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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. http://resolver.caltech.edu/CaltechAUTHORS:20190329-145231143

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Abstract

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
Related URLs:
URLURL TypeDescription
https://doi.org/10.1007/978-3-319-97379-1_11DOIArticle
https://arxiv.org/abs/1806.04925arXivDiscussion Paper
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.
Funders:
Funding AgencyGrant Number
NSFUNSPECIFIED
Simons FoundationUNSPECIFIED
Classification Code:2010 Mathematics Subject Classification: 11F03; 11F67; 11G05; 11G55
Record Number:CaltechAUTHORS:20190329-145231143
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20190329-145231143
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:94296
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:29 Mar 2019 22:30
Last Modified:29 Mar 2019 22:30

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