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Periods and algebraicity

Abstract

This project is devoted to some algebraicity problems concerning periods, bringing together algebro-geometric and number-theoretic aspects. Periods are complex numbers that can be written as an integral of an algebraic differential form. They are often expected to be transcendental numbers, but their study is of utmost importance to algebraic number theory, as they appear as special values of L-functions. One would like to understand, for instance, if a given period is an algebraic number (or the logarithm of an algebraic number), the algebraic relations periods satisfy, their “symmetries”, etc. The relation between the theory of periods with Algebraic Geometry usually allows us to rely on geometric methods to obtain such arithmetic results.We shall address the following problems: (i) special values of higher Green's functions, and the Gross-Zagier conjecture (joint with Francis Brown); (ii) transcendence conjectures on single-valued periods; (iii) motivic interpretation of multiple elliptic polylogarithms (joint with Nils Matthes). Our approach is essentially geometric and will involve tools such as: algebraic de Rham cohomology, moduli stacks of elliptic curves, pro-unipotent fundamental groups, motives of modular forms, and the universal vector extension of an abelian variety. (AU)

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