Results from MathSciNet: Mathematical Reviews on the Web \begin{matrix}c \\ ◯ \end{matrix} Copyright American Mathematical Society 2000

82e:14033 14G10 10D24 12A70 14G25 14K07

Mazur, B. [Mazur, Barry]

On the arithmetic of special values of L functions. Invent. Math. 55 (1979), no. 3, 207–240. From the introduction: “The conjectures of Birch and Swinnerton-Dyer imply that the special values of L functions attached to the parabolic modular forms of weight two for X_{0}(N) provide much information about the arithmetic of the Jacobian of X_{0}(N) over abelian number fields. For example, if ω is a parabolic newform of weight two for Γ_{0}(N), A the factor of the Jacobian of X_{0}(N) determined by ω, and χ a quadratic Dirichlet character with K = K_{χ}, its associated quadratic number field, then L(ω, χ, 1) is conjectured to vanish if and only if the rank of the Mordell-Weil group of A over K is greater than the rank of the Mordell-Weil group of A over Q. The special values L(ω, χ, 1) can be expressed as the product of two numbers: (1) the real or imaginary period of ω, the choice being dependent upon whether χ(−1) is +1 or -1, respectively; (2) an algebraic number which lies in the field generated by the Hecke eigenvalues of ω and the values of χ. This number can be computed by means of modular symbols.”

The author investigates the “algebraic part” of L(ω, χ, 1) for prime level N. His main theorem in part IV, Section 1 shows that the residue class of the “algebraic part” modulo a certain ideal provides some information concerning the arithmetic of X_{0}(N), and it may be regarded as a weak analogue of the conjectures of Birch and Swinnerton-Dyer. An interesting case arises which requires special treatment: suppose the rational prime N splits in the quadratic field K, and the class number of K is prime to p. The conjecture of Birch and Swinnerton-Dyer predicts the existence of a point of infinite order in the Mordell-Weil group of the p-Eisenstein quotient of the Jacobian of X_{0}(N) in this situation. The author shows how we may produce a point e_{K} by a simple construction of the type considered by Birch and Heegner, and proves e_{K} to be of infinite order. The concluding part of the introduction explains the specific nature of the obtained results by specializing to the first nontrivial case (N = 11) where X_{0}(N) is an elliptic curve. Another approach to the problem of Birch and Swinnerton-Dyer was earlier proposed by J. Coates and A. Wiles [same journal 39 (1977), no. 3, 223–251; MR 57#3134].

V. V. Shokurov (Moscow)