Twisted characteristic p zeta functions

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

doi:10.1016/j.jnt.2016.05.002

Cited by 9 publications

(10 citation statements)

References 42 publications

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“…IV]. We should also compare (7) with older results of Carlitz [15,Eq. (9.09)] on reciprocal sums for F q [θ] (see (75)).…”

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confidence: 95%

“…1], namely (4) L(F q [t]; 1) = − π (t − θ)ω C , which serves to interpolate values of Goss L-series of Dirichlet type as well as Carlitz zeta values at some negative integers. There has been an abundance of research in recent years on Pellarin L-series, including special value formulas and multivariable generalizations (e.g., see [7], [9]- [12], [25], [36]- [38]).…”

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confidence: 99%

“…For example, in Proposition 6.4 we find for i ≥ 2, (1) · · · f (i−1) , where ν is the numerator of the shtuka function in (18) and g i is given by a specific linear polynomial in t, y in (81). We further determine a formula for S i itself in Theorem 6.5, namely for i ≥ 2, (7) S i = ν (i) g (1) i · f (1) · · · f (i) Ξ , which provides a functional interpretation of a previous formula of Thakur [43,Thm. IV].…”

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confidence: 99%

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

Green

Papanikolas

2018

Res Math Sci

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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.

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“…IV]. We should also compare (7) with older results of Carlitz [15,Eq. (9.09)] on reciprocal sums for F q [θ] (see (75)).…”

mentioning

confidence: 95%

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confidence: 99%

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confidence: 99%

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

Green

Papanikolas

2018

Res Math Sci

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“…Define the canonical isomorphisms (6) ι : K → K, χ : K → K such that ι(t) = θ and ι(y) = η and so on. For convenience, for x ∈ K we will sometimes refer to χ(x) = x.…”

Section: Background and Notationmentioning

confidence: 99%

Tensor powers of rank 1 Drinfeld modules and periods

Green

2022

Journal of Number Theory

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We study tensor powers of rank 1 sign-normalized Drinfeld A-modules, where A is the coordinate ring of an elliptic curve over a finite field. Using the theory of A-motives, we find explicit formulas for the A-action of these modules. Then, by developing the theory of vector-valued Anderson generating functions, we give formulas for the period lattice of the associated exponential function. the Carlitz module and [32, §3] for tensor powers of the Carlitz module). Further, we have a nice product formula for π, the Carlitz period, and a formula for the bottom coordinate of the fundamental period associated with tensor powers of the Carlitz module (see [4, §2.5]).As a generalization for the Carlitz module, Drinfeld introduced the notion of Drinfeld modules (see also [21], [28] or [45] for a thorough account of Drinfeld modules). Since their introduction, many researchers have worked to developed an explicit theory for Drinfeld modules which parallels that for the Carlitz module, notably Anderson in [2] and [3], Thakur in [43] and [44], Dummit and Hayes in [17], and Hayes in [27]. To discuss the results of the present paper, we first recall a few basic facts about rank 1 sign-normalized Drinfeld modules over rings A, where A is the affine coordinate ring of an elliptic curve E/F q (see §3 for a more thorough review of Drinfeld modules). Define A = F q [t, y], where t and y are related via a cubic Weierstrass equation for E. Also define an isomorphic copy of A, which we denote A = F q [θ, η], where θ and η satisfy the same cubic Weierstrass equation as t and y. Let K be the fraction field of A, let K ∞ be the completion of K at the infinite place, let Date: October 4, 2018.

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“…In particular, we obtain a rationality result similar to that of Pellarin (Theorem 5.6). Our result involves twisted Lseries (see [5]) and a generalization of the Anderson-Thakur special function. The involved techniques are based on ideas developed in [6] where an analogue of Stark Conjectures is proved for sign-normalized rank one Drinfeld modules.…”

Section: Introductionmentioning

confidence: 96%