebras. This allows us to understand the wave-front sets for “general position” positive-depth representations, whose behavior is pretty different from those of depth-zero representations previously studied via affine Hecke algebras.
13:00-13:50
Yeongseong Jo (Ewha Womans University)
Title: Gauss Sums and Local Gamma Factors
Abstract: Straightedge and compass construction in geometry goes back at least to ancient Greek. Afterword, Gauss introduced what we call the Gauss period which is a kind of sums of roots of unity. The period is an ingredient for constructible polygons as well as explicit calculation in cyclotomic fields related with harmonic analysis. In particular, the difference of two half periods presents a special case of what is known as Gauss sums. In this talk, we will give expressions of Rankin-Selberg type gamma factors for cuspidal representations over finite fields and level zero representations in terms of local Gauss sums. We will also highlight some interesting features that explain the equivalence between poles of L-factors, non-vanishing of certain residual integrals, distinguished representations, and the absolute value of finite gamma factors.
14:20-15:10
Yao Cheng (Tamkang University)
Title: Local newforms for generic representations of unramified U2n+1 and Rankin-Selberg integrals
Abstract: Newforms have their root in the classical theory of Atkin-Lehner for modular forms. In the modular form setting, newforms are cusp forms which are simultaneously eigenfunctions of all Hecke operators. Consequently, their Fourier coefficients satisfy strong recurrence rela-tions and their L-functions are well behaved. Probably inspired by the connection between modular forms and representations of p-adic GL2, W. Casselman developed the theory of local newforms for generic representations of p-adic GL2. His result was subsequently extended to other p-adic classical groups by many authors. In a recent preprint, Atobe-Oi-Yasuda established the theory of local newforms for generic tempered representations of unramified p-adic U2n+1. Building on their results, we extended the theory to incorporate non-tempered generic representations. In this talk, we will introduce these works with sketchy proofs. If time permits, we will also mention possible arithmetic applications.
15:40-16:30
Title: APPLICATIONS OF THE ARTHUR-SELBERG TRACE FORMULA
Abstract: The Arthur-Selberg trace formula, developed by Arthur and Selberg, provides informa-tion for the Laplace eigenvalues and Hecke eigenvalues of Maass forms. In this talk, I will introduce two results on the Laplace eigenvalues and Hecke eigenvalues of Maass forms over a general number field, which were obtained using the Arthur-Selberg trace formula.
In 1965, Selberg asserted that the Laplace eigenvalues of Maass forms are larger than or equal to 1/4. If the Laplace eigenvalue is less than 1/4, then it is called an exceptional eigenvalue. First, I will focus on the result for the number of Maass forms over a general number field with exceptional eigenvalues. The other one is the result on the distribution of the Hecke eigenvalues of Maass forms. For a fixed prime p, Sarnak proved that the p-th Hecke eigenvalues of Maass forms over the field of rational numbers are equidistributed with respect to some measure. I will introduce extended results for an arbitrary number field. This work is joint with Dohoon Choi, Min Lee, and Subong Lim.
16:40-17:30
Yeansu Kim (Chonnam National University)
Title: Langlands functoriality conjecture for GSpin groups in the positive characteristic case and its applications
Abstract: Langlands functoriality conjecture is one main conjecture in the Langlands program. The conjecture describes the relationship between automorphic representations of two related dif-ferent groups. We introduce the case of a globally generic cuspidal automorphic representation of GSpin groups in positive characteristic case. There are two main tools: the Langlands-Shahidi L-functions and the generic unitary duals. First tool is available due to Lomeli and we further develop several properties of those L-functions. The second tool is included in the current project. We further describe local Langlands correspondence in those cases. This is work in progress with Castillo and Lomeli.
January 10th (Wed)
09:30-10:20
Ming-Lun Hsieh (National Taiwan University)
Title: Yoshida congruence and Rankin-Selberg L-functions
Abstract: We report on our progress towards main conjectures for Rankin-Selberg convolutions via the congruence between Yoshida lifts and non-Yoshida lifts on GSp(4). We give a construction of a particular Hida family of Yoshida lifts and its non-vanishing modulo p. We explain the general strategy of using the congruence between this Hida family and other Λ-adic Siegel modular forms to construct non-trivial element in Selmer groups of Rankin-Selberg convolutions. This talk is based joint works with Zheng Liu.
10:50-11:40
Dohyeong Kim (Seoul National University)
Title: A Diophantine application of Iwasawa’s conjecture
Abstract: Iwasawa’s conjecture states that for a prime p and a number field F , the µ-invariant vanishes for the cyclotomic Zp-extension of F . In this talk, I will explain how Iwasawa’s conjecture implies the dimension hypothesis in the nonabelian Chabauty program. Since Iwasawa’s conjecture in full generality remains unsolved, this proof is conditional. Although there are other conditional proofs, ours has a practical advantage in that it allows one to verify the dimension hypothesis for a given curve. We exploit this to generate some numerical examples.
11:50-12:40
Liang-Chung Hsia (National Taiwan Normal University)
Title: On Artin-Mazur zeta functions associated to non-archimedean dynamics
Abstract: In this talk, our aim is to discuss the Artin-Mazur zeta functions associated to dynamical systems over a non-archimedean field. To be precise, let K be a non-archimedean field which is complete with respect to a discrete valuation and let f(z) ∈ K[z] with degree deg(f) ≥ 2. Then, f induces a (discrete) dynamical system by acting on P1(K). Assume that the “K-rational Julia set” Jf (K) is nonempty, then Jf (K) is invariant under the action of f. We’re interested in the Artin-Mazur zeta function associated to the dynamical system arising from the restriction of f on Jf (K). We will show that the Artin-Mazur zeta function is a rational function provided that the set of critical points of f that are in Jf (K) have finite orbits under the action of f. This is a joint work with Hongming Nie and Chenxi Wu.
January 11th (Thu)
09:30-10:20
Yifan Yang (National Taiwan University)
Title: CONFORMAL METRICS ON THE RIEMANN SPHERE AND MODULAR DIFFERENTIAL EQUATIONS
Abstract: In this talk we consider the problem of existence of conformal metrics of constant curvature on the Riemann sphere with conic singularities of given angles at given points. In some cases we will see that this problem can be solved using modular forms. This talk is based on the joint works with Zhijie Chen and Chang-Shou Lin.
10:50-11:40
Subong Lim (Sungkyun Kwan University)
Title: SCHNEIDER-SIEGEL THEOREM FOR HARMONIC MAASS FORMS
Abstract: Hermite and Lindemann proved that values of expenential function at nonzero alge-braic points are transcendental. After this, it was proved by Schneider and Siegel that j-invariant takes transcendental values at non-CM algebraic points. In this talk, we explain an extension of this result to harmonic Maass forms.
13:00-13:50
Title: Anderson t-motives - dimensions of cohomology groups
Abstract: Anderson t-motives are high-dimensional generalizations of Drinfeld modules. Also, Anderson t-motives of dimension $n$ and rank $r$ are the finite characteristic analogs of abelian varieties with multiplication by an imaginary quadratic field, of dimension $r$ and signature $(n, r-n)$. We give an algorithm of calculation of their (co)homology groups $H^1$, $H_1$; we show that unlike the existence of the canonical pairing between $H^1$ and $H_1$, their dimensions (which are $\le r$) can be different.
14:20-15:10
Seokho Jin (Chung-Ang University)
Title: On modularity of some combinatorial sums arising from partitions
Abstract: In this paper, We give an infinite family of examples of combina-torial sums in fact related to meromorphic Jacobi forms and Maass forms. This is on the modularity completion of the k-rank-generating function gener-alizing Kathrin Bringmann’s previous result on the ranks of partitions. One interesting point is that in the “non-holomorphic” part the notion of gener-alized Rogers-Ramanujan functions emerges in a natural way. This work is a joint work with Sihun Jo of Woosuk university.
15:40-16:30
Title: Murmurations of holomorphic modular forms in the weight aspect
Abstract: In April 2022, He, Lee, Oliver and Pozdnyakov made an interesting discovery using ma-chine learning - a surprising correlation between the root numbers of elliptic curves and the coefficients of their L-functions. They coined this correlation ‘murmurations of elliptic curves’. Naturally, one might wonder whether we can identify a common thread of ‘murmurations’ in other families of L-functions. In this talk, I will introduce a joint work with Jonathan Bober, Andrew R. Booker and David Lowry-Duda, demonstrating murmurations in holomorphic mod-ular forms in the weight aspect.