Arithmetic aspects of automorphic forms: Petersson norms and special values of L-functions

自守形式的算术方面：Petersson 范数和 L 函数的特殊值

• 批准号: EP/L025515/1
• 负责人: Abhishek Saha
• 金额: $ 11.67万
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2014
• 资助国家: 英国
• 起止时间: 2014 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FL025515%2F1
• 关键词: Arithmetic; aspects; automorphic; forms; Petersson

The Langlands program has been an area of active research in mathematics for the last forty years and consists of a vast web of theorems and conjectures connecting objects in number theory, representation theory, analysis and geometry. Central to the Langlands program are automorphic forms and their associated L-functions. These objects arise from analysis and representation theory. Despite slow and steady progress, many fundamental questions about the relationship between automorphic forms and number theory/arithmetic geometry remain unanswered. The goal of this project is gain new insights into some of these questions by making a deep investigation of three key topics: a) Nearly holomorphic Siegel modular forms, b) Deligne's conjecture on algebraicity of critical L-values, c) Ratios of Petersson norms for functorially related automorphic forms.Nearly holomorphic Siegel modular forms were first introduced by Shimura and have been indispensable for studying special values of automorphic L-functions. However, despite their ubiquity, their arithmetic properties and place in the Langlands framework have not yet been fully understood. This project will study their representation-theoretic and arithmetic properties and prove a close link between them and "vector valued Siegel modular forms".The insights gained from the study of nearly holomorphic modular forms will be used to tackle some special cases of a famous conjecture made by the Fields medal winning mathematician Pierre Deligne. The simplest example of Deligne's conjecture is the classical fact that the value of the Riemann zeta function at all positive even integers is a power of pi times a rational number. This research will prove the expected rationality results for much more complicated L-functions that are of great importance in several fields of mathematics. This research will also aim to prove a well-known conjecture made by Ibukiyama and Katsurada concerning ratios of Petersson norms for certain Siegel modular forms. This problem has deep significance as it concerns the behavior of arithmetic properties of automorphic forms under Langlands lifting. Solving this will require innovative adaptation to existing methods.The methods used in this project will be a powerful mix of classical techniques, modern representation theoretic methods, and group cohomology. This research will open several new avenues for further exploration.

在过去的四十年中，Langlands计划一直是数学研究的领域，由庞大的定理和猜想组成，这些定理和猜想连接了数字理论，表示理论，分析和几何形状中的对象。 Langlands计划的中心是自动形式及其相关的L功能。这些对象来自分析和表示理论。尽管进步缓慢而稳定，但许多基本问题关于自动形式与数字理论/算术几何形状之间的关系仍未得到答复。 The goal of this project is gain new insights into some of these questions by making a deep investigation of three key topics: a) Nearly holomorphic Siegel modular forms, b) Deligne's conjecture on algebraicity of critical L-values, c) Ratios of Petersson norms for functorially related automorphic forms.Nearly holomorphic Siegel modular forms were first introduced by Shimura and have been必不可少的，不需要研究自多态L功能的特殊值。然而，尽管它们无处不在，但它们的算术特性和在兰兰兹框架中的位置尚未完全理解。该项目将研究其代表性理论和算术特性，并证明它们与“矢量有价值的Siegel模块化形式”之间的紧密联系。从研究几乎全体态度模块化形式的研究中获得的见解将用于解决一些特殊的案例，以表明田野奖得主的冠军冠军数学冠军Pierredignee。 Deligne猜想的最简单例子是经典的事实，即Riemann Zeta函数的值根本甚至是正整数也是PI TIMES有理数的力量。这项研究将证明对更复杂的L功能的预期合理性结果在数学的几个领域都非常重要。这项研究还将旨在证明Ibukiyama和Katsurada对某些Siegel模块化形式的彼得森规范的比率。这个问题具有深厚的意义，因为它涉及兰兰德举重下自动形式的算术特性的行为。解决此问题将需要对现有方法的创新适应。该项目中使用的方法将是经典技术，现代表示理论方法和群体共同体的有力结合。这项研究将为进一步探索开放几种新途径。

项目成果

Hybrid sup-norm bounds for Maass newforms of powerful level

强大水平的马斯新形式的混合超范数界限

• DOI: 10.2140/ant.2017.11.1009
• 发表时间: 2017
• 期刊: Algebra & Number Theory
• 影响因子: 1.3
• 作者: Saha A

Representations of SL_2(R) and nearly holomorphic modular forms

SL_2(R) 和近全纯模形式的表示

• 发表时间: 2016
• 作者: Pitale A

Lowest weight modules of Sp4(R) and nearly holomorphic Siegel modular forms

Sp4(R) 的最低重量模块和近全纯 Siegel 模块化形式

• DOI: 10.1215/21562261-2021-0012
• 发表时间: 2021
• 期刊: Kyoto Journal of Mathematics
• 影响因子: 0.6
• 作者: Pitale A

Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

Böcherer 对无平方水平的西格尔模形式的猜想的显式改进

• DOI: 10.2969/jmsj/78657865
• 发表时间: 2020
• 期刊: Journal of the Mathematical Society of Japan
• 影响因子: 0.7
• 作者: DICKSON M

On sup-norms of cusp forms of powerful level

论强力级尖点形式的超范数

• DOI: 10.4171/jems/746
• 发表时间: 2017
• 期刊: Journal of the European Mathematical Society
• 影响因子: 2.6
• 作者: Saha A