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2017 | Buch

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L-Functions and Automorphic Forms

LAF, Heidelberg, February 22-26, 2016

herausgegeben von: Prof. Dr. Jan Hendrik Bruinier, Prof. Winfried Kohnen

Verlag: Springer International Publishing

Buchreihe : Contributions in Mathematical and Computational Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book presents a collection of carefully refereed research articles and lecture notes stemming from the Conference "Automorphic Forms and L-Functions", held at the University of Heidelberg in 2016.

The theory of automorphic forms and their associated L-functions is one of the central research areas in modern number theory, linking number theory, arithmetic geometry, representation theory, and complex analysis in many profound ways.

The 19 papers cover a wide range of topics within the scope of the conference, including automorphic L-functions and their special values, p-adic modular forms, Eisenstein series, Borcherds products, automorphic periods, and many more.

Inhaltsverzeichnis

Frontmatter

Sturm-Like Bound for Square-Free Fourier Coefficients

Abstract

In this short article, we show the existence of an analogue of the classical Sturm’s bound in the context of the square-free Fourier coefficients for cusp forms of square-free levels. This number is a cut-off to determine a cusp form from its initial few square-free Fourier coefficients. We also mention some questions in this regard.

Pramath Anamby, Soumya Das

Images of Maass-Poincaré Series in the Lower Half-Plane

Abstract

In this note we extend integral weight harmonic Maass forms to functions defined on the upper and lower half-planes using the method of Poincaré series. This relates to Rademacher’s “expansion of zero” principle, which was recently employed by Rhoades to link mock theta functions and partial theta functions.

Nickolas Andersen, Kathrin Bringmann, Larry Rolen

On Denominators of Values of Certain L-Functions When Twisted by Characters

Abstract

We try to bound the denominators of standard L-functions attached to Siegel modular forms when we twist them by Dirichlet characters. Main tools are our modification of the doubling method (Böcherer et al., Ann. Inst. Fourier 50:1375–1443, 2000) together with its application to congruences by the method of Katsurada (Math. Z. 259:97–111, 2008) and integrality properties of Bernoulli numbers with characters.

Siegfried Böcherer

First Order p-Adic Deformations of Weight One Newforms

Abstract

This article studies the first-order p-adic deformations of classical weight one newforms, relating their fourier coefficients to the p-adic logarithms of algebraic numbers in the field cut out by the associated projective Galois representation.

Henri Darmon, Alan Lauder, Victor Rotger

Computing Invariants of the Weil Representation

Abstract

We propose an algorithm for computing bases and dimensions of spaces of invariants of Weil representations of \({\mathrm {SL}}_2(\mathbb {Z})\) associated to finite quadratic modules. We prove that these spaces are defined over \(\mathbb {Z}\), and that their dimension remains stable if we replace the base field by suitable finite prime fields.

Stephan Ehlen, Nils-Peter Skoruppa

The Metaplectic Tensor Product as an Instance of Langlands Functoriality

Abstract

We interpret the metaplectic tensor product construction of Mezo for the genuine representations of the Kazhdan-Patterson covering groups in terms of the L-group formalism of Weissman.

Wee Teck Gan

On Scattering Constants of Congruence Subgroups

Abstract

Let Γ be a congruence subgroup of level N. The scattering constant of Γ at two cusps is given by the constant term at s = 1 in the Laurent expansion of the scattering function of Γ at these cusps. Scattering constants arise in Arakelov theory when establishing asymptotics for Arakelov invariants of the modular curve associated to Γ, as the level N tends to infinity. More precisely, in the known cases, scattering constants essentially contribute to the leading term of the asymptotics for the self-intersection of the relative dualizing sheaf. In this article, we prove an identity relating the scattering constants of Γ to certain scattering constants of the principal congruence subgroup \(\overline {\Gamma }(N)\). Providing an explicit formula for the latter, in case that N = 2 or N ≥ 3 is odd and square-free, we thereby present a systematic way of computing the scattering constants of Γ in these cases.

Miguel Grados, Anna-Maria von Pippich

The Bruinier–Funke Pairing and the Orthogonal Complement of Unary Theta Functions

Abstract

We describe an algorithm for computing the inner product between a holomorphic modular form and a unary theta function, in order to determine whether the form is orthogonal to unary theta functions without needing a basis of the entire space of modular forms and without needing to use linear algebra to decompose this space completely.

Ben Kane, Siu Hang Man

Bounds for Fourier-Jacobi Coefficients of Siegel Cusp Forms of Degree Two

Abstract

We discuss and prove several estimates involving Peterrson norms of Fourier-Jacobi coefficients of Siegel cusp forms of degree two.

Winfried Kohnen, Jyoti Sengupta

Harmonic Eisenstein Series of Weight One

Abstract

In this short note, we will construct a harmonic Eisenstein series of weight one, whose image under the ξ-operator is a weight one Eisenstein series studied by Hecke (Math Ann 97(1):210–242, 1927).

Yingkun Li

A Note on the Growth of Nearly Holomorphic Vector-Valued Siegel Modular Forms

Abstract

Let F be a nearly holomorphic vector-valued Siegel modular form of weight ρ with respect to some congruence subgroup of \(\mathrm {Sp}_{2n}({{\mathbb Q}})\). In this note, we prove that the function on \(\mathrm {Sp}_{2n}({\mathbb R})\) obtained by lifting F has the moderate growth (or “slowly increasing”) property. This is a consequence of the following bound that we prove: \(\|\rho (Y^{1/2})F(Z) \| \ll \prod _{i=1}^n (\mu _i(Y)^{\lambda _1/2} + \mu _i(Y)^{-\lambda _1/2})\) where λ ₁ ≥… ≥ λ _n is the highest weight of ρ and μ _i(Y ) are the eigenvalues of the matrix Y .

Ameya Pitale, Abhishek Saha, Ralf Schmidt

Critical Values of L-Functions for GL 3 ×GL 1 over a Totally Real Field

Abstract

We prove an algebraicity result for all the critical values of L-functions for GL₃ ×GL₁ over a totally real field F, which we derive from the theory of Rankin–Selberg L-functions attached to pairs of automorphic representations on GL₃ ×GL₂. This is a generalization and refinement of the results of Mahnkopf (J. Reine Angew. Math. 497:91–112, 1998) and Geroldinger (Ramanujan J. 38(3):641–682, 2015).

A. Raghuram, Gunja Sachdeva

Indecomposable Harish-Chandra Modules for Jacobi Groups

Abstract

We describe some indecomposable \(({\mathfrak {g}},{\mathrm {K}})\)-modules for Jacobi groups that admit an automorphic realization with possible singularities. A particular tensor product decomposition of universal enveloping algebras of Jacobi Lie algebras, which does not lift to the groups, allows us to study distinguished highest weight modules for the Heisenberg group. We encounter modified theta series as components of vector-valued Jacobi forms, whose arithmetic type is not completely reducible.

Martin Raum

Multiplicity One for Certain Paramodular Forms of Genus Two

Abstract

We show that certain paramodular cuspidal automorphic irreducible representations of \({\mathrm {GSp}}(4,\mathbb {A}_{\mathbb {Q}})\), which are not CAP, are globally generic. This implies a multiplicity one theorem for paramodular cuspidal automorphic representations. Our proof relies on a reasonable hypothesis concerning the non-vanishing of central values of automorphic L-series.

Mirko Rösner, Rainer Weissauer

Restriction of Hecke Eigenforms to Horocycles

Abstract

We prove a sharp upper bound on the L ²-norm of Hecke eigenforms restricted to a horocycle, as the weight tends to infinity.

Ho Chung Siu, Kannan Soundararajan

On the Triple Product Formula: Real Local Calculations

Abstract

We consider a triple of admissible representations π _j for j = 1, 2, 3 of \({\mathrm {GL}}_2(\mathbb R)\) of weights k _j with k ₁ ≥ k ₂ + k ₃. Test vectors are given, and using a formula of Michel-Venkatesh explicit values for local trilinear forms are computed for these vectors. Using this we determine the real archimedean local factors in Ichino’s formula for the triple product L-function. Applications both new and old to subconvexity, quantum chaos and p-adic modular forms are discussed.

Michael Woodbury

An Introduction to the Theory of Harmonic Maass Forms

Abstract

In this note we give a short introduction to the theory of harmonic Maass forms. We start by introducing modular forms and Maass forms and then present the notion of (vector valued) harmonic Maass forms as developed by Bruinier and Funke in [4]. We end by giving two recent applications of this theory.

Claudia Alfes-Neumann

Elementary Introduction to p-Adic Siegel Modular Forms

Abstract

We give an introduction to the theory of Siegel modular forms mod p and their p-adic refinement from an elementary point of view, following the lines of Serre’s presentation (J.-P. Serre, Formes modulaires et fonctions zeta p-adiques. In: Modular Functions of One Variable III. Lecture Notes in Mathematics, vol. 350. Springer, New York, 1973) of the case SL(2).

Siegfried Böcherer

Liftings and Borcherds Products

Abstract

This chapter serves as a brief introduction to the theory of theta-liftings with the main focus on Borcherds’ singular theta-lift and the construction of Borcherds products. Thus, after a few initial examples for liftings, we proceed to develop the tools needed to understand how the Borcherds lift works. Namely, we go through the construction of symmetric domains for orthogonal groups, introduce vector-valued modular forms and explain the definition of the Siegel theta-function. Then, we give a detailed treatment of the regularization recipe for the theta-integral and of the proof for the key properties of the additive lift: the location and type of its singularities. Finally, in the closing section, we sketch how to obtain a multiplicative lifting and the Borcherds’ products.

Eric Hofmann

Titel: L-Functions and Automorphic Forms
herausgegeben von: Prof. Dr. Jan Hendrik Bruinier
Prof. Winfried Kohnen
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-69712-3
Print ISBN: 978-3-319-69711-6
DOI: https://doi.org/10.1007/978-3-319-69712-3