Stark units in positive characteristic

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

doi:10.1112/plms.12051

Cited by 21 publications

(54 citation statements)

References 31 publications

(152 reference statements)

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“…Let q be a fixed power of a prime p, and let F q be the field with q elements. We let E be an elliptic curve defined over F q , given by the equation (8) E : y 2 + a 1 ty + a 3 y = t 3 + a 2 t 2 + a 4 t + a 6 , a i ∈ F q , with point at infinity set to be ∞. We let A = F q [t, y] be the coordinate ring of functions on E regular away from ∞, and we let K = F q (t, y) be its fraction field.…”

Section: Setting and Notationmentioning

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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Section: Setting and Notationmentioning

confidence: 99%

“…, where the variables t and y satisfy equation (8). The ring T is complete with respect to the Gauss norm · , where for g = c n t n ∈ T,…”

Section: Setting and Notationmentioning

confidence: 99%

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

Green

Papanikolas

2018

Res Math Sci

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“…where λ is the invariant differential of E from (5). We remark that in defining the maps T and RES Ξ (i) , we were partially inspired by ideas of Sinha in [40, §4.6.6].…”

Section: Anderson Generating Functions and Periodsmentioning

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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“…Proof. In [4], Lemma 4.6, we only gave a sketch of the proof of the above results. We give here a detailed proof for the convenience of the reader.…”

Section: Basic Properties Of a Shtuka Functionmentioning

confidence: 99%