Modular group

Summary

In mathematics, the modular group is the projective special linear group \operatorname{PSL}(2,\mathbb Z) 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. Definition The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane, which have the form :z\mapsto\frac{az+b}{cz+d}, where a, b, c, d are integers, and ad − bc = 1. The group operation is function composition. This group of transformations is isomorphic to the projective special linear group PSL(2, Z), which is the quotient of the 2-dimensional special linear group SL(2, Z) over the integers by its center {I, −I}. In other words, PSL(2, Z

Official source

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

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This year's topic is "Adelic Number Theory" or how the language of adeles and ideles and harmonic analysis on the corresponding spaces can be used to revisit classical questions in algebraic number theory.

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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

Trace coordinates of Teichmüller space of Riemann surfaces of signature $(0,4)$

Thomas Gauglhofer, Klaus-Dieter Semmler

Summary: We explicitly give

calT

, the Teichmüller space of four-holed spheres (which we call

X

pieces) in trace coordinates, as well as its modular group and a fundamental domain for the action of this group on

calT

which is its moduli space. As a consequence, we see that on any hyperbolic Riemann surface, two closed geodesics of lengths smaller than

2operatornamearccosh(2)

intersect at most once; two closed geodesics of lengths smaller than

2operatornamearccosh(3)

are both non-dividing or intersect at most once.

2005

We study p-adic families of cohomological automorphic forms for GL(2) over imaginary quadratic fields and prove that families interpolating a Zariski-dense set of classical cuspidal automorphic forms only occur under very restrictive conditions. We show how to computationally determine when this is not the case and establish concrete examples of families interpolating only finitely many Bianchi modular forms. (C)& 2021 Elsevier Inc. All rights reserved.

ACADEMIC PRESS INC ELSEVIER SCIENCE2022