L-functions and the Birch–Swinnerton-Dyer conjecture

At a good prime p, set a_p=p+1 minus the number of points on the smooth reduction, including O. The local Euler polynomial is one minus a_p T plus pT². At bad primes use the local factor for a minimal model: a_p is one for split multiplicative reduction, minus one for nonsplit multiplicative reduction, and zero for additive reduction, with polynomial one minus a_p T. The degree drop records bad reduction rather than a missing arbitrary data point.

The product of reciprocal local polynomials evaluated at p to the minus s converges absolutely for the real part of s greater than three halves. For elliptic curves over Q, modularity gives an entire continuation and a functional equation after the conductor, gamma factor and powers of two pi are included. The center is s=1. Substituting s=1 into a nonconvergent raw Euler product is not a justified numerical evaluation of the continued function.

Choose a global minimal Weierstrass equation and its Néron differential omega. Let Omega be the integral of its absolute value over all real components. Let c_p be the index of the subgroup reducing to a nonsingular point inside E(Q_p). Use a basis of the free Mordell–Weil quotient and the full-difference pairing B from the preceding height lesson to define Reg_BSD. With that lesson's half-limit height, the diagonal of B is twice the height; replacing B by its half would change the determinant by two to the rank.

The full conjecture predicts finiteness of Sha and the displayed leading coefficient. Torsion contributes its squared order in the denominator, while the finite local indices, real period and regulator measure different arithmetic features. At rank zero the empty regulator determinant is one. A subgroup of finite index d in the free group has regulator d² times that of a basis, so an unsaturated point list gives the wrong coefficient even when it spans the correct rational vector space.

For elliptic curves over Q, analytic rank zero or one is known to equal algebraic rank through the modularity, Gross–Zagier and Kolyvagin theory. This statement about rank does not by itself establish every prime-part of the refined coefficient formula. Further results have additional hypotheses and stronger arithmetic conclusions. A calculation of a few Euler factors, an approximate zero, or a Selmer upper bound does not replace the analytic certification and arithmetic arguments used in those theorems.

Status checked October 18, 2026: the Clay Mathematics Institute's official BSD page labels the general problem unsolved. Its official Wiles description distinguishes the rank assertion from the refined leading term. This dated classification concerns the unrestricted conjecture; it does not deny the proved cases or ongoing refinements. Reports about a particular family must retain their field, analytic-rank and local hypotheses before being read as a theorem about all rational elliptic curves.