[Paper Review] The de Rham realization of the elliptic polylogarithm in families

René Achim Scheider|arXiv (Cornell University)|Jan 1, 2014

Algebraic Geometry and Number Theory26 references3 citations

TL;DR

This paper constructs the de Rham realization of the elliptic polylogarithm in families using logarithm sheaves and D-modules on abelian schemes, proving an explicit analytic characterization of the D-variant of the polylogarithm in terms of Eisenstein series and Siegel units. The key result identifies de Rham Eisenstein classes with modular forms via specialization along torsion sections, providing a motivic and cohomological interpretation of these classes in terms of modular forms and the Kodaira-Spencer map.

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.

Motivation & Objective

• To construct the de Rham realization of the elliptic polylogarithm in families over abelian schemes, particularly the universal elliptic curve.
• To establish a motivic and cohomological interpretation of the D-variant of the polylogarithm using logarithm sheaves and unipotent D-modules.
• To explicitly describe the specialization of the D-variant along torsion sections in terms of Eisenstein series and Siegel units.
• To relate the resulting cohomology classes to modular forms via the Kodaira-Spencer isomorphism and Poincaré duality.
• To provide a complete analytic characterization of the polylogarithm and its D-variant using Fourier-Mukai transforms and meromorphic Jacobi forms.

Proposed method

• Constructs logarithm sheaves as universal unipotent extensions of de Rham cohomology sheaves on abelian schemes.
• Applies the Fourier-Mukai transform on the universal vectorial extension to relate logarithm sheaves to the Poincaré bundle.
• Uses the theory of D-modules and integrable connections to define and analyze the D-variant of the polylogarithm.
• Employs meromorphic Jacobi forms and Eisenstein series to describe the analytic structure of the logarithm sheaves.
• Applies the analytification functor to relate algebraic de Rham cohomology to analytic objects on the universal elliptic curve.
• Uses contraction maps and Poincaré duality to define and compute de Rham Eisenstein classes from the polylogarithm.

Experimental results

Research questions

• RQ1 How can the elliptic polylogarithm be realized in the de Rham cohomology of families of abelian schemes?
• RQ2 What is the explicit analytic description of the D-variant of the polylogarithm on the universal elliptic curve?
• RQ3 How do the specializations of the D-variant along torsion sections relate to modular forms and Eisenstein series?
• RQ4 What is the motivic interpretation of the first logarithm extension class in terms of 1-motives and the de Rham-Manin map?
• RQ5 How can the de Rham Eisenstein classes be explicitly computed and related to Siegel units and modular forms?

Key findings

• The D-variant of the polylogarithm specializes along torsion sections to cohomology classes that are explicitly described via Eisenstein series and modular forms.
• For D ≡ 1 mod N, the n-th de Rham Eisenstein class Eisn(ta,b) is given by Eisn(ta,b) = −N^{n−1} (−1)^n / n! · F^{(n+2)}(a/N, b/N) in H^1_dR(S/Q, Sym^n H^1), where F^{(n+2)} is the modular form associated to the Eisenstein series.
• The degree 0 specialization Eis0(ta,b) equals −N^{−1} dlog(g(a/N, b/N)) in H^1_dR(S/Q), where g(a/N, b/N) is the Siegel unit.
• The D-variant of the polylogarithm at degree n is related to the de Rham Eisenstein classes via the formula: t^*_{a,b}(pol^n_D) = −N^{1−n} (D^2 Eis^n(ta,b) − D^{−n} Eis^n(t_{Da,Db})).
• The analytic characterization of the logarithm sheaves is established via a commutative diagram involving analytification and the fundamental meromorphic Jacobi form.
• The motivic de Rham-Manin map identifies the first logarithm extension class with the image of the modular form F^{(2)}(a/N, b/N) under the map (3.8.38), linking it to the Kodaira-Spencer class.