Identities for Anderson generating functions for Drinfeld modules

El-Guindy, Ahmad; Papanikolas, Matthew A.

doi:10.1007/s00605-013-0543-9

Cited by 19 publications

(47 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These functions serve similar purposes as the functions B n (t) in [21,Eq. (6.4)], and in particular via (3.2.7) they specialize at z = θ as logarithm coefficients: that is, γ n (θ) = β n , where log φ = n 0 β n τ n ∈ T s [[τ ]].…”

Section: Proof Of the De Rham Isomorphismmentioning

confidence: 97%

The de Rham isomorphism for Drinfeld modules over Tate algebras

Gezmiş

Papanikolas

2019

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

Introduced by Anglès, Pellarin, and Tavares Ribeiro, Drinfeld modules over Tate algebras are closely connected to Anderson log-algebraicity identities, Pellarin L-series, and Taelman class modules. In the present paper we define the de Rham map for Drinfeld modules over Tate algebras, and we prove that it is an isomorphism under natural hypotheses. As part of this investigation we determine further criteria for the uniformizability and rigid analytic triviality of Drinfeld modules over Tate algebras.

show abstract

Section: Proof Of the De Rham Isomorphismmentioning

confidence: 97%

The de Rham isomorphism for Drinfeld modules over Tate algebras

Gezmiş

Papanikolas

2019

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…We turn M into an L[t, y, τ ]-module and N into an L[t, y, σ]-module by defining the action for a ∈ M and b ∈ N as (18) τ a = f n a (1) and σb = f n b (−1) .…”

Section: Tensor Powers Of A-motivesmentioning

confidence: 99%

Tensor powers of rank 1 Drinfeld modules and periods

Green

2022

Journal of Number Theory

View full text Add to dashboard Cite

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.

show abstract

“…where we recall the definition of N θ and N η from (21). Next, we denote the matrix M D = η q i I − θ q i M m − M d , and note that it is diagonal and invertible.…”

Section: Coefficients Of the Exponential Functionmentioning

confidence: 99%

Special zeta values using tensor powers of Drinfeld modules

Green

2019

Mathematical Research Letters

View full text Add to dashboard Cite

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 vector valued Anderson generating functions, we give formulas for the coefficients of the logarithm and exponential functions associated to these A-modules. We then show that there exists a vector whose bottom coordinate contains a Goss zeta value, whose evaluation under the exponential function is defined over the Hilbert class field. This allows us to prove the transcendence of Goss zeta values and periods of Drinfeld modules as well as the transcendence of certain ratios of those quantities.

show abstract

Identities for Anderson generating functions for Drinfeld modules

Cited by 19 publications

References 21 publications

The de Rham isomorphism for Drinfeld modules over Tate algebras

The de Rham isomorphism for Drinfeld modules over Tate algebras

Tensor powers of rank 1 Drinfeld modules and periods

Special zeta values using tensor powers of Drinfeld modules

Contact Info

Product

Resources

About