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The Birch-Swinnerton-Dyer conjecture from a naive point of view


Zagier,  Don
Max Planck Institute for Mathematics, Max Planck Society;

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Zagier, D. (1991). The Birch-Swinnerton-Dyer conjecture from a naive point of view. In Arithmetic algebraic geometry (pp. 377-389). Boston: Birkhäuser.

Cite as: https://hdl.handle.net/21.11116/0000-0004-010A-0
[For the entire collection see Zbl 0711.00011.]\\par Let E denote an elliptic curve defined over \\bbfQ, given in \\bbfP\\sp 2 by an equation f(x,y,z)=0, where f\\in\\bbfZ[x,y,z] is homogeneous of degree 3 and of minimal discriminant Δ. The L-series of E is L(E,s)=\\prod\\sb p(1-a(p)p\\sp-s+\\varepsilon(p)p\\sp1-2s)\\sp-1, \\textRe sgt;3/2, where the product is over all primes, a(p)=p+1- \\textCard(E(\\bbfZ/p\\bbfZ)), \\varepsilon(p)=1 or 0 according as p\\nmidΔ or p\\midΔ. The Birch-Swinnerton-Dyer conjecture consists of two statements:\\par (A) L(E,s) continues meromorphically to s=1, where it has a zero of order r=\\textrank\\sb \\bbfZ E(\\bbfQ).\\par (B) Assuming (A), let λ be defined by L(E,s)\\approxλ(s- 1)\\sp r, then λ=cRΩ T\\sp-2\\textCard(\\cyr Sh), where R is the determinant of the Néron-Tate height pairing, Ω is the real period, T=\\textCard(E\\sb\\textTors(\\bbfQ)), c=\\prod\\sbp\\midΔc\\sb p, c\\sb p=[E(\\bbfZ\\sb p):E\\sp 0(\\bbfZ\\sb p)], E\\sp 0(\\bbfZ\\sb p) is the set of points reducing to non-singular points of E(\\bbfQ/p\\bbfZ) and \\cyr Sh is the Tate-Shafarevich group of E, conjectured to be finite.\\par In this attractively written paper, whose ``naive point of view'' was readily adopted by the reviewer, the author seeks to interpret the invariants occurring in the formula for λ by counting solutions of f(x,y,z)=0 in \\bbfQ and in \\bbfZ/n\\bbfZ. --- He begins by comparing the function L(E,s) with the zeta-function of a quadratic number field, in which \\sqrt R corresponds to the regulator, T to the number of roots of unity, \\cyr Sh to the class-group and in which r,\\ RT\\sp-2 and RT\\sp-2Ω may be interpreted in terms of counting solutions of equations. That leads to the Dirichlet class number formula and the analogue of the Birch-Swinnerton-Dyer conjecture.\\par The remainder of the paper is devoted to a reconsideration and reformulation of (A) and (B) above from the point of view of counting solutions of f(x,y,z)=0 in the light of the analogue in quadratic fields. To that end, the author defines two functions Z(E,s)=\\sum\\sb(x,y,z)\\in\\bbfZ\\sp 3/\\pm 1\\atopgcd(x,y,z)=1\\atop f(x,y,z)=0(x\\sp 2+y\\sp 2+z\\sp 2)\\sp-s/2,\\quad (\\textRe sgt;0), D(E,s)=\\sum\\sp ∞\\sbn=1≤ft(\\sum\\sb(x,y,z)\\in(\\bbfZ/n\\bbfZ)\\sp 3/(\\bbfZ/n\\bbfZ )\\sp ×\\atopgcd(x,y,z,n)=1\\atop f(x,y,z)\\equiv 0\\pmod n1\\right)n\\sp-s,\\quad(\\textRe sgt;2), and numbers κ\\in\\bbfR\\sbgt;0 and r\\in\\bbfZ\\sb≥0 by Ω\\sp- 1Z(E,s)\\sp 2∼κ(π/s)\\sp r (s→ 0).\\par With the foregoing notation, the Birch-Swinnerton-Dyer conjecture is equivalent to the statement that D(E,s) continues meromorphically to s=1 and satisfies \\textCard(\\cyr Sh)D(E,s)∼-κ(s-1)\\sp- r\\quad(s→ 1). The number \\textCard(\\cyr Sh) has an interpretation in terms of points on E and on principal homogeneous spaces over E over the ring \\varprojlim(\\bbfZ/n\\bbfZ).