arXiv subjects

Martin Raum

Publications and source records attributed to Martin Raum.

At least 37 records · Page 2Linked to original sources

arXiv2022

The Bernstein-Gelfand Tensor Product Functor and the Weight-2 Eisenstein Series

The Bernstein-Gelfand tensor product functors are endofunctors of the category of Harish-Chandra modules provided by tensor products with finite dimensional modules. We provide an automorphic analogue of these tensor product functors, implemented by vector-valued automorphic representations that are trivial at all finite places. They naturally explain the role of vector-valued modular forms in recent work by Bringmann-Kudla on Harish-Chandra modules associated with harmonic weak Maaß forms. We give a detailed account of the image $\mathrm{sym}^1 \otimes \varpi(E_2)$ of the automorphic representation $\varpi(E_2)$ generated by the Eisenstein series of weight $2$ under one of those tensor product functors. This builds upon work by Roy-Schmidt-Yi, who recently determined the structure of $\varpi(E_2)$. They found that $\varpi(E_2)$ does not decompose as a restricted tensor product over all places of $\mathbb{Q}$, while we discover that $\mathrm{sym}^1 \otimes \varpi(E_2)$ has a direct summand that does. This summand corresponds to a holomorphic and modular, vector-valued analogue of $E_2$. The complement in $\mathrm{sym}^1 \otimes \varpi(E_2)$ arises from one of the vector-valued examples in the work of Bringmann-Kudla. Our approach allows us to determine its structure at the finite places.

math.NT↗

arXiv2022

Eichler integrals and generalized second order Eisenstein series

We show that all Eichler integrals, and more generally all "generalized second order modular forms" can be expressed as linear combinations of corresponding generalized second order Eisenstein series with coefficients in classical modular forms. We determine the Fourier series expansions of generalized second order Eisenstein series in level one, and provide tail estimates via convexity bounds for additively twisted $\mathrm{L}$-functions. As an application, we illustrate a bootstrapping procedure that yields numerical evaluations of, for instance, Eichler integrals from merely the associated cocycle. The proof of our main results rests on a filtration argument that is largely rooted in previous work on vector-valued modular forms, which we here formulate in classical terms.

math.NT↗

arXiv2022

Congruences of Hurwitz class numbers on square classes

We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us to establish Ramanujan-type congruences for Hurwitz class numbers on square classes, where the holomorphic case parallels previous work by Radu on partition congruences. We offer two applications. The first application demonstrates common divisibility features of Ramanujan-type congruences for Hurwitz class numbers. The second application provides a dichotomy between congruences for class numbers of imaginary quadratic fields and Ramanujan-type congruences for Hurwitz class numbers.

math.NT↗

arXiv2022

A classification of harmonic weak Maaß forms of half-integral weight

We classify Harish-Chandra modules generated by the pullback to the metaplectic group of harmonic weak Maaß forms with exponential growth allowed at the cusps. This extends work by Schulze-Pillot and parallels recent work by Bringmann-Kudla, who investigated the case of integral weights. We realize each of our cases via a regularized theta lift of an integral weight harmonic weak Maaß form. Harish-Chandra modules in both integral and half-integral weight that occur need not be irreducible. Therefore, our display of the role that the theta lifting takes in this picture, we hope, contributes to an initial understanding of a theta correspondence for extensions of Harish-Chandra modules.

math.NT↗

arXiv2021

The maximal discrete extension of the Hermitian modular group

Let $Γ_n(\mathcal{\scriptstyle{O}}_\mathbb{K})$ denote the Hermitian modular group of degree $n$ over an imaginary-quadratic number field $\mathbb{K}$. In this paper we determine its maximal discrete extension in $SU(n,n;\mathbb{C})$, which coincides with the normalizer of $Γ_n(\mathcal{\scriptstyle{O}}_{\mathbb{K}})$. The description involves the $n$-torsion subgroup of the ideal class group of $\mathbb{K}$. This group is defined over a particular number field $\widehat{\mathbb{K}}_n$ and we can describe the ramified primes in it. In the case $n=2$ we give an explicit description, which involves generalized Atkin-Lehner involutions. Moreover we find a natural characterization of this group in $SO(2,4)$.

math.NT↗

arXiv2021

Relations among Ramanujan-Type Congruences I

We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific congruences for Hecke eigenvalues. In particular, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra. More generally, we show for weakly holomorphic modular forms of integral weight, that Ramanujan-type congruences naturally occur for shifts in the union of two square-classes as opposed to single square-classes that appear in the literature on the partition function. We also rule out the possibility of square-free periods, whose scarcity in the case of the partition function was investigated recently. We complement our obstructions on maximal Ramanujan-type congruences with several existence statements. Our results are based on a framework that leverages classical results on integral models of modular curves via modular representation theory, and applies to congruences of all weakly holomorphic modular forms. Steinberg representations govern all maximal Ramanujan-type congruences for integral weights. We discern the scope of our framework in the case of half-integral weights through example calculations.

math.NT↗

arXiv2021

Modular forms of virtually real-arithmetic type I -- Mixed mock modular forms yield vector-valued modular forms

The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated Eichler-Shimura integrals. Applications beyond number theory range from combinatorics, geometry, and representation theory to string theory and conformal field theory. We unify these relaxed notions in the framework of vector-valued modular forms by introducing a new class of $\mathrm{SL}_{2}(\mathbb{Z})$-representations: virtually real-arithmetic types. The key point of the paper is that virtually real-arithmetic types are in general not completely reducible. We obtain a rationality result for Fourier and Taylor coefficients of associated modular forms.

math.NT↗

arXiv2020

Non-Holomorphic Ramanujan-type Congruences for Hurwitz Class Numbers

In contrast to all other known Ramanujan-type congruences, we discover that Ramanujan-type congruences for Hurwitz class numbers can be supported on non-holomorphic generating series. We establish a divisibility result for such non-holomorphic congruences of Hurwitz class numbers. The two keys tools in our proof are the holomorphic projection of products of theta series with a Hurwitz class number generating series and a theorem by Serre, which allows us to rule out certain congruences.

math.NT↗

arXiv2019

Congruences on Square-Classes for the Partition Function

We considerably improve Ono's and Ahlgren-Ono's work on the frequent occurrence of Ramanujan-type congruences for the partition function, and demonstrate that Ramanujan-type congruences occur in families that are governed by square-classes. We thus elucidate for the first time an exemplary family of congruences found by Atkin-O'Brien. Our results are based on a novel framework that leverages available results on integral models of modular curves via representations of finite quotients of $\mathrm{SL}_2(\mathbb{Z})$ or $\mathrm{Mp}_1(\mathbb{Z})$. This framework applies to congruences of all weakly holomorphic modular forms.

math.NT↗

arXiv2019

All modular forms of weight 2 can be expressed by Eisenstein series

We show that every elliptic modular form of integral weight greater than $1$ can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $\mathrm{L}$-values present in all previous work. For weights greater than $2$, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.

math.NT↗

arXiv2019

The skew-Maass lift I

The classical Maass lift is a map from holomorphic Jacobi forms to holomorphic scalar-valued Siegel modular forms. Automorphic representation theory predicts a non-holomorphic and vector-valued analogue for Hecke eigenforms. This paper is the first part of a series of papers. In this series of papers, we provide an explicit construction of the non-holomorphic Maass lift that is linear and also applies to non-eigenforms. In this first part, we develop new techniques to study Fourier series expansions of Siegel modular forms, which allow us to construct a Maass lift from harmonic Maass-Jacobi forms to scalar-valued Maass-Siegel forms.

math.NT↗

arXiv2018

Hyper-Algebras of Vector-Valued Modular Forms

We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on these hyper-algebras. These definitions bridge the classical and representation theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein series. As a byproduct, we obtain inclusions of cuspidal automorphic representations into the tensor product of global principal series.

math.NT↗

arXiv2018

Hyperelliptic Curves over Small Finite Fields and GPU Accelerators

We present a hardware-accelerated computation of Hasse-Weil invariants of all hyperelliptic curves of given genus over a fixed finite field. Our main motivation is the determination of traces of Frobenius on cohomology corresponding moduli stacks à la Bergström-Faber-van-der-Geer. This paper also constitutes a case study of the performance of lookup table based Zech arithmetic on Graphics Processing Units (GPUs). GPUs are by now ubiquitous in numerics, to an extend that their development itself is propelled by scientific computing. Algebraic computing has profited very little from the advancement of hardware design. We suggest that specific computations can be benificially adjusted to GPUs with comparatively little effort.

math.NT↗

arXiv2017

HLinear: Exact Dense Linear Algebra in Haskell

We present an implementation in the functional programming language Haskell of the PLE decomposition of matrices over division rings. Our benchmarks indicate that it is competitive with the C-based implementation provided in Flint. Describing the guiding principles of our work, we introduce the reader to basic ideas from high-performance functional programming.

cs.MS↗

arXiv2015

Harmonic Maaß-Jacobi forms of degree 1 with higher rank indices

We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of order 4. In ranks exceeding 1, the notions of H-harmonicity and semi-holomorphicity are the same.

math.NT↗

arXiv2014

Computing Genus 1 Jacobi Forms

We develop an algorithm to compute Fourier expansions of vector valued modular for Weil representations. As an application, we compute explicit linear equivalences of special divisors on modular varieties of orthogonal type. We define three families of Hecke operators for Jacobi forms, and analyze the induced action on vector valued modular forms. The newspaces attached to one of these families are used to give a more memory efficient version of our algorithm.

math.NT↗

arXiv2014

Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition

Real-analytic Jacobi forms play key roles in different areas of mathematics and physics, but a satisfactory theory of such Jacobi forms has been lacking. In this paper, we fill this gap by introducing a space of harmonic Maass-Jacobi forms with singularities which includes the real-analytic Jacobi forms from Zwegers's PhD thesis. We provide several structure results for the space of such Jacobi forms, and we employ Zwegers's $\widehatμ$-functions to establish a theta-like decomposition.

math.NT↗

arXiv2014

Holomorphic projections and Ramanujan's mock theta functions

We employ spectral methods of automorphic forms to establish a holomorphic projection operator for tensor products of vector-valued harmonic weak Maass forms and vector-valued modular forms. We apply this operator to discover simple recursions for Fourier series coefficients of Ramanujan's mock theta functions.

math.NT↗