Hyperderivatives of periods and quasi-periods for Anderson $t$-modules

Abstract

We investigate periods, quasi-periods, logarithms, and quasi-logarithms of Anderson $t$-modules, as well as their hyperderivatives. We develop a comprehensive account of how these values can be obtained through rigid analytic trivializations of abelian and $\mathbf{A}$-finite $t$-modules. To do this we build on the exponentiation theorem of Anderson and investigate quasi-periodic extensions of $t$-modules through Anderson generating functions. By applying these results to prolongation $t$-modules of Maurischat, we integrate hyperderivatives of these values together with previous work of Brownawell and Denis in this framework.

arXiv comment111 pages

References in corpus (18)

• Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms
• Determination of the algebraic relations among special Gamma-values in positive characteristic
• Linear independence of monomials of multizeta values in positive characteristic
• Values of certain L-series in positive characteristic
• Linear independence of Gamma values in positive characteristic
• A Weil-Barsotti formula for Drinfeld modules
• An effective criterion for Eulerian multizeta values in positive characteristic
• Artin t-Motifs
• The de Rham isomorphism for Drinfeld modules over Tate algebras
• Periods of $t$-modules as special values
• Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic
• Taylor coefficients of Anderson-Thakur series and explicit formulae
• Vectorial Drinfeld modular forms over Tate algebras
• Algebraic independence of the Carlitz period and its hyperderivatives
• Anderson t-modules with thin t-adic Galois representations
• Algebraic relations among hyperderivatives of periods and logarithms of Drinfeld modules
• Geometric Gamma values and zeta values in positive characteristic
• Hyperderivative power sums, Vandermonde matrices, and Carlitz multiplication coefficients

Cited by in corpus (5)

• Effective rigid analytic trivializations for Drinfeld modules
• Algebraic relations among hyperderivatives of periods and logarithms of Drinfeld modules
• Linear equations on Drinfeld modules
• Linear relations among algebraic points on tensor powers of the Carlitz module
• Abelian equals A-finite for Anderson A-modules