×

Singularités des fonctions obtenues par intégration sur la fibre \(X^ 2-Y^ 3=s,\) et identités modulaires. (Singularities of functions obtained by integration on the fiber \(X^ 2-Y^ 3=s,\) and modular identities). (French) Zbl 0563.32003

The starting point of this paper is the determination of the singular terms of the asymptotic expansion, when s tends to zero, of the functions obtained by integration on the complex torus: \(X^ 2-Y^ 3=s\). Afterwards, by an analytic computation, we specify the coefficients associated to these singular terms. And, a uniformization of the elliptic curve \(X^ 2-Y^ 3=s\) (s\(\neq 0)\) allows a topological computation which gives us new expressions for the desired coefficients. At last, we identify the results furnished by the two methods and obtain modular identities. Particularly, we have an explicit Chowla-Selberg formula in the canonical elliptic case described above.

MSC:

32C30 Integration on analytic sets and spaces, currents
32S05 Local complex singularities
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
14H52 Elliptic curves
PDFBibTeX XMLCite