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.

在过去的四十年里，朗兰兹程序一直是数学研究的一个活跃领域，它包含了数论、表示论、分析和几何中连接对象的定理和猜想的庞大网络。朗兰兹程序的核心是自同构形式及其相关的l函数。这些对象产生于分析和表征理论。尽管取得了缓慢而稳定的进展，但关于自同构形式与数论/算术几何之间关系的许多基本问题仍未得到解答。本项目的目标是通过对以下三个关键主题的深入研究：a)近全纯Siegel模形式，b) Deligne关于临界l值的代数性猜想，c)功能相关自同构形式的Petersson范数比率，从而获得对这些问题的新见解。近全纯Siegel模形式是由Shimura首先提出的，对于研究自同构l函数的特殊值是必不可少的。然而，尽管它们无处不在，但它们的算术性质和在朗兰兹框架中的位置尚未被完全理解。本项目将研究它们的表示理论和算术性质，并证明它们与“向量值西格尔模形式”之间的密切联系。从近全纯模形式的研究中获得的见解将用于解决由菲尔兹奖获得者数学家皮埃尔·德列涅提出的一个著名猜想的一些特殊情况。Deligne猜想最简单的例子是一个经典的事实，即黎曼ζ函数在所有正偶数上的值是π乘以有理数的幂。本研究将证明更复杂的l函数的预期合理性结果，这些函数在数学的许多领域具有重要意义。本研究还旨在证明Ibukiyama和Katsurada关于某些Siegel模形式的Petersson范数比率的著名猜想。这个问题涉及到自同构形式在朗兰兹提升下的算术性质的行为，具有深刻的意义。解决这个问题需要对现有方法进行创新适应。在这个项目中使用的方法将是经典技术、现代表示理论方法和群上同学的有力结合。这项研究将为进一步探索开辟几条新的途径。

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