The Birch-Swinnerton-Dyer Conjecture from a Naive Point of View

Abstract

Throughout this paper, E will denote an elliptic curve defined over Q which we suppose given in P² by an equation

$$ f(x,y,x) = 0\quad \quad (f \in \mathbb{Z}[x,y,z] homogeneous of degree 3) $$

((1))

with f of minimal discriminant Δ. If R is any ring with unit, then E(R) denotes the set of solutions of f = 0 in P²(R)={(x, y, z) ∈ R³ : xR + yR + zR = R}/ R^x. (In particular, E(Z)) is the same as the Mordell-Weil group E(Q) and not, as sometimes in the literature, the finite set of integral points in the affine model f(x, y, 1) = 0 of E over Z.) The L-series of E is the Dirichlet series given by

$$ L(E,s) = \prod\limits_p {\frac{1}{{1 - a(p){p^{{ - s}}} + \varepsilon (p){p^{{1 - 2s}}}}}} = \sum\limits_{{n - 1}}^{\infty } {\frac{{a(n)}}{{{n^s}}}} \quad \quad ({\rm Re} (s) >\frac{3}{2}), $$

(2)

where the product is over all primes, a(p) = p + 1 — E(Z/pZ) (• denotes cardinality) and ε(p) = 1 or 0 depending whether p ∤ Δ or pΔ.