Elliptic curves with a torsion point

Abstract

Let E be an elliptic curve defined over the field Q of rational numbers, then the torsion subgroup of the Mordell-Weil group E(Q) is finite and it is known that there exist the elliptic curves whose torsion subgroups E(Q)t are of the following types: (1), (2), (3), (2, 2), (4), (5), (2, 3), (7), (2, 4), (8), (9), (2, 5), (2, 2, 3), (3, 4) and (2, 8). It has been conjectured from various reasons that E(Q)t is exhausted by the above types only. If E has a torsion point of order precisely n, then it is known that E has an n-isogeny, that is to say, an isogeny of degree n.

A conjecture of Gross and Zagier: Case E(ℚ)tor≅ℤ/3ℤ

“…Now consider the local images of Sel φ d (E d /Q) as follows. 12 Note that for any odd primes ℓ dividing ∆, we get…”

“…[12], Theorem 1.1). The quotient curve E ′ := E/T has a rational point of order 3 if and only if b is a cube t 3 with t > 0.…”

ON ELLIPTIC CURVES WHOSE 3-TORSION SUBGROUP SPLITS AS μ3⊕ℤ/3ℤ

“…We note that the Weierstrass equation of E(α, b) given by the equation (1) is minimal if (α, β) ∈ C 27b (Z) with gcd(α, β) = 1 (see [2,Section 1]). …”

“…Since the point P is rational over Q, this map gives an exact sequence (2) 0 → Z/3Z → E [3] → µ 3 → 0 of Gal(Q/Q)-modules. The purpose of this paper is to study elliptic curves E such that E [3] is split as µ 3 ⊕ Z/3Z.…”

Normal forms, K3 surface moduli, and modular parametrizations

“…with the singular fibers of types I 1 , I 2 , I 3 , I 6 over the points t = −8, 1, 0, ∞ respectively. Another model for this surface is given in [16]:…”