Abstract
We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to -X whose class numbers are indivisible by a given prime ell and whose imaginary quadratic fields satisfy any given set of local conditions. This estimate matches the best results in the direction of the Cohen–Lenstra heuristics for the number of imaginary quadratic fields with class number indivisible by a given prime. This general result is applied to study rank 0 twists of certain elliptic curves.
Highlights
1 Background Ideal class numbers of imaginary quadratic fields have been studied since Gauss, who conjectured that for any given h, there are only finitely many negative fundamental discriminants D such that h(D) = h
Using the half-integral weight modular forms established by Waldspurger and a theorem of Frey [8], James [15] showed that the elliptic curve with Cremona label 14B satisfies ME0(X) X
Wiles [31] established the existence of imaginary quadratic fields with prescribed local data whose class numbers are indivisible by a given odd prime
Summary
1 Background Ideal class numbers of imaginary quadratic fields have been studied since Gauss, who conjectured that for any given h, there are only finitely many negative fundamental discriminants D such that h(D) = h. Strong results on Goldfeld’s conjecture have been obtained for special elliptic curves by making use of the aforementioned theorem of Davenport and Heilbronn on the 3indivisibility of class numbers. Using the half-integral weight modular forms established by Waldspurger and a theorem of Frey [8], James [15] showed that the elliptic curve with Cremona label 14B satisfies ME0(X) X.
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