Primes, Elliptic Curves and Cyclic Groups: a Synopsis
نویسنده
چکیده
The main question addressed in this paper focuses on the frequency with which the reductions modulo primes of a rational elliptic curve give rise to cyclic groups. This question is part of a broad theme of investigations about the distribution of Frobenius in an infinite family of division fields defined by an elliptic curve over a global field (or in variations of such families in other arithmetic-geometric contexts). Illustrative of many of the ideas, methods and obstacles that occur in the broader theme, the investigations of the cyclicity question are foundational for a researcher interested in pursuing analytic studies of primes in arithmetic-geometric contexts. While most of the paper is a survey of prior results, the statement and detailed outline of proof of the more refined version of part (i) of Theorem 48, given in Section 10.1, are new.
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