Finding Large Selmer Groups

Raoul Bott has inspired many of us by the magnificence of his ideas, by the way he approaches and explains mathematics, and by his warmth, friendship, and humor. In celebration of Raoul’s eightieth birthday we offer this brief article in which we will explain how the recent cohomological ideas of Jan Nekovár̆ [N2] imply (under mild hypotheses plus the Shafarevich-Tate conjecture) systematic growth of the ranks of the group of rational points in certain elliptic curves as one ascends the finite layers of appropriate towers of number fields. LetK/k be a quadratic extension of number fields, and denote by σ the nontrivial automorphism of K/k. Let p be an odd prime number. By a Zp-power extension of K we mean an abelian extension L/K with Galois group Zp for some d. If L/K is a Zp-power extension and L/k is Galois, then σ acts on Gal(L/K) and we will say that L/K is k-positive (resp. k-negative) if σ acts trivially (resp. by the scalar −1) on Gal(L/K). Thus L/k is abelian if L/K is k-positive, and Gal(L/k) is a generalized dihedral group if L/K is k-negative. For any such K/k there is a maximal k-positive Zp-power extension K , and a maximal k-negative one K −. The extension K /K is always nontrivial because K + contains the cyclotomic Zp-extension of K. The extension K −/K is nontrivial if K is not totally real (see Lemma 3.2). If E is an elliptic curve defined over K and L is a (possibly infinite) extension of K, say that E has Mordell-Weil growth relative to L/K if for every finite extension F of K in L, the rank of the Mordell-Weil group E(F ) is at least [F : K]. In particular, if [L : K] is infinite this property will imply that the Mordell-Weil rank of E over L is infinite. Say that E has p-Selmer growth relative to L/K if the pro-p-Selmer rank of E over F is at least [F : K] for all finite extensions F of K in L. Recent work of Nekovár̆ ([N2], especially §10.7) shows that under extremely mild hypotheses, if E is an elliptic curve over k that has odd pro-p-Selmer rank over K and that is of good ordinary reduction at the primes above p, then E has p-Selmer growth relative to K −/K. Assuming the Shafarevich-Tate conjecture, this is equivalent to the statement that (under the same hypotheses) if E has odd Mordell-Weil rank overK, then it has Mordell-Weil growth relative to either K −/K. In this paper we do two things. First, we give a somewhat different exposition of Nekovár̆’s theorem, in the hope of making this important result more accessible and widely known. Namely, we will show how to derive a weaker version of Nekovár̆’s theorem (Theorem 3.1 below) from the main result of [MR2] (which in turn relies