Root numbers and parity of ranks of elliptic curves

Abstract

Abstract. The purpose of the paper is to complete several global and local results concerning parity of ranks of elliptic curves. Primarily, we show that the Shafarevich-Tate conjecture implies the parity conjecture for all elliptic curves over number fields, give a formula for local and global root numbers of elliptic curves and complete the proof of a conjecture of Kramer and Tunnell in characteristic 0. The method is to settle the outstanding local formulae by deforming from local fields to totally real number fields and then using global parity results.

Growth of in towers for isogenous curves

“…The Tamagawa numbers for types II and II * are always 1 ([30] §IV.9, Table 4.1), and they turn out to be 3 for the IV, IV * cases, so that the Tamagawa quotient alternates between 3 and 1 3 . (To see that they are 3, we use the fact that over a local field K/Q 3 , the parity of ord 3 c(E ′ /K) c(E/K) can be recovered Om φ (K), the local root number w(E/K), and the Artin symbol (−1, K(ker φ t )/K); see [8] Thm 5.7. In our case, Om φ (K) is computed as in Remark 4.8, the local root number is +1 over Q 3 and is unchanged in odd degree Galois extensions, and the points in ker φ t are defined over Q, so that the Artin symbol is trivial.…”

A note on the Mordell–Weil rank modulo n

“…If the Birch-Swinnerton-Dyer conjecture holds (or if Ш is finite, see [4]), then the MordellWeil rank modulo 2 is a sum of local invariants with values in Z/2Z. Specifically, for an elliptic curve E over a local field k write w(E/k) = ±1 for its local root number, and define λ by…”

A remark on the Diophantine equation f(x)=g(y)

“…Recently, B. Mazur and K. Rubin ( [6]) obtained several general results on the rank of the quadratic twists of E/K based on a detailed analysis of the 2-Selmer groups (Lemma 2.2). Using their results and a characterization of elliptic curves with constant 2-Selmer parity from [2], we show that the answer to Question 1.1. for (m, n) = (2, 3) is positive over algebraic number fields (Theorem 2.3. ).…”

“…Let K be a field of algebraic numbers and let E be an elliptic curve over K such that E(K) [2] = 0. Then:…”