Abstract
Noriyuki Otsubo: The adelic Gaussian hypergeometric function and Anderson-Ihara theory
This is a joint work with Masanori Asakura. The classical Gaussian hypergeometric functions
over the complex numbers and their analogues over ﬁnite ﬁelds are diﬀerent realizations of thesame motives, the former being the complex periods and the latter being the Frobenius traces.
Here we only consider those functions F (a, b; c; x) with c = 1. By the Euler-Gauss summation
formula and its ﬁnite ﬁeld analogue, their special values at x = 1 are respectively the betafunction and Jacobi sums. We give a deﬁnition of the adelic Gaussian hypergeometric function
using the adelic étale cohomology of the tower of such motives with respect to levels. Thisfunction interpolates all the Gaussian hypergeometric functions (with c = 1) over all ﬁnite ﬁelds.
Specializing at x = 1, we recover the adelic beta function of Anderson-Ihara; this is an adelic
analogue of the Euler-Gauss summation formula.
Masanori Asakura: Frobenius structure on hypergeometric equations, p-adic polygamma values and p-adic L-values
Recently, according to the idea of Dwork, Kedlaya provides certain formula on the Frobenius
structure on a hypergeometric equation for F (a1 , ..., an ; b1 , ..., bn ; z) with bi − bj ̸∈ Z. His formula
describes the Frobenius matrix on the equation in terms of a product of p-adic gamma values.
In this talk, we extend it to the cases b1 = ... = bs = 1, s ≤ n. In such a case, the Frobenius is nolonger described by p-adic gamma function, and then we describe it by the p-adic polygammafunctions. Since the p-adic polygamma values are linear combinations of p-adic L-values ofDirichlet characters, it turns out that the Frobenius is described by p-adic L-values. Our formula
has a geometric application. Let X be a smooth projective family over Gm − {1} such that thePicard-Fuchs equation is a hypergeometric equation. Then it follows from our formula that theFrobenius matrix on the log-crystalline cohomology of the ﬁber at z = 0 is described by p-adic
gamma values and p-adic L-values. If the ﬁber is a total degeneration (hence a mixed Tatemotive), then it can be described by p-adic L-values (no p-adic gamma value appears), this is adistinguished application from Kedlaya’s result. This is a joint work with Kei Hagihara.
Hironori Shiga: Back stage of K3-hypergeometric Arithmetic-Geometric Means
Roman time Platon-school philosopher Plotinus said : the Prima Causa of the world is “the1 . But the European civilization didn’t ﬁnd “0 . This concept goes back to the Indian Buddhist
philosopher Nagarjuna (c.150 – c.250). These two numbers are the units of two operation systemsof C, and they (+ and ·) are linkaged by the exponential function.
Many interesting number theoretic hypotheses (like the ABC conjecture) appear as the incarnation of the mystery of this linkage. The speaker believes the Gauss AGM theorem is anotherincarnation. So he is eager to make up a deep and detailed story of AGM-type theorems especiallycoming from hypergeometric functions (those are extensions of the exponential function).
In this talk he tells about the backstage of new coming AGM theorems induced from a familyof K3 surfaces on the 4-dimensional Hermitian domain(∼= the 4-dimensional type IV domain).
Fuetaro Yobuko: Quasi-F -splitting and klt singularities
Quasi-F -splitting is an extension of the notion of F -splitting, which is deﬁned for schemesof characteristic p. For K3 surfaces, F -splitting is equivalent to ordinary and quasi-F -splitting
is equivalent to nonsupersingular. These notions are applied to singularities. It is known thattwo dimensional klt singularities are F-split if the characteristic p is greater than 5, but someof them are not when p is 2, 3 or 5. In this talk, I will present a joint work with Tatsuro
Kawakami, Teppei Takamatsu, Hiromu Tanaka, Jakub Witaszek and Shou Yoshikawa showingthat two dimensional klt singularities are quasi-F -split in any characteristic.
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Shuji Yamamoto: On the Ohno-Zagier type formulasFirst I review the Ohno-Zagier formula, which represents the generating function of certain
sums of multiple polylogarithms, together with some generalizations. Then I show a proof ofit which only uses series manipulations, instead of the original one using diﬀerential equation.
Finally, if possible, I would like to discuss the reduction modulo p of the truncated series andthe relation to the ﬁnite multiple zeta values.
Masaki Kato: Functional relations for elliptic q-multiple polylogarithms
The elliptic q-multiple polylogarithm is a common generalization of elliptic and q-multiple
polylogarithms. It is also closely related to the double sine function introduced and studied
by Shintani and Kurokawa. In this talk, we study functional relations for elliptic q-multiple
polylogarithms. We establish p and q-diﬀerence relations and inversion formulas satisﬁed byelliptic q-multiple polylogarithms.
Furusho Hidekazu: p-adic hypergeometric function related with p-adic multiple polylogarithms
I introduce a (maybe new) p-adic analogue of Gauss hypergeometric function, which is constructed from p-adic multiple polylogarithms. I explain my method of a residue-wise analytic
prolongation of the function and then show that a p-adic analogue of Gauss hypergeometrictheorem holds.
Makoto Kawashima: Hypergeometric series and the Rodrigues’ formula
Hypergeometric series is a testing ground for phenomena in irrationality theory. In this talkwe shall discuss linear independence of values concerning hypergeometric series. Our results areobtained by a generalization of the classical Rodrigues’ formula for the Legendre polynomials.
Part of my talk is a joint work with S. David and N. Hirata.

Takeshi Saito: On the Hasse–Arf theorem
The classical Hasse–Arf theorem aﬃrms the integrality of the conductor of an abelian characterof the Galois group of a local ﬁeld. Kato reﬁned this in 1980’s by introducing a ﬁltration on
the dual group of the abelianized absolute Galois group by using the cup-product with values in
the Brauer group and proved a variant of the HA theorem in some cases where the extension of
the residue ﬁeld is inseparable. We reformulate the HA theorem as equivalent conditions for aninequality on the conductor to be an equality. We also discuss a generalization of Kato’s theoryof Swan conductors.
Ashay Burungale: The BSD conjecture for certain CM elliptic curves and applications
We describe a few results towards the BSD conjecture for CM elliptic curves over the last ﬁveyears. Some of the results have found applications to classical Diophantine problems such as thecube sum problem. (Joint with M. Flach, S. Kobayashi, K. Ota, C. Skinner, Y. Tian)

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Ken Sato: Higher Chow cycles on K3 surfaces and their images under the regulator map
The higher Chow group CHp (X, q) is a generalization of the classical Chow group CHp (X).
It satisﬁes many interesting properties, but its explicit structure is still mysterious for almost all
varieties when p is greater than 1.
In this talk, I will explain an explicit construction of non-trivial higher Chow cycles in
CH2 (X, 1) on a K3 surface X. Then we show that these cycles generate a subgroup in CH2 (X, 1)
whose rank is at least 18 for very general cases. More strongly, the image of the subgroup in
CH2 (X, 1)ind is also at least rank 18 for very general cases where CH2 (X, 1)ind is the quotient ofCH2 (X, 1) by the images of the intersection product maps.
The key for showing non-triviality of these cycles is calculating the image of the cycles underthe Beilinson regulator map. The images of the regulator map are related to a special values of acertain type of normal functions associated with the cycle families. These normal functions haveintegral representations and satisfy inhomogeneous diﬀerential equations which are analogous to
integral representations and diﬀerential equations of a product of hypergeometric functions.
Yu Katagiri: p-adic properties of division polynomials and algebraic sigma functions
For an elliptic curve over a ﬁeld, the n-th division polynomial Fn is deﬁned to be a rational
function whose zeros are the non-trivial n-torsion points and whose pole is the point at inﬁnity.
Silverman considered a sequence (Fn (P )) for a ﬁxed rational point P and proved that thissequence has a convergent subsequence in the case that the elliptic curve is deﬁned over a p-adicﬁeld. In this talk, we describe the limit of this subsequence explicitly using a special value of acertain sigma function.
Nobuo Tsuzuki: Congruence of Galois representations and application to mirror symmetries ofCalabi-Yau in Dwork’s families
We introduce a notion of congruence, “mod ℓ reciprocity", between two representations ofabsolute Galois group of number ﬁelds. If one of the images of Galois group is large enough,the two mod ℓ representations will be equivalent. We will give examples of mod ℓ reciprocity
in which one of two representations arises from mirror symmetries of Calabi-Yau varieties inarithmetic Dwork’s family. We will explain our motivation of the study. This is an ongoing workwith Takuya Yamauchi.

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