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HECKE ALGEBRAS, GENERATING SERIES AND APPLICATIONS

Abstract : The main result in presented work consists of explicit computation of the generating power series of Hecke operators in local Hecke algebra for the symplectic groups of genus 3 and 4. The computation algorithm is based on the Satake isomorphism, which allows to carry out all operations in the algebra of polynomials in multiple variables. This is the first time when this expression was computed in genus 4. In order to obtain the main result, the method of symbolic computation was developed. This algorithmic approach is also applied to other types of Hecke series. In particular, we formulate and prove the analog of Rankin's Lemma in higher genus. We also computed the symmetric squares and symmetric cubes generating series.

Based on our computational results we formulate a modularity lifting conjecture for convolutions of L-functions attached to Siegel modular forms. We review other important conjectures related to Siegel modular forms and their L-functions. We use these constructions to compute the rational algebraic factors in critical values of the spinor L-function attached to F12 of Miyawaki. To our knowledge this is the first example of a spinor L-function of Siegel cusp forms of degree 3, when the special values can be computed explicitly.

Finally, we apply the theory of Hecke algebras to constructions of algebraic cryptosystems on some finite sets of left cosets in Hecke algebra. We use a relation between left cosets and points on certain projective algebraic varieties.
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Submitted on : Sunday, January 4, 2009 - 1:31:43 PM
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Kirill Vankov. HECKE ALGEBRAS, GENERATING SERIES AND APPLICATIONS. Mathematics [math]. Université Joseph-Fourier - Grenoble I, 2008. English. ⟨tel-00349767⟩

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