Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on arithmetic progressions modular forms.

Let p(n) denote the number of partitions of n. Let A,B ∈ N with A > B and ≥ 5 a prime, such that p(An+B) ≡ 0 (mod ), n ∈ N. Then we will prove that |A and ( 24B−1 ) = (−1 ) . This settles an open problem by Scott Ahlgren and Ken Ono. Our proof is based on results by Deligne and Rapoport.

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A Course in Arithmetic

Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences

A generalization of the conjecture of Birch and Swinnerton-Dyer is investigated using complex multiplication cycles on a particular Kuga fiber variety. A weak finiteness result consistent with the conjecture is proved. The image of complex multiplication cycles under the Abel-Jacobi map is computed explicitly. The results provide numerical evidence supporting the conjecture. They also give evidence for a relationship between complex multiplication cycles and a modular form of weight 5/2 and raise questions for further investigation

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Complex Multiplication Cycles and a Conjecture of Beilinson and Bloch

/pdf/algebraic-number-theory-a-computational-approach-4kizc2mzqe.pdf

Algebraic Number Theory, a Computational Approach

Jacobi forms of lattice index, whose theory can be viewed as extension of the theory of classical Jacobi forms, play an important role in various theories, like the theory of orthogonal modular forms or the theory of vertex operator algebras. Every Jacobi form of lattice index has a theta expansion which implies, for index of odd rank, a connection to half integral weight modular forms and then via Shimura lifting to modular forms of integral weight, and implies a direct connection to modular forms of integral weight if the rank is even. The aim of this thesis is to develop a Hecke theory for Jacobi forms of lattice index extending the Hecke theory for the classical Jacobi forms, and to study how the indicated relations to elliptic modular forms behave under Hecke operators.
After defining Hecke operators as double coset operators, we determine their action on the Fourier coefficients of Jacobi forms, and we determine the multiplicative relations satisfied by the Hecke operators, i.e. we study the structural constants of the algebra generated by the Hecke operators. As a consequence we show that the vector space of Jacobi forms of lattice index has a basis consisting of simultaneous eigenforms for our Hecke operators, and we discover the precise relation between our Hecke algebras and the Hecke algebras for modular forms of integral weight. The latter supports the expectation that there exist equivariant isomorphisms between spaces of Jacobi forms of lattice index and spaces of integral weight modular forms. We make this precise and prove the existence of such liftings in certain cases. Moreover, we give further evidence for the existence of such liftings in general by studying numerical examples. 
Jacobiformen von Gitterindex, deren Theorie als Erweiterung der Theorie klassischer Jacobiformen betrachtet werden kann, spielen in unterschiedlichen Theorien, wie der Theorie der Modulformen auf orthogonalen Gruppen oder der Theorie der Vertexoperatoralgebren, eine bedeutende Rolle. Jede Jacobiform mit Gitterindex hat eine Theta-Entwicklung, die bei ungeradem Index eine Verbindung zu Modulformen von halbganzen Gewichten herstellt und daher via Shimura-Liftung eine Beziehung zu Modulformen ganzen Gewichts impliziert, und bei geradem Index eine direkte Verbindung zu Moldulformen ganzzahligen Gewicht suggeriert. Das Ziel dieser Dissertation ist, eine Hecke-Theorie fur Jacobiformen mit Gitterindex zu entwickeln, indem die Hecke-Theorie fur die klassischen Jacobiformen erweitert wird, und zu untersuchen, wie sich die angedeuteten Beziehungen zu elliptischen Moldulformen unter Hecke-Operatoren verhalten. 
Nachdem die Hecke-Operatoren als Doppelnebenklassen-Operatoren definiert werden, wird deren Wirkung auf die Fourier-Koeffizienten der Jacobiformen und die multiplikativen Relationen, welche von Hecke-Operatoren erfullt werden, untersucht, indem z.B. die Struktur-Konstanten der Algebra berechnet werden, die von den Hecke-Operatoren erzeugt werden. Daraufhin wird gezeigt, dass der Vektorraum der Jacobiformen mit Gitterindex uber eine Basis von simultanen Eigenformen fur die Hecke-Operatoren verfugt und es wird die prazise Beziehung zwischen der Hecke-Algebra und der Hecke-Algebra fur Moldulformen ganzzahligen Gewichts aufgezeigt. Letztgenannte Beziehung stutzt die Erwartung, dass aquivariante Isomorphismen zwischen den Raumen der Jacobiformen mit Gitterindex und Raumen Moldulformen ganzzahligen Gewichts existieren. Es erfolgt eine Prazisierung und der Beweis, dass solche Liftungen in bestimmten Fallen existieren. Zudem geben wir weitere Argumente fur die Existenz solcher Liftungen im allgemeunen Fall durch das Studium numerischer Beispiele.

background

Hecke operators on Jacobi forms of lattice index and the relation to elliptic modular forms

The theory of 'zeta functions of fractal strings' has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called `distance zeta functions', which since then, has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta functions is the class of 'tube zeta functions', defined using the tube function of a fractal set. These three classes of zeta functions, under the name of 'fractal zeta functions', exhibit deep connections with Minkowski contents and upper box dimensions, as well as, more generally, with the complex dimensions of fractal sets. Further extensions include zeta functions of relative fractal drums, the box dimension of which can assume negative values, including minus infinity. We also survey some results concerning the existence of the meromorphic extensions of the spectral zeta functions of fractal drums, based in an essential way on earlier results of the first author on the spectral (or eigenvalue) asymptotics of fractal drums. It follows from these results that the associated spectral zeta function has a (nontrivial) meromorphic extension, and we use some of our results about fractal zeta functions to show the new fact according to which the upper bound obtained for the corresponding abscissa of meromorphic convergence is optimal. Finally, we conclude this survey article by proposing several open problems and directions for future research in this area.

A Course in Arithmetic

Citations

Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences

Complex Multiplication Cycles and a Conjecture of Beilinson and Bloch

Algebraic Number Theory, a Computational Approach

Hecke operators on Jacobi forms of lattice index and the relation to elliptic modular forms

Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums

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