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

@article{Green2016SpecialLA,
  title={Special L-values and shtuka functions for Drinfeld modules on elliptic curves},
  author={Nathan Green and Matthew A. Papanikolas},
  journal={Research in the Mathematical Sciences},
  year={2016},
  volume={5},
  pages={1-47}
}
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 $$\mathbb {F}_q[t]$$Fq[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… 

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References

SHOWING 1-10 OF 44 REFERENCES

Algebraic relations among periods and logarithms of rank 2 Drinfeld modules

TLDR
It is shown that the transcendence degree of the field generated by the entries of the Drinfeld logarithms of algebraic functions which are linearly independent over ${Bbb F}_q(\theta)$ is $4$.

Universal Gauss-Thakur sums and $$L$$L-series

We prove that the maximal abelian extension tamely ramified at infinity of the rational function field over $$\mathbb {F}_q$$Fq is generated by the values at the points in the algebraic closure of

Rank-one Drinfeld modules on elliptic curves

The sgn-normalized rank-one Drinfeld modules 0 associated with all elliptic curves E over Fq for 4 < q < 13 are computed in explicit form. (Such 0 for q < 4 were computed previously.) These

Explicit formulae for $$L$$L-values in positive characteristic

We focus on the generating series for the rational special values of Pellarin’s $$L$$L-series in $$1 \le s \le 2(q-1)$$1≤s≤2(q-1) indeterminates, and using interpolation polynomials we prove a closed

On Pellarin’s $L$-series

Necessary and sufficient conditions are given for a negative integer to be a trivial zero of a new type of $L$-series recently discovered by F. Pellarin, and it is shown that any such trivial zero is

Log-Algebraicity of TwistedA-Harmonic Series and Special Values ofL-Series in Characteristicp

Abstract We find special points in the Carlitz module related, on the one hand, to the values at s =1 of characteristic p Dirichlet L -function analogues, and on the other hand, to the values at

Gamma functions for function fields and Drinfeld modules

The purpose of this paper is to study gamma functions in the context of the theory of function fields of one variable over a finite field. In particular, we explore, among other things, their

Drinfeld modules and arithmetic in the function fields

q: a power of a prime p; K" a function field of one variable over its field of constants F; 'a place of K; A" the ring of elements of K integral outside Koo" the completion of K at f: the completion

ALGEBRAIC INDEPENDENCE OF PERIODS AND LOGARITHMS OF DRINFELD MODULES CHIEH-YU CHANG AND MATTHEW A. PAPANIKOLAS, WITH AN APPENDIX BY BRIAN CONRAD

1.1. Drinfeld logarithms. In this paper we prove algebraic independence results about periods, quasi-periods, logarithms, and quasi-logarithms on Drinfeld modules, which are inspired by conjectures