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We make a detailed account of sign-normalized rank 1 Drinfeld A-modules, forA the coordinate ring of an elliptic curve over a finite field, in order toprovide a parallel theory to the Carlitz module for F_q[t]. Using preciseformulas for the shtuka function for A, we obtain a product formula for thefundamental period of the Drinfeld module. Using the shtuka function we findidentities for deformations of reciprocal sums and as a result prove specialvalue 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.

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