
TL;DR
This paper investigates the product of small Elkies primes for elliptic curves over finite fields, showing that for infinitely many pairs, this product can be significantly larger than heuristic estimates suggest, impacting point-counting algorithms.
Contribution
It proves the existence of infinitely many elliptic curves over finite fields with a large product of Elkies primes, surpassing heuristic bounds, thus providing new insights into the distribution of Elkies primes.
Findings
Existence of infinitely many pairs with large Elkies prime products
Demonstrates that $L_p(E)$ can grow faster than heuristic estimates
Implications for the efficiency of point-counting algorithms
Abstract
Given an elliptic curve over a finite field of elements, we say that an odd prime is an Elkies prime for if is a quadratic residue modulo , where t_E = q+1 - #E(\F_q) and #E(\F_q) is the number of -rational points on . These primes are used in the presently most efficient algorithm to compute #E(\F_q). In particular, the bound such that the product of all Elkies primes for up to exceeds is a crucial parameter of this algorithm. We show that there are infinitely many pairs of primes and curves over with for some absolute constant , while a naive heuristic estimate suggests that . This complements recent results of Galbraith and Satoh (2002), conditional under the Generalised Riemann Hypothesis, and of…
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