Arithmetic of elliptic curves upon quadratic extension

Abstract

This paper is a study of variations in the rank of the Mordell-Weil group of an elliptic curve E defined over a number field F as one passes to quadratic extensions K of F. Let S(K) be the Selmer group for multiplication by 2 on E(K). In analogy with genus theory, we describe S(K) in terms of various objects defined over F and the local norm indices <" = dimF2£(Ft))/Norm{£(ÄH,)} for each completion Fv of F. In particular we show that dim S(K) + dim E(K)2 has the same parity as Zi". We compute i" when E has good or multiplicative reduction modulo v. Assuming that the 2-primary component of the Tate-Shafarevitch group U1(K) is finite, as conjectured, we obtain the parity of rank E(K). For semistable elliptic curves defined over Q and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of Birch and Swinnerton-Dyer.

On ranks of twists of elliptic curves and power-free values of binary forms

“…We refer to §2 of [5] for a discussion of this conjecture; see [22], [23] for some recent work on this problem. We also have corresponding conjectures for cubic, quartic and sextic twists.…”

“…Further, in 1960 Honda [18] (see also [38], p. 162) conjectured that the rank of any twist of a given elliptic curve E over a number field K is bounded by a constant which depends on E and K only. Some related work on ranks of twists may be found in [12], [22] and [35].…”

On 2‐Selmer groups of twists after quadratic extension

“…In Section 4, we study algebraically the behaviour of 2-Selmer groups of elliptic curves in quadratic extensions, building on work of Kramer [20]. Along the way we give two reinterpretations of Kramer's work, one in the language of Selmer structures, and another in terms of the Weil restriction of scalars.…”

“…At primes 𝑝 ∈ Σ the cokernel of the local norm map is more complicated and depends on the reduction type of 𝐸 𝑑 ∕ℚ 𝑝 . See [20] or [19] for more details. However, since the isomorphism class of 𝐸 𝑑 over ℚ 𝑝 depends only on the class of 𝑑 in ℚ × 𝑝 ∕ℚ ×2 𝑝 , the same is true of the cokernel of the local norm map.…”

Untitled

“…It had been proved earlier that if the Tate-Shafarevich Conjecture is true, then the Parity Conjecture holds for semistable elliptic curves (combining [33] and Theorem 5.8) and for the curves y 2 = x 3 − d 2 x (see Monsky's appendix to [24]).…”