On the vanishing of twists of L-functions of elliptic curves Chantal ...
On the vanishing of twists of L-functions of elliptic curves Chantal ...
On the vanishing of twists of L-functions of elliptic curves Chantal ...
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Quadratic charactersLet χ d be a quadratic character <strong>of</strong> discriminantd. Then,χ d = χ d and L E (s, χ d ) = L Ed (s).The sign <strong>of</strong> <strong>the</strong> functional equation isw E χ d (−N E ) = ±1and L Ed (1) which vanishes for about half <strong>the</strong>discriminants d.Conjecture (Birch and Swinnerton-Dyer)rank(E) = ord s=1 L E (s)We writeN ∗ (X) = # {|d| ≤ X : rank(E d ) is *}Conjecture (Goldfeld)N 0 (X) ∼ N 1 (X) ∼ 6 π 2 XN ≥2 (X) = o(X)In o<strong>the</strong>r words, <strong>the</strong> average rank is as small asit can be.2
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