arXiv subjects

Kenji Sakugawa

Publications and source records attributed to Kenji Sakugawa.

8 recordsLinked to original sources

arXiv2026

Symmetric multiple Eisenstein series

In this paper, we introduce the symmetric multiple Eisenstein series, a variant of the multiple Eisenstein series. As a fundamental result, we show that they satisfy the linear shuffle relation. As a case study, we investigate the vector space spanned by symmetric double Eisenstein series of weight $k$. When $k$ is even, it coincides with the space spanned by modular forms of weight $k$ and the derivative of the Eisenstein series of weight $k-2$. For $k$ odd, we prove that its dimension equals $\lfloor k/3\rfloor$. We further provide an explicit correspondence between the linear shuffle relation and the Fay-shuffle relation satisfied by elliptic double zeta values, which may be of independent interest. In connection with modular forms, we prove that every modular form can be expressed as a linear combination of symmetric triple Eisenstein series. This will serve as a first step toward understanding modular phenomena for symmeric multiple zeta values observed by Kaneko and Zagier.

math.NT↗

arXiv2024

On the infinite Frobenius action on de Rham fundamental groups of affine curves

We study the action of the infinite Frobenius on the de Rham fundamental groups of affine curves defined over $\bfR$. As an application, we compute extension classes of real mixed Hodge structures associated with the motivic fundamental groups of affine curves. In the case of modular curves, we relate our computation to special values of Rankin-Selberg L-functions, and show that the associated extensions of mixed Hodge structures are non-split. We compute local zeta integrals both at good primes and, in certain cases, at bad primes.

math.AG↗

arXiv2024

The depth-weight compatibility on the motivic fundamental Lie algebra and the Bloch-Kato conjecture for modular forms

Let $p$ be a prime number and let $V$ be a continuous representation of $\mathrm{Gal}(\overline {\mathbf Q}/\mathbf Q)$ on a finite dimensional $\mathbf Q_p$-vector space, which is geometric. One of the Bloch-Kato conjectures for $V$ predicts that the rank of the Hasse-Weil $L$-function of $V$ at $s=0$ coincides with the rank of Blcoh-Kato Selmer group of $V^\vee(1)$. In this paper, we prove that the depth-weight compatibility on the fundamental Lie algebra of the mixed Tate motives over $\mathbf Z$ implies the Bloch-Kato conjecture for the $p$-adic Galois representations associated with full-level Hecke eigen cuspforms.

math.NT↗

arXiv2024

Integrality of Hecke eigenvalues and the growth of Hecke fields

We prove that Hecke eigenvalues for any Hilbert and Siegel modular forms are algebraic integers. Our method does not rely on cohomologicality nor Galois representations. We apply the integrality of Hecke eigenvalues for Hilbert modular forms of non-parallel weight to the estimation of the growth of Hecke fields of Hilbert cusp forms with non-vanishing central $L$-values. As a further application, we give the growth of the fields of rationality of cuspidal automorphic representations of ${\rm GL}_{2d}(\mathbb{A}_\mathbb{Q})$ for a prime number $d$ with non-vanishing central $L$-values. We also apply the integrality of Hecke eigenvalues for holomorphic Siegel cusp forms of general degree in order to give the growth of the Hecke fields of those forms.

math.NT↗

arXiv2016

Finite and étale polylogarithms

We show an explicit formula relating étale polylogarithms introduced by Wojtkowiak and finite polylogarithms introduced by Elbaz-Vincent and Gangl. This formula is an étale analog of Besser's formula relating Coleman's $p$-adic polylogarithms and the finite polylogarithms.

math.NT↗

arXiv2016

On functional equations of finite multiple polylogarithms

Recently, several people study finite multiple zeta values (FMZVs) and finite polylogarithms (FPs). In this paper, we introduce finite multiple polylogarithms (FMPs), which are natural generalizations of FMZVs and FPs, and we establish functional equations of FMPs. As applications of these functional equations, we calculate special values of FMPs containing generalizations of congruences obtained by Meštrović, Z. W. Sun, Z. W. Sun-L. L. Zhao, and Tauraso-J. Zhao. We show supercongruences for certain generalized Bernoulli numbers and the Bernoulli numbers as an appendix.

math.NT↗

arXiv2015

Polylogarithmic analogue of the Coleman-Ihara formula, I

The Coleman-Ihara formula expresses Soule's $p$-adic characters restricted to $p$-local Galois group as the Coates-Wiles homomorphism multiplied by $p$-adic $L$-values at positive integers. In this paper, we show an analogous formula that $\ell$-adic polylogarithmic characters for $\ell=p$ restrict to the Coates-Wiles homomorphism multiplied by Coleman's $p$-adic polylogarithms at any roots of unity of order prime to $p$.

math.NT↗

arXiv2014

Construction of Hurwitz Spaces and Application to the Regular Inverse Problem

The author give a simple construction of Hurwitz spaces which is defined by Fried and Volklein, and generalize Hurwitz spaces. As a consequence of this construction, the author prove the regularities of the groups PSO^+_{n}(\mathbb F_{p^m}) if p is an odd prime which congruentes with 7 modulo 12, n is an even positive integer grater than 11 and m=1 or p is an odd prime which congruentes with 7 modulo 12, φ(p^m-1)/2+1\eqiv n/2 (\mod 2), p^m\equiv 3(\mod 4) and n>\max\{φ(p^m-1),7\}.

math.NT↗