An Equivalent Geometric Construction of the L-Function of an Elliptic Curve: From Toroidal Helix to the s=1 Critical Section

This paper gives an equivalent geometric construction of the L-function of an elliptic curve, in strict duality with the spherical-helix construction of the ζ function. The classical ℘-isomorphism realizes an elliptic curve E as a flat torus (Clifford torus in S³ ⊂ ℝ⁴); analytic continuation, guaranteed by modularity, places L(E,s) inside this geometry, so the complex variable s becomes the coordinate of a point on the toroidal helix. The center of symmetry s=1 of the functional equation (dual to σ=1/2 on the sphere) determines the critical section of the nested tori. Where this section meets the torus, each symmetric pair (Q,R) together with the common apex O forms an isosceles triangle, forced without exception by toroidal symmetry. This triangle carries a threefold identity: metric symmetry |OQ|=|OR| (arithmetic), symmetry s↔2−s about s=1 (analytic, since s is a point on the helix), and group-law balance Q+R+O=0 (the common substrate). Under the equivalent construction the arithmetic and analytic sides are already in bijection, so no bridge between rank and order of vanishing need be built — the isosceles triangle is that bridge. The dimension of the non-degenerate isosceles triangles cut by the s=1 section reads as the Mordell–Weil rank on the geometric side and as ord_{s=1} L(E,s) on the analytic side, yielding the rank identity ord_{s=1} L(E,s) = rank E(ℚ). The load-bearing point is the equivalent construction itself, resting on the ℘-isomorphism, the Mordell–Weil theorem, the Néron–Tate height, modularity, and the geometric construction of this paper. Bilingual (English and Chinese PDFs).