Lecture

Petersson Inner Product and Hecke Operators

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In course

MATH-511: Number theory II.b - Modular forms

In this course we will introduce core concepts of the theory of modular forms and consider several applications of this theory to combinatorics, harmonic analysis, and geometric optimization. ...

Description

This lecture introduces the Petersson inner product, a fundamental concept in the theory of modular forms, and explores the definition and properties of Hecke operators, crucial tools in the study of modular forms as functions on lattices. The lecture also covers the translation of Hecke operators to the language of matrices and their normalization. Additionally, it discusses the equivalence of different definitions of Hecke operators and their implications on Fourier coefficients, providing insights into the structure and completeness of certain collections of matrices.

Instructor

Maryna Viazovska

Maryna Viazovska did her bachelor studies at the Kyiv National Taras ShevchenkoUniversity and completed her MSc at the Technical University Kaiserslautern.She obtained her PhD in 2013 in Bonn.She was a postdoctoral researcher at the Institut des Hautes Etudes Scientifiquesand at the Humboldt University of Berlin, and in 2017 was a Minerva DistinguishedVisitor at Princeton University. She joined EPFL in 2017 as Tenure-Track AssistantProfessor and was promoted Full Professor in 2018.

Official source

Ontological neighbourhood

Mathematics

Number theory: Topics in number theory

Related concepts (32)

Modular form

In mathematics, a modular form is a (complex) analytic function on the upper half-plane that satisfies: a kind of functional equation with respect to the group action of the modular group, and a growth condition. The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

Hecke operator

In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations. used Hecke operators on modular forms in a paper on the special cusp form of Ramanujan, ahead of the general theory given by .

Siegel modular form

In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties (with some extra level structure) should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups.

Modular group

In mathematics, the modular group is the projective special linear group of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and −A are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic. The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane, which have the form where a, b, c, d are integers, and ad − bc = 1.

Petersson inner product

In mathematics the Petersson inner product is an inner product defined on the space of entire modular forms. It was introduced by the German mathematician Hans Petersson. Let be the space of entire modular forms of weight and the space of cusp forms. The mapping , is called Petersson inner product, where is a fundamental region of the modular group and for is the hyperbolic volume form. The integral is absolutely convergent and the Petersson inner product is a positive definite Hermitian form.

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