The de Rham realization of the elliptic polylogarithm in families

Abstract

This thesis establishes a geometric approach to the de Rham realization of the polylogarithm. As a central result we construct the logarithm sheaves of rational abelian schemes in terms of the birigidified Poincaré bundle with universal integrable connection on the product of the abelian scheme and the universal vectorial extension of its dual. This is achieved essentially by restricting the mentioned data of the Poincaré bundle along the infinitesimal neighborhoods of the zero section of the universal extension. We also clarify how these constructions naturally express within the language of the Fourier-Mukai transformation for $\mathcal D$-modules on abelian schemes. Our geometric perspective moreover permits an interpretation of fundamental formal properties of the logarithm sheaves within the standard theory of the Poincaré bundle. For a relative elliptic curve we additionally present a related viewpoint on the first logarithm extension via $1$-motives. Having developed in detail the outlined geometric understanding of the logarithm sheaves, we then exploit it systematically for an investigation of the polylogarithm for the universal family of elliptic curves with level $N$ structure. A main theorem of the work gives an explicit analytic description for a variant of the small elliptic polylogarithm via the coefficient functions appearing in the Laurent expansion of a meromorphic Jacobi form defined by Kronecker in the 19th century. Furthermore, using the previous result, we determine the specialization of the modified polylogarithm along torsion sections concretely in terms of certain algebraic Eisenstein series. From this we regain in particular the known expressions of the de Rham Eisenstein classes by algebraic modular forms.