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
AN - SCOPUS:77952540015
SN - 0033-5606
VL - 61
SP - 255
EP - 263
JO - Quarterly Journal of Mathematics
JF - Quarterly Journal of Mathematics
IS - 2
ER -