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

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

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Zagier, Don1, Author
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1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201

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Abstract: [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).

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Language(s): eng - English
Dates: 1991
Publication Status: Issued
Pages: -
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Rev. Type: Internal
Identifiers: eDoc: 744932
Other: 111
URI: https://doi.org/10.1007/978-1-4612-0457-2
Degree: -

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Title: Arithmetic algebraic geometry
Source Genre: Book
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Publ. Info: Boston : Birkhäuser
Pages: x, 444 S. Volume / Issue: - Sequence Number: - Start / End Page: 377 - 389 Identifier: ISBN: 0-8176-3513-0

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Title: Progress in mathematics
Source Genre: Series
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Pages: - Volume / Issue: 89 Sequence Number: - Start / End Page: - Identifier: -