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

  • Nathan Green
  • Matthew A. Papanikolas
Research
  • 90 Downloads

Abstract

We make a detailed account of sign-normalized rank 1 Drinfeld \(\mathbf {A}\)-modules, for \(\mathbf {A}\) the coordinate ring of an elliptic curve over a finite field, in order to provide a parallel theory to the Carlitz module for \(\mathbb {F}_q[t]\). Using precise formulas for the shtuka function for \(\mathbf {A}\), 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.

Keywords

Drinfeld modules Pellarin L-series Shtuka functions Reciprocal sums Anderson generating functions Log-algebraicity 

Mathematics Subject Classification

Primary 11G09 Secondary 11R58 11M38 12H10 

Notes

Author’s contributions

All contributions to this article were made jointly by NG and MAP. Both authors read and approved the final manuscript.

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Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

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