Modular form

Summary

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. Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group \mathrm{SL}_2(\mathbb Z) \subset \mathrm{SL}_2(\mathbb R). The term "modular form", as a systematic description, is usually attributed to Hecke. Each modular form is attached to a Galois representation. Definition In general, given a subgroup \Ga

Official source

https://en.wikipedia.org/wiki/Modular_form

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In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group G to the complex numbers (or complex vector space) which is invariant under the action

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Analytic twists of modular forms and applications

Alexandre François Peyrot

We are interested in the study of non-correlation of Fourier coefficients of Maass forms against a wide class of real analytic functions. In particular, the class of functions we are interested in should be thought of as some archimedean analogs of Frobenius trace functions. In the first part of the thesis, we give an axiomatic definition for this class, and prove that these functions satisfy properties similar to that of Frobenius trace functions. In particular, we prove non-correlation statements analogous to those given by Fouvry, Kowalski and Michel for algebraic trace functions. In the second part of the thesis, we establish the existence of large values of Hecke-Maass L-functions with prescribed argument. In studying these problems, one encounters sums of Fourier coefficients of Maass forms against real oscillatory functions. In some cases, one can prove that these functions satisfy the axioms discussed previously.

EPFL2017

Analytic twists of modular forms

Alexandre François Peyrot

POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN2018

Trace coordinates of Teichmüller spaces of Riemann surfaces

Thomas Gauglhofer

Using an algebraic formalism based on matrices in SL(2,R), we explicitly give the Teichmüller spaces of Riemann surfaces of signature (0,4) (X pieces), (1,2) ("Fish" pieces) and (2,0) in trace coordinates. The approach, based upon gluing together two building blocks (Q and Y pieces), is then extended to tree-like pants decomposition for higher signatures (g,n) and limit cases such as surfaces with cusps or cone-like singularities. Given the Teichmüller spaces, we establish a set of generators of their modular groups for signatures (0,4), (1,2) and (2,0) in trace coordinates using transformations acting separately on the building blocks and an algorithm on dividing geodesics. The fact that these generators act particularly nice in trace coordinates gives further motivation to this choice (rather then the one of Fenchel-Nielsen coordinates). This allows us to solve the Riemann moduli problem for X pieces, "Fish" pieces and surfaces of genus 2; i.e. to give the moduli spaces as the fundamental domains for the action of the modular groups on the Teichmüller spaces. In this context, we also give an algorithm deciding whether two Riemann surfaces of signatures (0,4), (1,2) or (2,0) given by points in the Teichmüller space are isometric or not. As a consequence, we show the following two results concerning simple closed geodesics: On any purely hyperbolic Riemann surface (containing neither cusps nor cone-like singularities), the longest of two simple closed geodesics that intersect one another n times is of length at least ln, a sharp constant independent of the surface. We explicitly give ln for n = 1,2,3 and study its behaviour when n goes to infinity. X pieces are spectrally rigid with respect to the length spectrum of simple closed geodesics.

EPFL2006