Periods of 1-motives and transcendence.
A number of recent speculations deal with the open problem of finding all algebraic relations between periods [\textit{M. Kontsevich,} and \textit{D. Zagier}, B. Engquist (ed.) et al., Mathematics unlimited - 2001 and beyond. Berlin: Springer, 771--808 (2001; Zbl 1039.11002)]. A far reaching statement, generalizing previous conjectures by Schanuel and Grothendieck, has been proposed by \textit{Y. André} [Quelques conjectures de transcendance issues de la géométrie algébrique, Institut Mathématique de Jussieu, preprint 121 (1997)]. The author specializes André's conjecture to \(1\)-motives and shows that it is equivalent to a statement which she calls the elliptico-toric conjecture. She produces a number of special cases involving the classical exponential function, elliptic Weierstraß\ \(\wp\) and \(\zeta\) functions, elliptic integrals and the modular invariant.
Transcendence, algebraic independence, transcendence degree, Schanuel's conjecture, periods, \(1\)-motives, Grothendieck conjecture, de Rham cohomology, Mumford-Tate groups
scientific article; zbMATH DE number 1891439
scientific article