The study of Galois deformations and an inductive structure of their automorphy

• Project/Area Number: 23740027
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Kagoshima University
• Principal Investigator: YAMAUCHI Takuya 鹿児島大学, 教育学部, 准教授 (90432707)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2014: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: ガロア表現 / 保型形式 / 保型性問題 / セール予想 / 法p保型形式 / GSp4 / 法pジーゲル保型形式 / Serre conjecture / Galois representations / Artin 予想 / 外積Ｌ関数 / Up作用素 / p安定化

Outline of Final Research Achievements

In this research project, the researcher carried out the program to study a correspondence between Galois representations (in the algebraic side) and automorphic forms (in the analytic side). Usually we understand it via p-adic Galois representations and it's reduction which is often called mod p representation. Then we have to divide the modularity problem into the type of mod p representation in question and plug them into the full modularity problem.
In this project He studied the property of modular mod p representation so that any such representation could be reduced to a manageable class of mod p representation.

Research Products

• [Journal Article] A conditional construction of Artin representations for real analytic Siegel cusp form sof weight (2,1) (2015)
  • Author(s): Henry H. Kim, and Takuya Yamauchi
  • Journal Title: Contemporary Mathematics Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Cusp forms for exceptional group of type E_7 (2015)
  • Author(s): Henry H. Kim, and Takuya Yamauchi
  • Journal Title: compositio math Volume: 印刷中
  • Peer Reviewed
• [Journal Article] On the rational K2 of a curve of GL2 type over a global field of positive characteristic. (2014)
  • Author(s): Chida, Masataka; Kondo, Satoshi; Yamauchi, Takuya
  • Journal Title: Journal of K-Theory Volume: 14
  • NAID: 120005676686
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Local newforms and formal exterior square L-functions (2013)
  • Author(s): Michitaka Miyauchi, Takuya Yamauchi
  • Journal Title: International Journal of Number Theory Volume: 9 Issue: 08 Pages: 1995-2010
  • DOI: 10.1142/s179304211350070x
  • Peer Reviewed
• [Presentation] The weight in Serre's conjecture for GSp4 (2015)
  • Author(s): 山内卓也
  • Organizer: 東北大学整数論セミナー
  • Place of Presentation: 東北大学理学研究科合同 A 棟 802 号室(仙台市)
  • Year and Date: 2015-03-18
  • Invited
• [Presentation] Cusp forms on E_7 (2014)
  • Author(s): 山内卓也 平成２６年１２月１日（月
  • Organizer: 金沢大学組合せ数学セミナー
  • Place of Presentation: 金沢大学自然科学研究棟５号館２階２３２室(金沢市)
  • Year and Date: 2014-12-01
  • Invited
• [Presentation] Artin representations for GSp(4) attached to real analytic Siegel modular forms (2013)
  • Author(s): Takuya Yamauchi
  • Organizer: University of Toronto Number Theory / Representation Theory Seminar Academic year 2013-2014
  • Place of Presentation: Paris (France)
  • Invited
• [Presentation] The L-function of some Siegel modular 3-folds and endoscopic lifts (2013)
  • Author(s): Takuya Yamauchi
  • Organizer: Seminar atJussieu in Paris
  • Place of Presentation: Toronto (Canada)
  • Invited
• [Presentation] Arithmetic Calabi-Yau families associated to generalized hypergeometric local systems and its applications (2013)
  • Author(s): 山内 卓也
  • Organizer: 第８回鹿児島代数解析幾何セミナー
  • Place of Presentation: 鹿児島大学理学部
  • Invited
• [Presentation] Endoscopic lift to the Siegel paramodular threefold related to Klein cubic threefold (2012)
  • Author(s): 山内 卓也
  • Organizer: University of Toronto Number Theory / Representation Theory Seminar
  • Place of Presentation: University of Toronto (Canada)
  • Invited
• [Presentation] L-function of some Siegel threefold of low level: non-neat case (2012)
  • Author(s): 山内卓也
  • Organizer: Workshop on p-adic arithmetic geometry and motives(招待講演)
  • Place of Presentation: Institute of Mathematics, Tohoku University, Kawai hall
• [Presentation] 超幾何層に付随するCalabi-Yau多様体とその応用 (2011)
  • Author(s): 山内卓也
  • Organizer: 第１４７回数理情報科学談話会(招待講演)
  • Place of Presentation: 鹿児島大学理学部棟2号館4階404号室（多目的ルーム）