Special functions and twisted L-series

Anglès, Bruno; Dac, Tuân Ngô; Ribeiro, Floric Tavares

doi:10.5802/jtnb.1007

Cited by 13 publications

(18 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have thus proved the following lemma. For other approaches to defining Anderson-Thakur functions for rank one Drinfeld modules on curves of arbitrary genera, the reader is directed to [6].…”

Section: Since Elements Ofmentioning

confidence: 99%

See 1 more Smart Citation

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

Green

Papanikolas

2018

Res Math Sci

View full text Add to dashboard Cite

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.

show abstract

Section: Since Elements Ofmentioning

confidence: 99%

“…Using different methods, more recently Anglès, Ngo Dac, and Tavares Ribeiro [6] have also shown that similar sums are rational multiples of π ρ /ω ρ , but although their arguments work for all genera, they do not obtain precise formulas.…”

mentioning

confidence: 99%

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

Green

Papanikolas

2018

Res Math Sci

View full text Add to dashboard Cite

show abstract

“…We have a natural sign function

\bar{prefixsgn} : {double-struckC}_{\infty} {(false(\frac{1}{t} false))}^{\times} \to {double-struckC}_{\infty}^{\times}

attached to

\frac{1}{t}

. When

d_{\infty} = 1

, we normalize

f

such that

\bar{prefixsgn} false(f false) = 1

(such a normalization does not hold when

d_{\infty} > 1

, see ). We set

false(\infty false) = \sum_{i = 0}^{d_{\infty} - 1} ({\bar{\infty}}^{false(i false)}),

W false(C_{\infty} false) = ⋃_{m ⩾ 0} L false(V + m (\infty) false),

where

L false(V + m (\infty) false) = \{x \in F^{\times}, (x) + V + m (\infty) ⩾ 0\} \cup \{0\} .

Observe that for

i > 0

(f f^{false(1 false)} \dots f^{false(i - 1 false)}) = V^{(i)} - V + false(ξ false) + \dots + (ξ^{false(i - 1})

…”

Section: Log‐algebraicity Theoremmentioning

confidence: 99%

“…Thus,

f = \sum_{i ⩾ k} f_{i} \frac{1}{t^{i}} \in H (((), \frac{1}{t})) \subset {double-struckC}_{\infty} (((), \frac{1}{t})),

where

f_{i} \in H, \forall i ⩾ k

f_{k} \neq 0

k ⩽ - 1

. We refer the interested reader to for a detailed study of the arithmetic of such a sthuka function

f

.…”

Section: Log‐algebraicity Theoremmentioning

confidence: 99%

Stark units in positive characteristic

Anglès

Dac

Ribeiro

2017

Proceedings of the London Mathematical Society

Self Cite

View full text Add to dashboard Cite

Abstract. We show that the module of Stark units associated to a signnormalized rank one Drinfeld module can be obtained from Anderson's equivariant A-harmonic series. We apply this to obtain a class formulaà la Taelman and to prove a several variable log-algebraicity theorem, generalizing Anderson's log-algebraicity theorem. We also give another proof of Anderson's log-algebraicity theorem using shtukas and obtain various results concerning the module of Stark units for Drinfeld modules of arbitrary rank.

show abstract

“…There are many recent generalizations and interesting important results in different directions due to Taelman, Papanikolas, Pellarin, Perkins, Angles, Ngo Dac, Tavares Riberio, Fang, Derby, see [9,10,20,22,36,38,41,43] and other papers on arXiv.…”

Section: General a L-functions Etcmentioning

confidence: 99%