L-functions and Motives in Niseko 2026

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

Payman Eskandari, TBA

Ryotaro Harada, TBA

Yasuhiro Ishitsuka, Exponential sums on singular binary forms

Bruno Kahn, TBA

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

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

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

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

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

• Hayato Kanno (Tohoku University), Algebra of Multiple Eisenstein Series

• 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), On the properties of quasi-highly Kummer-faithful fields

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