Special L-values and shtuka functions for Drinfeld modules on elliptic curves

Abstract

We make a detailed account of sign-normalized rank 1 Drinfeld A-modules, for A the coordinate ring of an elliptic curve over a finite field, in order to provide a parallel theory to the Carlitz module for F q [t]. Using precise formulas for the shtuka function for A, we obtain a product formula for the fundamental period of the Drinfeld module. Using the shtuka function we find identities for deformations of reciprocal sums and as a result prove special value formulas for Pellarin L-series in terms of an Anderson-Thakur function. We also give a new proof of a log-algebraicity theorem of Anderson.

Log‐algebraic identities on Drinfeld modules and special L‐values

“…since a q k ≡ a q k− d (mod f ) for all . By (19)- (20), it suffices to prove (c) for P = ax i . We recall (see [19, § 3.6]) that for 1,…”

“…These results were then generalized to rank 1 Drinfeld modules over more general rings by Anglès, Ngo Dac, and Tavares Ribeiro . Work of Green and the third author provides explicit formulas for log‐algebraic identities for rank 1 Drinfeld modules over coordinate rings of elliptic curves. Log‐algebraicity on tensor powers of the Carlitz module is investigated in a forthcoming paper by Papanikolas.…”

Special zeta values using tensor powers of Drinfeld modules

“…We then show that the sequence of matrices {C i } satisfies the recurrence in Lemma 4.3 for i ≥ 1. First observe that by Proposition 3.1(e) (45) d[θ]g = tg − f n E θ g (1) and d[η]g = yg − f n E η g (1) , with g defined as in (26). Using (45), we write…”

“…In the case of tensor powers of Drinfeld modules, however, the matrices involved are much more complicated and do not give clean formulas as they do in the Carlitz case. We develop new techniques to analyze the coefficients of the logarithm and exponential function inspired partially by work of Papanikolas and the author in [26] and partially by ideas of Sinha in [39]. Further, Anderson and Thakur use special polynomials (called Anderson-Thakur polynomials) in [5] to relate evaluations of the logarithm function to zeta values.…”

“…Further, Anderson and Thakur use special polynomials (called Anderson-Thakur polynomials) in [5] to relate evaluations of the logarithm function to zeta values. It is not yet clear how to generalize these Anderson-Thakur polynomials to tensor powers of Drinfeld modules, and so instead we use a generalization of techniques developed by Papanikolas and the author in [26] to prove formulas for zeta values. We comment that this technique allows us to study zeta values only for 1 ≤ s ≤ q − 1; developing techniques to study zeta values for all n ≥ 1 is a topic of ongoing study (see Remark 6.1).…”