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Preprint

• Takashi Hara and Tadashi Ochiai: On the p-adic L-function and Iwasawa main conjecture for an Artin motive over a CM field preprint   arXiv:2407.06983, accepted by Journal of the Mathematical Society of Japan.
  Abstract: For an algebraic Hecke character defined on a CM field F of degree 2d, Katz constructed a p-adic L-function of d+1+δ_F,p variables in his paper published in 1978, where δ_F,p denotes the Leopoldt defect for F and p. In the present article, we generalise the result of Katz under certain technical conditions (containing the absolute unramifiedness of F at p), and construct a p-adic Artin L-function of d+1+δ_F,p variables, which interpolates critical values of the Artin L-function associated to a p-unramified Artin representation of the absolute Galois group G_F. Our construction is an analogue over a CM field of Greenberg's construction over a totally real field, but there appear several new difficulties which do not matter in Greenberg's case.
• Takashi Hara, Kenji Sakugawa and Koji Tasaka: Symmetric multiple Eisenstein series in preparation
  Abstract (tentative): In this paper, we introduce and study the symmetric multiple Eisenstein series, a variant of the multiple Eisenstein series. As a fundamental result, we show that they satisfy the linear shuffle relations. As a case study, we determine the vector space spanned by symmetric double Eisenstein series of weight k. When k is even, this space 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 is [k/3], which coincides with the dimension of the space of elliptic double zeta values of weight k obtained by Matthes in 2017. As one explanation for this coincidence, we further provide an explicit correspondence between linear shuffle relations and Fay-shuffle relations 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 the “modular relations” for symmetric multiple zeta values, observed by Kaneko and Zagier.

Published / to be published

• Takashi Hara, Tadashi Miyazaki and Kenichi Namikawa: Uniform integrality of critical values of the Rankin–Selberg L-function for GL_n × GL_n-1 Representation Theory, 29, 527–615 (2025)  Link.
  Abstract: After introducing the notion of uniform integrality of critical values of the Rankin–Selberg L-functions for GL_n × GL_n-1, we study it when the base field is totally imaginary. For this purpose, we adopt specific models of highest weight representations of the general linear groups, construct the Eichler–Shimura classes for GL_n and GL_n-1 in an explicit manner, and then evaluate the cohomological cup product of them, by making the best use of Gel'fand–Tsetlin bases.
• Takashi Hara and Kenichi Namikawa: A motivic interpretation of Whittaker periods for GL_n manuscripta mathematica, 174, 303–353 (2024)   Link
  Abstract: Admitting the existence of conjectural motives attached to irreducible cohomological cuspidal automorphic representations of GL_n, we write down Raghuram and Shahidi's Whittaker periods in terms of Yoshida's fundamental periods when the base field is a totally real number field or a CM field.
• Takashi Hara and Kenichi Namikawa: A cohomological interpretation of archimedean zeta integrals for GL₃ × GL₂ Research in Number Theory, 7, Article Number:68 (2021),   Link
  Abstract: By studying an explicit form of the Eichler–Shimura map for GL₃, we describe a precise relation between critical values of the complete L-function for the Rankin–Selberg convolution GL₃ × GL₂ and the cohomological cup product of certain rational cohomology classes which are uniquely determined up to rational scalar multiples from the cuspidal automorphic representations under consideration. This refines rationality results on critical values due to Raghuram et al.
• Takashi Hara and Takahiro Kitayama: Character varieties of higher dimensional representations and splittings of 3-manifolds Geometriae Dedicata, 213, 433–466 (2021)   Link
  Abstract: In 1983 Culler and Shalen established a way to construct essential surfaces in a 3-manifold from ideal points of the SL₂-character variety associated to the 3-manifold group. We present in this article an analogous construction of certain kinds of branched surfaces (which we call essential tribranched surfaces) from ideal points of the SL_n-character variety for a natural number n greater than or equal to 3. Further we verify that such a branched surface induces a nontrivial presentation of the 3-manifold group in terms of the fundamental group of a certain 2-dimensional complex of groups.
• Takashi Hara and Tadashi Ochiai: The cyclotomic Iwasawa main conjecture for Hilbert cuspforms with complex multiplication Kyoto Journal of Mathematics, 58, No. 1, 1–100 (2018)   Link.
  Abstract: We deduce the cyclotomic Iwasawa main conjecture for Hilbert modular cuspforms with complex multiplication from the multivariable main conjecture for CM number fields. To this end, we study in detail the behaviour of the p-adic L-functions and the Selmer groups attached to CM number fields under specialisation procedures.
• Takashi Hara: Inductive construction of the p-adic zeta functions for non-commutative p-extensions of exponent p of totally real fields Duke Mathematical Journal, 158, No. 2, 247–305 (2011)   Link.
  Abstract: We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension of a totally real number field such that the finite part of its Galois group is a p-group of exponent p. We first calculate the Whitehead groups of the Iwasawa algebra and its canonical Ore localisation by using Oliver-Taylor's theory of integral logarithms. This calculation reduces the existence of the non-commutative p-adic zeta function to certain congruences between p-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne-Ribet's theory and a certain inductive technique. As an application we shall prove a special case of (the p-part of) the non-commutative equivariant Tamagawa number conjecture for critical Tate motives.
• Takashi Hara: Iwasawa theory of totally real fields for certain non-commutative p-extensions Journal of Number Theory, 130, Issue 4, 1068–1097 (2010)   Link.
  Abstract: In this paper, we will prove the non-commutative Iwasawa main conjecture ---formulated by John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha and Otmar Venjakob (2005)--- for certain specific non-commutative p-adic Lie extensions of totally real fields by using theory on integral logarithms introduced by Robert Oliver and Laurence R. Taylor, theory on Hilbert modular forms introduced by Pierre Deligne and Kenneth A. Ribet, and so on. Our results give certain generalization of the recent work of Kazuya Kato on the proof of the main conjecture for Galois extensions of Heisenberg type.