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.