The 11th(2024) NCTS–POSTECH-PMI Joint Workshop on Number Theory

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

Youngmin Lee (KIAS)

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.

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.

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

Dmitry Logachev (POSTECH)

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

Min Lee (University Bristol)

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.