Birch–Swinnerton-Dyer conjecture

Claim: complete proof of the Birch–Swinnerton-Dyer conjecture for all elliptic curves over ℚ.
Key step: a self-adjoint resonance operator whose zeta-determinant matches the Hasse–Weil L(E,s)L(E,s)L(E,s) forces

L(r)(E,1)  =  #\Sha(E) \Reg(E) ΩE ∏cp(#Etors)2  >  0,L^{(r)}(E,1)\;=\; \dfrac{\#\Sha(E)\,\Reg(E)\,\Omega_E\,\prod c_p}{(\#E_{\text{tors}})^2}\;>\;0,L(r)(E,1)=(#Etors)2#\Sha(E)\Reg(E)ΩE∏cp>0,

linking analytic rank rrr to the Mordell–Weil rank