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

Special L-values and shtuka functions for Drinfeld modules on elliptic curves We make a detailed account of sign-normalized rank 1 Drinfeld $$\mathbf {A}$$ A -modules, for  $$\mathbf {A}$$ A the coordinate ring of an elliptic curve over a finite field, in order to provide a parallel theory to the Carlitz module for $$\mathbb {F}_q[t]$$ F q [ t ] . Using precise formulas for the shtuka function for $$\mathbf {A}$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

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

, Volume 5 (1) – Jan 30, 2018
47 pages

/lp/springer_journal/special-l-values-and-shtuka-functions-for-drinfeld-modules-on-elliptic-LUKUYJWTiU
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis
eISSN
2197-9847
D.O.I.
10.1007/s40687-018-0122-8
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

Research in the Mathematical SciencesSpringer Journals

Published: Jan 30, 2018

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