L-functions and Motives in Niseko 2026

Hohto Bekki, Regulators and L-values of some Fermat hypersurfaces

The Beilinson conjectures predict that (the transcendental part of) special values of L-functions of motives are described by their regulators. Motivated by the work of Otsubo, who gave explicit computations of the regulators of Fermat curves, we aim to find some explicit formulas for the regulators of higher-dimensional Fermat hypersurfaces. In this talk, I would like to report on some curious integral representations of $\zeta(3)$, the Catalan constant, an L-value of a certain Asai L-function of a Hilbert modular form, etc., that we observed in the course of this project. This talk is based on ongoing joint projects with Lambert A'Campo, Steven Charlton, Aleksander Horawa, Finnley Paolella, Constantin Torp, and Antonia Zerbs.

Payman Eskandari, Mixed motives and linear forms in Catalan’s constant

According to a philosophy proposed by Francis Brown, geometric constructions of small motives with a given period might be useful in studying arithmetic properties of the period. In this talk, we follow this philosophy for Catalan’s constant G, which is the alternating sum of the reciprocals of squares of odd positive integers (1-1/9+1/25-…). We will describe a geometric construction of a 2-dimensional mixed motive over the field of rational numbers that has G as a period. The construction leads to a supply of linear forms in 1 and G. We will discuss some rational approximations of G that naturally arise from this motive, and compare them with some other rational approximations of G constructed previously by Rivoal, Zudilin, and Nesterenko via hypergeometric functions. The talk is partially based on joint work with Kumar Murty and Yusuke Nemoto.

Ryotaro Harada, On the t-motivic interpretation of special values of Thakur’s hypergeometric function

In 1995, Thakur introduced positive characteristic analogues of hypergeometric functions and established several of their properties, including specializations to the Carlitz exponential and Bessel–Carlitz functions, contiguous relations, and summation formulas. However, it remained unknown whether these functions were related to t-motives. In this talk, I will present a t-motivic interpretation of Thakur’s hypergeometric functions at algebraic points. As applications of this interpretation and Chang’s refined version of the Anderson–Brownawell–Papanikolas criterion, we establish linear independence results for the hypergeometric functions associated with certain distinct parameters and algebraic points. Furthermore, if time permits, I will discuss my recent observation concerning the algebraic independence of these values.

Yasuhiro Ishitsuka, Exponential sums on singular binary forms

I will present recent results on exponential sums taken over certain families of singular binary forms of a fixed degree. These exponential sums arose in joint work with Taniguchi, Thorne, and Xiao on the 2-Selmer groups of elliptic curves with almost-prime discriminants. In this talk, I will treat the estimation and explicit evaluation of these exponential sums as a problem in its own right, and present the main results together with some of the ideas underlying their computation.

Bruno Kahn, On the standard conjecture B for abelian schemes over a curve

Yves André has shown that the conjecture of  the title, for all such abelian schemes over the complex numbers, implies the Hodge conjecture for all complex abelian varieties. I will give several reformulations, hoping that some may be used towards a proof. 

Hidenori Katsurada, Harder’s conjecture

Harder’s conjecture asserts that the Fourier coefficients of a normalized Hecke eigenform f of one variable are related modulo a certain prime ideal to the Hecke eigenvalues of a Hecke eigenform of degree 2. This conjecture is interesting in its own right and plays a crucial role in constructing torsion elements of the Bloch-Kato Selmer group associated with f. It was proposed by G. Harder in 2003, arising from his profound considerations on the Eisenstein cohomology of Siegel modular varieties. Despite various attempts by many mathematicians, the only proven example of this conjecture around 2019 was that by G. Chenevier and J. Lannes. One of the reasons making this conjecture difficult to prove is that the Harder conjecture is not a conjecture about the congruence between two Hecke eigenforms in the space of automorphic forms of the same weight. In this talk, we reformulate this conjecture as a congruence between the lifts of two Hecke eigenforms belonging to the space of the same weight, aiming to provide a clue to solving the original conjecture. As an application, we prove the Harder conjecture in a certain case. This is a joint work with Hiraku Atobe, Masataka Chida, Tomoyoshi Ibukiyama and Takuya Yamauchi. If time permits, we will also discuss related topics.

Shu Kawaguchi, Reflective modular forms on the moduli space of Eisenstein K3 surfaces and analytic torsions

Using equivariant analytic torsion, we systematically construct reflective modular forms on the coarse moduli spaces of Eisenstein K3 surfaces, which are complex ball quotients. As a corollary, many of these moduli spaces are quasi-affine. We will also discuss the coarse moduli spaces of some other K3 surfaces. This is joint work in progress with Ken-Ichi Yoshikawa.

Naho Kawasaki, On weighted sums for multiple zeta values of level 2

There is a well-known dimension conjecture for the Q-vector spaces spanned by multiple zeta values of fixed weight. Multiple T-values are level 2 analogues of multiple zeta values, and analogous dimension conjectures exist for them. Unlike in the multiple zeta value case, however, not enough relations are currently known to reduce the number of generators to the conjectured dimension in each weight. Weighted sum formulas for multiple T-values were obtained in depth 2 by Kaneko--Tsumura and in depth 3 by Berger--Chandra--Jain--Xu--Xu--Zhao. In this talk, we present weighted sum formulas in depth 4. This is joint work with Hikaru Sata.

Yukako Kezuka, On the structure of anticyclotomic local units and CM elliptic curves

Rubin's work on local units in anticyclotomic extensions provides a fundamental local ingredient in the Iwasawa theory of CM elliptic curves. More recently, Burungale, Kobayashi, and Ota, along with Yan and Zhu, used these ideas to establish the direct-sum decomposition of these local units for all odd supersingular primes. The aim of this talk is to explore the structure of these local units at the remaining prime p=2 and to discuss its applications to CM Iwasawa theory. This is based on ongoing joint work with Ashay Burungale. 

Yusuke Nemoto, Elements in K_4 and regulator maps of Fermat curves

Algebraic $K$-theory is a fundamental theory that connects algebraic geometry and number theory. For example, it enters into the Beilinson conjecture on the special values of $L$-functions of varieties (or, more generally, motives) over number fields. In this talk, we construct explicit elements in the group $K_4$ of the Fermat curves $x^N+y^N=1$ for all $N \geq 3$. The construction, which is uniform in $N$, uses polylogarithmic complexes and a map of de Jeu to $K$-theory. We prove that the elements are non-trivial by showing that their images under Beilinson's regulator map are non-zero. Notably, we obtain explicit formulas for their regulator integrals involving special values of Zagier's trilogarithm function. As a corollary, we show that these regulator integrals are asymptotic to $\frac32 \zeta(3)N^2$ as $N \to + \infty$. Moreover, we numerically verify some cases of Beilinson's conjectures on special values of $L$-functions at $s=3$ for $N \in \{3, 4, 6 \}$. This is a joint work with F. Brunault and D. Lilienfeldt.

Yoshiaki Okumura, Torsion of A-motives with values in positive characteristic cyclotomic towers

In function field arithmetic, Drinfeld modules and abelian Anderson modules play a role of elliptic curves and higher-dimensional abelian varieties. Unlike abelian varieties, such objects can be embedded fully faithfully into the category of A-motives, and to this end, we can investigate the finiteness of torsion points of them through the Galois representations attached to A-motives. In this talk, for A-motives with small rank and good reduction defined over positive characteristic local fields, we show a finiteness theorem for Galois modules coming from A-motives and explain that it implies the finiteness of torsion points of good abelian Anderson modules with values in z-adic cyclotomic towers. This is an analogue of a theorem of Imai stating that abelian varieties with good reduction over p-adic fields have only finitely many torsion points with values in p-adic cyclotomic towers.

Shin-ichiro Seki, The ring of integers modulo infinitely large primes and transcendental numbers

In the ring of integers modulo infinitely large primes, finite algebraic numbers defined by Rosen and finite multiple zeta values defined by Kaneko and Zagier have been studied as periods. Transcendental number theory in this ring is also of considerable interest, and a few papers on this topic have appeared recently. In particular, Anzawa--Funakura and Luca--Zudilin studied the transcendence of numbers defined from q-Fibonacci sequences and numbers defined from traces of Frobenius of elliptic curves over the field of rational numbers. In the recent work, we obtain refined results on the transcendence of these numbers, improving upon previous research. We also prove the transcendence of several numbers that had not been treated in earlier work. In this talk, I will present these results. This is joint work with Toshiki Matsusaka (Kyushu University).

Fumiaki Suzuki, Potential vanishing of degree 3 unramified cohomology over finite fields

The degree 3 unramified cohomology group H^3_{nr}(k(X)/k,\Q_\ell/\Z_\ell(2)) is a birational invariant that plays an important role in the study of codimension 2 cycles. Over the complex numbers, this group may be infinite, and a simple example is provided by the triple self-product of the Fermat cubic curve, due to Schoen and Scavia. We show that over a finite field k of characteristic different from 3, if \ell > 3 and k is sufficiently large, the degree 3 unramified cohomology group of this product vanishes. We then prove that an analogous result for every smooth projective variety over any finite field of characteristic > 2 follows from the Tate conjecture. This is joint work with Federico Scavia.

Sho Yoshikawa, Diophantine stability and modularity of elliptic curves

Establishing the modularity of elliptic curves over totally real number fields is an important problem in arithmetic geometry, closely connected with the Langlands program. In this talk, I will explain a method for constructing many non-abelian totally real fields over which every elliptic curve is known to be modular. The starting point is a modularity criterion obtained by combining powerful modularity theorems of Kisin, Thorne, and others with an observation from ongoing work with Masataka Chida and Tetsushi Ito concerning the torsion of the modular curve X_0(15) over totally real extensions. Roughly speaking, this reduces the modularity problem to controlling the Mordell–Weil rank of X_0(15). We study this rank in dihedral extensions. Combining ideas from the diophantine-stability results of Mazur–Rubin and Pathak–Ray with class field theory, we construct infinitely many totally real dihedral extensions in which the Mordell–Weil rank does not grow. The modularity criterion then yields infinite families of non-abelian totally real fields over which every elliptic curve is modular. 

Place : Setsu Niseko, Ground Floor, "Park 90"

Speakers and Titles : 

• Kenichi Bannai (Keio University/RIKEN), Stochastic processes via stochastic points and the Borel topos

• Sudipa Das (Harish-Chandra Research Institute),  Galois module structure of square root of inverse different

• Hayato Kanno (Tohoku University), Algebra of multiple Eisenstein series

• Satoshi Kumabe (Tokyo University of Science), Algebraic hypergeometric functions over finite fields

• Itsuki Nakamura (Institute of Science Tokyo), p-adic double L-functions with tame Dirichlet characters

• Ryo Negishi (Rikkyo University), GKZ systems and Gauss-Manin connections of projective hypersurfaces

• Eisuke Otsuka (Tohoku University), Algebraic construction of polylogarithms on higher-genus Riemann surfaces

• Ken Sato (Chiba University), On symplectic action on higher Chow cycles

• Asuka Shiga (Tohoku University), Abundance and rarity of BSD twins of elliptic curves

• Densuke Shiraishi (National Institute of Technology, Kagawa College), On ℓ-adic Galois multiple polylogarithms and their Landen-type formulas

• Yoshiaki Yamamura (Hokkaido University), Finiteness of ℓ-primary torsion of semi-abelian varieties over higher local fields

• Mahiro Yokomizo (Tohoku Univeresity), Iterated integrals on Fermat curve

Schedule : 

• Monday 14 : Das, Kanno, Nakamura, Negishi

• Tuesday 15 : Otsuka, Shiga, Yamamura, Yokomizo

• Thursday 17 :  Bannai, Kumabe, Sato, Shiraishi