DescriptionSiegel modular forms (see also Wikipedia page) take their name from one of the most important mathematicians of the 20th century, Carl Ludwig Siegel, who introduced them in his study of quadratic forms. They are a natural, and vast, generalisation of classical modular forms, and one of the central fields of research in modern Number Theory, since they provide plenty of new phenomena which are not seen in the theory of modular forms.In this project we will start with learning some basics of the theory of Siegel modular forms such as the properties of the symplectic group, and its action to the Siegel upper half space. Then we will give the definition of a Siegel modular form as a holomorphic function on the Siegel upper half space which has enough symmetries (modularity property). Then the project may take one of the following directions (the list below is by all means not complete): 1) Eisenstein series: The aim of this direction is to define and study Eisenstein series which provide a plethora of examples of Siegel modular forms. Of particular interest is the structure of the space of Eisenstein series (Siegel type Eisenstein series, Klingen type Eisenstein series, and Siegel's Phi operator) and/or their Fourier coefficients which carry some very interesting arithmetic information (representation numbers). 2) Theta series: This is another family of examples of Siegel modular forms. The main aim will be study their properties and their connection to the theory of singular modular forms. They are also very important for the study of half-integral Siegel modular forms. 3) Hecke Algebra: The focus here will be the study of Hecke operators. These operators play a prominent role in the theory and their eigenfunctions are of special interest, which are related with some very important conjectures (paramodular conjecture). 4) L-functions: One can associate at least two interesting L-functions to a Siegel modular form, the standard L function and the spinor L-function. They are both very important and their properties have been extensively studied. A possible direction here will be to use the so-called Rankin-Selberg method to study the analytic properties of the standard L function. 5) Algebraic structure of Siegel modular forms: Despite their very analytic nature, Siegel modular forms enjoy many algebraic properties, which can be defined from the field of definition of their Fourier coefficients. This direction will explore this structure. 6) Maass differential operators and Dirichlet series: Maass (and Selberg) introduced various differential operators which have some important applications to the study of various Dirichlet series attached to Siegel modular forms. 7) Connections with the theory of Error-Correcting Codes: Surprisingly enough there are interesting connections with the theory of Error-Correcting Codes via the so-called Weight Distribution of Codes. Resources
[2] Anatoli Andrianov, Introduction to Siegel modular forms and Dirichlet series, Springer 2009. [3] Gerard van der Geer, Siegel modular forms, notes available online from the arXiv . [4] Winfried Kohnen, A short course on Siegel modular forms, available from his web page . [5] Hans Maass, Siegel's Modular Forms and Dirichlet Series, LNM 216.
Pre-requisites
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