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Eisenstein Series of Weight One, q-Averages of the 0-Logarithm and Periods of Elliptic Curves

Grayson, Daniel R. and Ramakrishnan, Dinakar (2018) Eisenstein Series of Weight One, q-Averages of the 0-Logarithm and Periods of Elliptic Curves. In: Geometry, Algebra, Number Theory, and Their Information Technology Applications. Springer Proceedings in Mathematics & Statistics. No.251. Springer , Cham, Switzerland, pp. 245-266. ISBN 978-3-319-97378-4.

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For any elliptic curve E over k ⊂ R with E(C) = C^×/q^Z, q = e^(2πiz),Im(z) >, we study the q-average D_(0,q), defined on E(C), of the function D_0(z) = Im(z/(1−z)). Let Ω+(E) denote the real period of E. We show that there is a rational function R ∈ Q(X_1(N)) such that for any non-cuspidal real point s ∈ X_1(N) (which defines an elliptic curve E(s) over R together with a point P(s) of order N), πD_(0,q)(P(s)) equals Ω+(E(s))R(s). In particular, if s is Q-rational point of X_1(N), a rare occurrence according to Mazur, R(s) is a rational number.

Item Type:Book Section
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Additional Information:© 2018 Springer Nature Switzerland AG. First Online: 18 September 2018. D. R. Grayson and D. Ramakrishnan Research supported by the NSF; D. Ramakrishnan supported by a Simons Fellowship.
Funding AgencyGrant Number
Simons FoundationUNSPECIFIED
Series Name:Springer Proceedings in Mathematics & Statistics
Issue or Number:251
Classification Code:2010 Mathematics Subject Classification: 11F03; 11F67; 11G05; 11G55
Record Number:CaltechAUTHORS:20190329-145231143
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:94296
Deposited By: Tony Diaz
Deposited On:29 Mar 2019 22:30
Last Modified:16 Nov 2021 17:04

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