On elliptic curves whose conductor is a product of two prime powers

Abstract

We find all elliptic curves defined over Q that have a rational point of order N, N ≥ 4, and whose conductor is of the form p a q b , where p, q are two distinct primes, a, b are two positive integers. In particular, we prove that Szpiro's conjecture holds for these elliptic curves.problem under the condition that E has a rational torsion point. For example,in [6] the elliptic curves with conductor p m , p is prime, and 2-torsion points were listed.It was shown in [5] that all elliptic curves with conductor 2 m p n where p ≡ 3 or 5 mod 8, p = 3, that have a rational point of order 2, are effectively determined under the truth of the conjecture of Ankeny-Artin-Chowla.It is worth mentioning that the complete list of elliptic curves with a prime conductor has already been produced. The following theorem gives this list explicitly.Theorem 1.1 (Theorem 5.3.2, [10]). Let E be an elliptic curve over Q with prime conductor p. Then either |∆ E | = p or p 2 , or else p = 11 and ∆ E = 11 5 , or p = 17 and ∆ E = 17 4 , or p = 19 and ∆ E = 19 3 , or p = 37 and ∆ E = 37 3 . In particular, ∆ E | p 5 .The elliptic curves in Theorem 1.1 turn out to satisfy Szpiro's conjecture which is stated below for the convenience of the reader.Conjecture 1.2. If E is an elliptic curve over Q, thenOne of the popular strategies to find elliptic curves E/Q with a given conductor is to solve certain Diophantine equations obtained by equating the discriminant of E to the product of powers of the prime divisors of the conductor.Mazur gave a complete classification of the torsion subgroup E tors (Q) of E(Q), see (Theorem 7.5, §8, Chapter VIII, [11]). More precisely, E tors (Q) is isomorphic to one of the following fifteen groups: Z/nZ, 1 ≤ n ≤ 12, n = 11; or Z/2Z × Z/2nZ, 1 ≤ n ≤ 4.Given that P ∈ E(Q)[m], m = 2, 3, it is known that there exist b, c ∈ Q such that the following Weierstrass equation defines an elliptic curve E b,c isomorphic to E E b,c : y 2 + (1 − c)xy − by = x 3 − bx 2 with the image of P being (0, 0). The discriminant ∆(b, c) of E b,c is given as follows:By taking m to be an integer in {4, 5, 6, 7, 8, 9, 10, 12}, one finds an explicit relation between b, c, see for example §2 of [7].In this article, we generalize Theorem 1.1 to elliptic curves whose conductors have two distinct prime divisors only. More precisely, we generate the list of all elliptic curves

On higher congruences between cusp forms and Eisenstein series. II

“…Elliptic curves with conductor N a product of two primes were partially classified in [19]. This result allows us to discard the congruences with r ∈ {8, 9}.…”

“…Then the inequality |a 3 (f )| ≤ 2 √ 3 < 4 and the congruence a 3 (f ) ≡ 4 (mod r ) holds, hence r | (4 − a 3 (f )) < 8. For r = 7 by [19,Theorem 3.6] it follows that N = 26. We compute that the space S 2 (26) new is of dimension 2 and spanned by the forms f 1 , f 2 with the following Fourier expansions f 1 = q − q 2 + q 3 + q 4 − 3q 5 − q 6 − q 7 − q 8 − 2q 9 + 3q 10 + 6q 11 + .…”

Genus 2 curves with bad reduction at one odd prime

“…Ivorra [11] classified elliptic curves over Q of conductor 2 k p (p odd prime) with a rational point of order 2. Bennett, Vatsal and Yazdani [1] classified all elliptic curves over Q with a rational 3-torsion point and good reduction outside the set {3, p}, for a fixed prime p. Let us also mention the papers by Howe [9], Sadek [22] and Dąbrowski-Jędrzejak [7] concerning classification of elliptic curves over Q with good reduction outside two distinct primes and with a rational point of fixed order ≥ 4.…”

Elliptic curves over the rational numbers with semi-abelian reduction and two-division points

“…We just mention a few: Ogg himself analyzed elliptic curves with discriminant ±2 ν • 3 and ±2 ν • 3 2 in [15]. The case ∆ = ±2 ν • p was studied by Ivorra [11], those with ∆ = p 1 p 2 by Sadek [17]. Ogg's results where extended to imaginary quadratic number fields by Setzer [20], Stroecker [21] and Kida [12].…”