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 Poincar\'e 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 Poincar\'e 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 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
Poincar\'e 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.Comment: Doctoral thesis, University of Regensbur