Seyfi's Talk, September 25: Elliptic Curves with all quadratic twists of positive rank Definition: E/Q an elliptic curve. L(E/Q,s) = \prod_{good p} \frac{1}{1-a_p p^{-s} + p^{1-2s}} \prod_{p bad} \frac{1}{1-a_p p^{-s}} a_p = p+1 - #E(F_p), if E has good reduction = 1 , split multiplicative reduction = -1 , nonsplit multiplicative = 0 , p cuspidal Reference: p. 360, Silverman's book AEC [analogous definition over a number field] \Lambda(E/Q,s) = N_{E/Q}^(s/2) (2\pi)^{-s} \Gamma(s) L(E/Q,s) N_{E/Q} = conductor of the elliptic curve = \prod p^{f_p} f_p = 0 if p is good, 1 if p is nodal, 2 + \delta_p, p cuspidal, \delta_p measures wild ramification \Gamma(s) = \int_0^{\infty} t^{s-1} e^{-t} dt extends meromorphically to all of C. Let E/K an elliptic curve, F/K a quadratic extension L(E/F,s) = L(E/K,s) L(E^F/K,s) Conjecture (Hasse-Wweil): \Lambda(E/K,S) has an analytic continuation of all of C and satisfies \Lambda(E/K,s) = w(E/K) \Lambda(E/K,2-s) Thus the root number is +1 if the order of vanishing at s = 1 is even, -1 if the order of vanishing at s = 1 is odd. Example: If the L-function does not vanish at s = 1, then the root number is equal to 1. Definition: E/K is Lawful if w(E/F) = 1 for every quadratic extension F/K. Remark: E is lawful if for all quadratic twists F/K, w(E^F/K) = w(E/K). This follows from the elementary fact together with the conjectured analytic continuation. Definition: w(E/K) = \prod_v w(E/K_v) The local root number at every Archimedean place is -1. For all v, w(E/K_v) := \epsilon(\sigma^1_{E,v},\psi,dx)/|\epsilon(\sigma^{1_{E,v},\psi,dx)| \epsilon-factor of canonical representation of the Weil-Deligne group of K_v \psi = unitary character of K_v dx = Haar measure on K_v Thm (Rohrlich 1996, p. 329) K = Q w(E/\R) = -1 For a prime p, if E has multiplicative reduction at p, then w(E/Q_v) = \chi(-1), \chi is the character of Q associated to F(\sqrt{-c_6})/F v_p(j(E)) < 0, then w(E/Q_v) = \chi(-1) Definition: E is lawful good w(E/K) = 1 lawful evil w(E/K) = -1 Chaotic: not lawful. BSD: Lawful evil: w(E/K) = w(E^F/K) = -1, then rk(E/F) = rk(E/K) + rk(E^F/K) > rk(E/K) for all quadratic F/K! Theorem 1: E/K an elliptic curve. TFAE: \\ (i) E/K is lawful. (ii) E/K_v is lawful for all places v. (iii) w(E/F) = w(E/K)^{[F:K]} for all finite extensions F/K (iv) K has no real places and E acquires everywhere good reduction over an abelian extension L of K. (v) K has no real places and for all primes p and all places v not dividing p f K, the action of G(\overline{K_v}/K_v) on T_p(E) is abelian.