Iwasawa theory for Asai L-functions

2021 Fiscal Year Final Research Report

• Project/Area Number: 17K14174
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Kyushu University (2018-2021); Tokyo Denki University (2017)
• Principal Investigator: Namikawa Kenichi 九州大学, 数理学研究院, 助教 (10757066)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Keywords: 岩澤理論 / L関数 / 保型表現論 / p進L関数

Outline of Final Research Achievements

We studied about period integrals for Asai representations and Rankin-Selberg products in an explicit manner. In particular, we constructed theta series on GSp(4) and we proved the Bessel period formulas and inner product formulas for these theta series, which give fundamental properties for modular Iwasawa theory for Asai representations. We also constructed p-adic Asai L-functions for GL(2) if the base field is a CM field. Furthermore, considering an analogue of Asai representations, we proved an explicit period integral formulas for Rankin-Selberg L-functions and we clarify a motivic background of Whittaker periods for GL(n) under standard conjectures such as an existence of motives corresponding automorphic representations.

Free Research Field

数論

Academic Significance and Societal Importance of the Research Achievements

保型形式に関する周期積分の研究は, L関数の特殊値の研究において基本的である. 本研究により, 特殊値の数論的性質を調べるために周期積分の明示的な公式をいくつか得ることが出来た. これらはとくに浅井表現, Rankin-Selberg積のL関数に関する今後の岩澤理論的な研究にとっての基礎となっており, 今後のこの方面への研究の発展が期待される. また本研究の手法を他の表現, 代数群に対して適用することも興味深い.