44 research outputs found
Theta operators, Goss polynomials, and v-adic modular forms
We investigate hyperderivatives of Drinfeld modular forms and determine
formulas for these derivatives in terms of Goss polynomials for the kernel of
the Carlitz exponential. As a consequence we prove that v-adic modular forms in
the sense of Serre, as defined by Goss and Vincent, are preserved under
hyperdifferentiation. Moreover, upon multiplication by a Carlitz factorial,
hyperdifferentiation preserves v-integrality.Comment: 20 page
Identities for Anderson generating functions for Drinfeld modules
Anderson generating functions are generating series for division values of
points on Drinfeld modules, and they serve as important tools for capturing
periods, quasi-periods, and logarithms. They have been fundamental in recent
work on special values of positive characteristic L-series and in transcendence
and algebraic independence problems. In the present paper we investigate
techniques for expressing Anderson generating functions in terms of the
defining polynomial of the Drinfeld module and determine new formulas for
periods and quasi-periods.Comment: 18 page
An effective criterion for Eulerian multizeta values in positive characteristic
Characteristic p multizeta values were initially studied by Thakur, who
defined them as analogues of classical multiple zeta values of Euler. In the
present paper we establish an effective criterion for Eulerian multizeta
values, which characterizes when a multizeta value is a rational multiple of a
power of the Carlitz period. The resulting "t-motivic" algorithm can tell
whether any given multizeta value is Eulerian or not. We also prove that if
zeta_A(s_1,...,s_r) is Eulerian, then zeta_A(s_2,...,s_r) has to be Eulerian.
When r=2, this was conjectured (and later on conjectured for arbitrary r) by
Lara Rodriguez and Thakur for the zeta-like case from numerical data. Our
methods apply equally well to values of Carlitz multiple polylogarithms at
algebraic points and zeta-like multizeta values.Comment: 32 page