Rank-One Drinfeld Modules on Elliptic Curves

Dummit, David S.; Hayes, David R.

doi:10.2307/2153547

Cited by 10 publications

(15 citation statements)

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“…where the sum is over all prime ideals p ⊆ A of degree 1. Now, evaluating λ (1) i at Ξ, we see from (17), (18), and (82) that for i ≥ 2,…”

Section: Proof Of a Theorem Of Andersonmentioning

confidence: 99%

“…We specify that V be chosen to be in the formal group of E at the infinite place of K and that f have sign 1 (see §2 for details on signs), and in this way f is uniquely determined. In (18) we write f = ν/δ for explicit ν, δ ∈ H [t, y]. See §3 for the precise ways one uses f to construct ρ, as well as its exponential and logarithm functions exp ρ and log ρ .…”

mentioning

confidence: 99%

“…We apply the methods of Anglès, Pellarin, and Simon to find exact formulas for each of these quantities. 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.…”

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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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“…where the sum is over all prime ideals p ⊆ A of degree 1. Now, evaluating λ (1) i at Ξ, we see from (17), (18), and (82) that for i ≥ 2,…”

Section: Proof Of a Theorem Of Andersonmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

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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“…Some special cases, in both even and odd characteristic, have also been studied by Dummit and Hayes in [6]. In this section we apply Brown's method to determine the signs of all singular moduli in the case of even characteristic.…”

Section: Determining the Signsmentioning

confidence: 99%

On Singular Moduli of Drinfeld Modules in Characteristic Two

Hsia

1998

Journal of Number Theory

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“…As further examples see his two papers with David Dummit [1] and [2]. Early last year, when he was already very ill, his wife informed me that there were page proofs lying around the house.…”

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confidence: 99%

David R. Hayes: Some remarks on his life and work

Rosen

2013

Journal of Number Theory

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Some remarks on his life and work I am sure that I must have met David Hayes for the first time at two summer conferences that were held at Bowdoin College in the mid 1960s. One was about algebraic number theory and the other about algebraic geometry, subjects of great interest to both of us. However, try as I might, I cannot recall a specific interaction during that distant period.I do remember quite vividly reading a paper of his entitled "Explicit class field theory for rational function fields" [10] which appeared in the Transactions of the American Mathematical Society in 1974. In this beautiful work, David exposits and carries forward some important work of his thesis advisor Leonard Carlitz. Carlitz wrote over 770 papers (which may be a record). The one which David wrote about had the uninformative title "A class of polynomials". In this paper Carlitz introduces a mathematical object which would later be renamed the Carlitz module. He showed that torsion points on the Carlitz module generate abelian extensions of F(T ), the rational function field over a finite field. These extensions are analogous to cyclotomic extensions of the rational numbers Q. In addition to giving an elegant exposition of Carlitz's theory, David added the very important theorem that if one adjoins all the torsion points the resulting field is "almost" the maximal abelian extension of F(T ). Moreover, he shows how to modify the construction to give the whole maximal abelian extension, thus creating a function field version of the famous Kronecker-Weber theorem. In the same year 1974, Vladimir Drinfeld published a paper (in Russian) entitled "Elliptic Modules". In future years elliptic modules were called, for obvious reasons, Drinfeld modules. As it turned out, the Carlitz module is a very special case of a Drinfeld module. 0022-314X/2012 Published by Elsevier Inc. http://dx.

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Rank-One Drinfeld Modules on Elliptic Curves

Cited by 10 publications

References 4 publications

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

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

On Singular Moduli of Drinfeld Modules in Characteristic Two

David R. Hayes: Some remarks on his life and work

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