Counting elliptic curves with bad reduction over a prescribed set of primes

Abstract

Let p ≥ 5 be a prime and T a Kodaira type of the special fiber of an elliptic curve. We estimate the number of elliptic curves over Q up to height X with Kodaira type T at p. This enables us find the proportion of elliptic curves over Q, when ordered by height, with Kodaira type T at a prime p ≥ 5 inside the set of all elliptic curves. This proportion is a rational function in p. For instance, we show that p 8 (p − 1) p 9 − 1 of all elliptic curves with bad reduction at p are of multiplicative reduction. Furthermore, we prove that the prime-to-6 part of the conductors of a majority (= ζ(10)/ζ(2) ≈ 0.6) of elliptic curves are squarefree, where ζ is the Riemann-zeta function.

Local and global densities for Weierstrass models of elliptic curves

“…This paper grew out of independent work of each of the authors: unpublished notes on purely local densities (at arbitrary primes) by Cremona, and a 2017 preprint [28] on global densities (excluding conditions at the primes p = 2 and p = 3) by Sadek. After the first version of the current paper appeared online, we noticed a new preprint [11] by Cho and Jeong, whose subject matter has some overlap with the current paper, but with several differences: conditions at the primes 2 and 3 are excluded in [11], and only conditions at finitely many primes are considered, through the use of short Weierstrass equations.…”

Statistics for Iwasawa invariants of elliptic curves

“…(2) E κ : y 2 = x 3 + ax + b has a point over F p of order p. For the primes p in the range 5 ≤ p < 500, computations on sage show that d(p) ≤ 1 and d(p) = 1 for p ∈ {5, 7, 61}. The estimate (4.4) follows from the method of M. Sadek [30], or the results of J. Cremona and Sadek, see [8].…”

“…Average results on Tamagawa numbers. Let p ≥ 5 be a fixed prime, and l be a prime different from p. In this section, we estimate the proportion of elliptic curves E/Q up to height X with Kodaira type I p at l. These estimates are well known, but we include them for the sake of completeness, see [30,8].…”

“…We include both upper and lower bounds, however, we only apply upper bounds in our analysis. The following calculations have been done in the preprint [30,Lemma 4.1]. We clarify the arguments and include them here for completeness.…”