Tate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves

Details Tate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves Tate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves

2007-07-19

arxiv.org/abs/0707.2976v3

Mathematics

Number Theory

Scientific paper

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

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