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Nils-Peter Skoruppa

Publications and source records attributed to Nils-Peter Skoruppa.

16 recordsLinked to original sources
arXiv2023

Jacobi Forms of Lattice Index I. Basic Theory

This is the first one of a series of articles in which we develop the theory of Jacobi forms of lattice index, their close interplay with the arithmetic theory of lattices and the theory of Weil representations. We hope to publish this series eventually in an extended and combined way as a monograph. In this part we present the basic theory and first structure theorems. We deduce explicit dimension formulas and give non-trivial explicit examples

math.NT↗
arXiv2022

A classical approach to relative quadratic extensions

We show that we can develop from scratch and using only classical language a theory of relative quadratic extensions of a given number field $K$ which is as explicit and easy as for the well-known case that $K$ is the field of rational numbers. As an application we prove a reciprocity law which expresses the number of solutions of a given quadratic equation modulo an integral ideal $\mathfrak{a}$ of $K$ in terms of $\mathfrak{a}$ modulo the discriminant of the equation. We study various $L$-functions associated to relative quadratic extensions. In particular, we define, for totally negative algebraic integers $Δ$ of a totally real number field $K$ which are squares modulo~$4$, numbers $H(Δ,K)$, which share important properties of classical Hurwitz class numbers. In an appendix we give a quick elementary proof of certain deeper properties of the Hilbert symbol on higher unit groups of dyadic local number fields.

math.NT↗
arXiv2019

Theta Blocks

We define theta blocks as products of Jacobi theta functions divided by powers of the Dedekind eta-function and show that they give a powerful new method to construct Jacobi forms and Siegel modular forms, with applications also in lattice theory and algebraic geometry. One of the central questions is when a theta block defines a Jacobi form. It turns out that this seemingly simple question is connected to various deep problems in different fields ranging from Fourier analysis over infinite-dimensional Lie algebras to the theory of moduli spaces in algebraic geometry. We give several answers to this question.

math.NT↗
arXiv2017

Computing Invariants of the Weil representation

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.

math.NT↗
arXiv2016

Computing Jacobi Forms

We describe an implementation for computing holomorphic and skew-holomorphic Jacobi forms of integral weight and scalar index on the full modular group. This implementation is based on formulas derived by one of the authors which express Jacobi forms in terms of modular symbols of elliptic modular forms. Since this method allows to generate a Jacobi eigenform directly from a given modular eigensymbol without reference to the whole ambient space of Jacobi forms it makes it possible to compute Jacobi Hecke eigenforms of large index. We illustrate our method with several examples.

math.NT↗
arXiv2012

Explicit computations of Siegel modular forms of degree two

Unlike classical modular forms, there is currently no general way to implement the computation of Siegel modular forms of arbitrary weight, level and character, even in degree two. There is however, a way to do it in a unified way. After providing a survey of known computations we describe the implementation of a class modeling Siegel modular forms of degree two in Sage. In particular, we describe algorithms to compute a variety of rings of Siegel modular forms, many of which are implemented in our class. A wide variety of Siegel modular forms (e.g., both vector- and scalar-valued) can be modeled via this class and we unify these via a construct we call a formal Siegel modular form. We define this notion and discuss it in detail.

math.NT↗
arXiv2012

Linear characters of SL_2 over Dedekind domains

For an important class of arithmetic Dedekind domains O including the ring of integers of not totally complex number fields, we describe explicitly the group of linear characters of SL_2(O). For this, we determine, for arbitrary Dedekind domains O, the group of linear characters of SL_2(O) whose kernel is a congruence subgroup.

math.NT↗
arXiv2010

Numerical Computation of a Certain Dirichlet Series Attached to Siegel Modular Forms of Degree Two

The Rankin convolution type Dirichlet series $D_{F,G}(s)$ of Siegel modular forms $F$ and $G$ of degree two, which was introduced by Kohnen and the second author, is computed numerically for various $F$ and $G$. In particular, we prove that the series $D_{F,G}(s)$, which share the same functional equation and analytic behavior with the spinor $L$-functions of eigenforms of the same weight are not linear combinations of those. In order to conduct these experiments a numerical method to compute the Petersson scalar products of Jacobi Forms is developed and discussed in detail.

math.NT↗
arXiv2008

Reduction mod $\ell$ of Theta Series of Level $\ell^n$

It is proved that the theta series of an even lattice whose level is a power of a prime $\ell$ is congruent modulo $\ell$ to an elliptic modular form of level~1. The proof uses arithmetic and algebraic properties of lattices rather than methods from the theory of modular forms. The methods presented here may therefore be especially pleasing to those working in the theory of quadratic forms, and they admit generalizations to more general types of theta series as they occur e.g. in the theory of Siegel or Hilbert modular forms.

math.NT↗
arXiv2007

Jacobi Forms of Degree One and Weil Representations

We discuss the notion of Jacobi forms of degree one with matrix index, we state dimension formulas, give explicit examples, and indicate how closely their theory is connected to the theory of invariants of Weil representations associated to finite quadratic modules.

math.NT↗
arXiv2007

Jacobi Forms of Critical Weight and Weil Representations

Jacobi forms can be considered as vector valued modular forms, and Jacobi forms of critical weight correspond to vector valued modular forms of weight $\frac12$. Since the only modular forms of weight $\frac12$ on congruence subgroups of $\SL$ are theta series the theory of Jacobi forms of critical weight is intimately related to the theory of Weil representations of finite quadratic modules. This article explains this relation in detail, gives an account of various facts about Weil representations which are useful in this context, and it gives some applications of the theory developed herein by proving various vanishing theorems and by proving a conjecture on Jacobi forms of weight one on $\SL$ with character.

math.NT↗
arXiv2005

SL(2,Z)-Invariant Spaces Spanned by Modular Units

Characters of rational vertex operator algebras (RVOAs) arising in 2-dimensional conformal field theories often belong (after suitable normalization) to the (multiplicative) semigroup E^+ of modular units whose Fourier expansions are in 1+q Z_{>=0}[[q]], up to a fractional power of q. If even all characters of a RVOA share this property then we have an example of what we call modular sets, i.e. finite subsets of E^+ whose elements (additively) span a vector space which is invariant under the usual action of SL(2,Z). The classification of modular sets and RVOAs seem to be closely related. In this article we give an explicit description of the group of modular units generated by E^+, we prove a certain finiteness result for modular sets contained in a natural semi-subgroup E_* of E^+, and we discuss consequences.

q-alg↗
arXiv1996

Product Expansions of Conformal Characters

We describe several infinite series of rational conformal field theories whose conformal characters are modular units, i.e. which are modular functions having no zeros or poles in the upper complex half plane, and which thus possess simple product expansions. We conjecture that certain infinite series of rational models of Casimir W-algebras always have this property. Furthermore, we describe an algorithm which can be used to prove whether a modular function is a modular unit or not.

hep-th↗
arXiv1994

Conformal Characters and Theta Series

We describe the construction of vector valued modular forms transforming under a given congruence representation of the modular group SL(2,Z) in terms of theta series. We apply this general setup to obtain closed and easily computable formulas for conformal characters of rational models of W-algebras.

hep-th↗
arXiv1994

Modular Invariance and Uniqueness of Conformal Characters

We show that the conformal characters of various rational models of W-algebras can be already uniquely determined if one merely knows the central charge and the conformal dimensions. As a side result we develop several tools for studying representations of SL(2,Z) on spaces of modular functions. These methods, applied here only to certain rational conformal field theories, may be useful for the analysis of many others.

hep-th↗