Exponents of Class Groups and Elliptic Curves

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.

show abstract

On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments

Pierce

Turnage-Butterbaugh

Wood

2021

Mathematical Research Letters

View full text Add to dashboard Cite

It is conjectured that within the class group of any number field, for every integer ℓ ≥ 1, the ℓ-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength of this conjecture remains open, and even partial progress is limited. Significant recent progress toward average versions of the ℓ-torsion conjecture has relied crucially on counts for number fields, raising interest in how these two types of question relate. In this paper we make explicit the quantitative relationships between the ℓ-torsion conjecture and other well-known conjectures: the Cohen-Lenstra heuristics, counts for number fields of fixed discriminant, counts for number fields of bounded discriminant (or related invariants), counts for elliptic curves with fixed conductor. All of these considerations reinforce that we expect the ℓ-torsion conjecture to be true, despite limited progress toward it. Our perspective focuses on the relation between pointwise bounds, averages, and higher moments, and demonstrates the broad utility of the "method of moments." Pierce, Turnage-Butterbaugh, and Wood 7 Known results toward Conjecture 1.1, and relation to GRH 606 8 Appendices 611 References 614 1/2+ε K , 1 We will use Vinogradov's notation: A ≪ B denotes that there exists a constant C such that |A| ≤ CB, and A ≪ κ B denotes that C may depend on κ.

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Exponents of Class Groups and Elliptic Curves

Cited by 3 publications

References 8 publications

On signatures of elliptic curves and modular forms

On signatures of elliptic curves and modular forms

Computing the Modular Degree of an Elliptic Curve

On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments

Contact Info

Product

Resources

About