Let E be a CM elliptic curve over the rationals and p an odd prime ordinary for E. If the ℤp-corank of p∞ Selmer group for E equals 1, then we show that the analytic rank of E also equals 1. This is joint work with Ashay Burungale.
Tag - Elliptic curves
The generalised Kato classes of Darmon-Rotger arise as p-adic limits of diagonal cycles on triple products of modular curves, and in some cases, they are predicted to have a bearing on the arithmetic of elliptic curves over ℚ of rank 2. In this talk, we will report on a joint work in progress with Ming-Lun Hsieh concerning a special case of the conjectures of Darmon-Rotger.
Take E/ℚ to be an elliptic curve with full rational 2-torsion (satisfying some extra technical assumptions). In this talk, we will show that 100% of the quadratic twists of E have rank less than two, thus proving that the BSD conjecture implies Goldfeld's conjecture in these families. To do this, we will extend Kane's distributional results on the 2-Selmer groups in these families to 2k-Selmer groups for any k>1. In addition, using the close analogy between 2k-Selmer groups and 2k+1-class groups, we will prove that the 2k+1-class groups of the quadratic imaginary fields are distributed as predicted by the Cohen-Lenstra heuristics for all k>1.
Given a non-isotrivial elliptic curve E over K=𝔽q(t), there is always a finite extension L of K which is itself a rational function field such that E(L) has large rank. The situation is completely different over complex function fields: For "most" E over K=ℂ(t), the rank E(L) is zero for any rational function field L=ℂ(u). The yoga that suggests this theorem leads to other remarkable statements about rational curves on surfaces generalizing a conjecture of Lang.
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