TY - JOUR

T1 - Tate-shafarevich groups and frobenius fields of reductions of elliptic curves

AU - Shparlinski, Igor E.

PY - 2010/6

Y1 - 2010/6

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77952540015&partnerID=8YFLogxK

U2 - 10.1093/qmath/hap001

DO - 10.1093/qmath/hap001

M3 - Article

VL - 61

SP - 255

EP - 263

JO - Quarterly Journal of Mathematics

JF - Quarterly Journal of Mathematics

SN - 0033-5606

IS - 2

ER -