Selmer's Conjecture and Families of Elliptic Curves

Abstract

1. In this note we consider the elliptic curve d:y 2 = x { x 2 -( l + f 4 ) 2 } (1.1) over the field C(t) of rational functions with complex coefficients. It turns out that all C(0-r ational points on si satisfy x , y e QU/2, i, t). As a consequence, the curvehas only points of order 2 over Q(t). On the other hand, the famous conjecture of Selmer [7] is shown to imply that for every rational r the curve 28 r \y 2 = x{x 2 -(7 + 7r 4 ) 2 } (1.3) has infinitely many rational points.To place this phenomenon in a more general setting, let R = Q[t] and let F E R[x, y] be irreducible over the algebraic closure of Q{t). If x, y are regarded as coordinates of affine space, the equation F = 0 defines an affine curve over Q(t) and also a family of affine curves F(x, y, r) = 0 over Q, as r runs over the elements of Q such that F(x, y, r) is absolutely irreducible (the remaining values of r form a finite set £).Let # be the projective curve defined over Q(t) obtained from normalization of the completion of the curve F = 0 and let ( € r be the normalization of the completion of the affine curve F(x, y, r) = 0 for r e Q \ £ .If ( ti is a curve of genus 1, then for all but finitely many values r e Q \ £ the curve # r is also of genus 1. Without further comment we ignore the exceptional values of r. Let ^(Q(0) denote the set of Q(t)-rational points on # , and # r (Q) the set of Q-rational points on # r .If %>(Q(t)) and ^r(Q) are nonempty, they have natural structure as groups when one point is selected as origin. They are finitely generated by virtue, respectively, of the theorems of Neron ([6, Theoreme 3]) and of Mordell. Let g{^) (respectively gi^r)) be the reduced rank of #(Q(t)) (respectively # r (Q)). Neron has shown (ibid., Theoreme 6) that there are infinitely many r e Q such that g(^r) $s g{^). Our example shows that there can be strict inequality for every rational r.An analogous result where r is restricted to values in Z has been obtained by D.J. Lewis and A. Schinzel [4]. The curve # given by y 2 = x 4 -( 8 t 2 + 5) 2 has g(<%) = 0, but Selmer's Conjecture implies that g{%> r ) ^ 1 for every integer r. In this example, however, g{^r) = 0 for infinitely many rational r.

Root numbers and ranks in positive characteristic

“…Thus, k is separably closed with char(k) = 2, so must be a square in K × since v( ) is even. By the 2-torsion hypothesis, for any Weierstrass K-model y 2 = f (x) for E there is at least one K-rational root of the cubic f. The discriminant of f is a square in K × , so f splits over K and hence E [2] is K-split.…”

“…attached to E. Since E [2] is K-split, has trivial reduction modulo 2. Thus, the Galois action must be pro-2, and hence tame.…”

“…We first address the properties of e in the cases with potentially good reduction. By the method of proof of Lemma 2.1 (2), if E has potentially good reduction then it acquires good reduction over a quadratic extension of a splitting field for E [2]. This splitting field is a tame Galois extension with degree dividing 6, so Lemma 2.1 implies that e must divide 12 and moreover that if p = 3 then 3 e. The tameness and Lemma 2.1 ensure that e is the order of the image of inertia in the -adic representation for E (any = p).…”

“…The discriminant and parameters c 4 and c 6 of such a model are given by the following formulas [31, p. 46]: [2], the point [2]P has coordinates given by [31, pp. 58-59]:…”

On the rank of the fibers of rational elliptic surfaces

“…There are examples for which W (E t ) is constant, but they all correspond to isotrivial surfaces; see for example [Cassels and Schinzel 1982]. In the nonisotrivial case and for B ‫ސ‬ 1 , H. Helfgott [2003] has shown, under classical conjectures, that the sets U ± (k) are infinite.…”