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We introduce the concept of quadratic modular symbol and study how these symbols are related to quadratic $p$ -adic $L$ -functions. These objects were introduced in [3] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic $p$ -adic $L$ -functions to more general Shimura curves.

Quadratic polynomials, period polynomials, and Hecke operators

Marie Jameson, Wissam Raji (2013)

Acta Arithmetica

For any non-square 1 < D ≡ 0,1 (mod 4), Zagier defined $F_{k} (D; x) : = \sum_{\begin{matrix} a, b, c \in ℤ, a < 0 \\ b^{2} - 4 a c = D \end{matrix}} m a x (0, {(a x^{2} + b x + c)}^{k - 1})$ . Here we use the theory of periods to give identities and congruences which relate various values of $F_{k} (D; x)$ .

Quadratic Relations for Generators of Units in the Modular Function Field.

Dan Kubert (1977)

Mathematische Annalen

Quadratische Formen und Modulfunktionen

M Eichler (1958)

Acta Arithmetica

Quantitative analysis of the Satake parameters of GL₂ representations with prescribed local representations

Yuk-Kam Lau, Charles Li, Yingnan Wang (2014)

Acta Arithmetica

We investigate the vertical version of the Sato-Tate conjecture for some GL₂ automorphic representations over totally real fields with specified local components at a finite set of finite places.

Quantitative spectral gap for thin groups of hyperbolic isometries

Michael Magee (2015)

Journal of the European Mathematical Society

Let $Λ$ be a subgroup of an arithmetic lattice in $SO (n + 1, 1)$ . The quotient $ℍ^{n + 1} / Λ$ has a natural family of congruence covers corresponding to ideals in a ring of integers. We establish a super-strong approximation result for Zariski-dense $Λ$ with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).

Quantum ergodicity of Eigenfunctions on $P S L_{2} (𝐙) / 𝐇^{2}$

Wenzhi Luo, Peter Sarnak (1995)

Publications Mathématiques de l'IHÉS

Quantum unique ergodicity for Eisenstein series on $P S L_{2} (ℤ ∖ P S L_{2} (ℝ)$

Dmitry Jakobson (1994)

Annales de l'institut Fourier

In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on $P S L_{2} (ℤ) ∖ P S L_{2} (ℝ)$ . This generalizes a recent result of W. Luo and P. Sarnak who proves equidistribution for $P S L_{2} (ℤ) ∖ ℍ$ . The averaged versions of these results have been proven by Zelditch for an arbitrary finite-volume surface, but our proof depends essentially on the presence of Hecke operators and works only for congruence subgroups of $S L_{2} (ℝ)$ . In the proof the key estimates come from applying...

Quantum variance for Hecke eigenforms

Wenzhi Luo, Peter Sarnak (2004)

Annales scientifiques de l'École Normale Supérieure

Quasimodular forms: an introduction

Emmanuel Royer (2012)

Annales mathématiques Blaise Pascal

Quasimodular forms were the heroes of a Summer school held June 20 to 26, 2010 at Besse et Saint-Anastaise, France. We give a short introduction to quasimodular forms. More details on this topics may be found in [1].

Quasimodular forms and Poincaré series

Min Ho Lee (2009)

Acta Arithmetica

Quasimodular forms and quasimodular polynomials

Min Ho Lee (2012)

Annales mathématiques Blaise Pascal

This paper is based on lectures delivered at the Workshop on quasimodular forms held in June, 2010 in Besse, France, and it provides a survey of some recent work on quasimodular forms.

Quasi-modular forms attached to elliptic curves, I

Hossein Movasati (2012)

Annales mathématiques Blaise Pascal

In the present text we give a geometric interpretation of quasi-modular forms using moduli of elliptic curves with marked elements in their de Rham cohomologies. In this way differential equations of modular and quasi-modular forms are interpreted as vector fields on such moduli spaces and they can be calculated from the Gauss-Manin connection of the corresponding universal family of elliptic curves. For the full modular group such a differential equation is calculated and it turns out to be the...

Quasi-periodic functions and Drinfeld modular forms

Ernst-Ulrich Gekeler (1989)

Compositio Mathematica

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