We derive an explicit zero-free region for symmetric square L-functions of
elliptic curves, and use this to derive an explicit lower bound for the modular
degree of rational elliptic curves. The techniques are similar to those used in
the classical derivation of zero-free regions for Dirichlet L-functions, but
here, due to the work of Goldfield-Hoffstein-Lieman, we know that there are no
Siegel zeros, which leads to a strengthened result

Watkins, Mark

English

arXiv

Abstract. We give explicit lower bounds on the modular degree of a rational elliptic curve. The technique is via a convolution-type formula involving the symmetric-square L-function, for which an analogue of “no Siegel zeros ” is known due to a result of Goldfeld, Hoffstein, and Lieman; our main task is to determine an explicit constant for their bound. Combined with an easy bound on the Faltings height in terms of the discriminant, this gives an explicit lower bound on the modular degree that is À N7/6 logN log logN where N is the conductor. This improves previous explicit bounds that were linear in N. In an appendix, we calculate the Euler factors and local conductors for symmetric power L-functions of an elliptic curve, a topic of independent interest. 1

Mark Watkins

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Explicit lower bounds on the modular degree of an elliptic curve

We derive an explicit zero-free region for symmetric square L-functions of elliptic curves, and use this to derive an explicit lower bound for the modular degree of rational elliptic curves. The techniques are similar to those used in the classical derivation of zero-free regions for Dirichlet L-functions, but here, due to the work of Goldfield-Hoffstein-Lieman, we know that there are no Siegel zeros, which leads to a strengthened result

Syracuse University Research Facility and Collaborative Environment

Explicit Lower Bounds on the Modular Degree of an Elliptic Curve

https://surface.syr.edu/cgi/viewcontent.cgi?article=1110&amp;context=mat

Explicit lower bounds on the modular degree of an elliptic curve

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