Tate-shafarevich groups and frobenius fields of reductions of elliptic curves

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Abstract

Let E/ℚ be a fixed elliptic curve over ℚ which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, Cojocaru and Duke have obtained an asymptotic formula for the number of primes p ≤ x such that the reduction of E modulo p has a trivial Tate-Shafarevich group. Recent results of Cojocaru and David lead to a better error term. We introduce a new argument in the scheme of the proof, which gives a further improvement.

Original languageEnglish
Pages (from-to)255-263
Number of pages9
JournalQuarterly Journal of Mathematics
Volume61
Issue number2
DOIs
Publication statusPublished - Jun 2010

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