Periods of third kind for rank 2 Drinfeld modules and algebraic independence of logarithms

2009/09/01 by Chieh-Yu Chang, Chang, Chieh-Yu
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #math.NT #msc:11G09 #msc:11J93 #msc:11M38

paper · pdf · doi:10.48550/arxiv.0909.0101

19 pages

arxiv created 2009/09/01 · arxiv updated 2009/12/01

Abstract

In analogy with the periods of abelian integrals of differentials of third kind for an elliptic curve defined over a number field, we introduce a notion of periods of third kind for a rank 2 Drinfeld Fq[t]-module rho defined over an algebraic function field and derive explicit formulae for them. When rho has complex multiplication by a separable extension, we prove the algebraic independence of rho-logarithms of algebraic points that are linearly independent over the CM field of rho. Together with the main result in [CP08], we completely determine all the algebraic relations among the periods of first, second and third kinds for rank 2 Drinfeld Fq[t]-modules in odd characteristic.

Related

• On the third kind periods for abelian t-modules
• Algebraic relations among hyperderivatives of periods and logarithms of Drinfeld modules
• On Furusho's analytic continuation of Drinfeld logarithms
• Drinfeld Module and Weil pairing over Dedekind domain of class number two
• Tensor powers of rank 1 Drinfeld modules and periods