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1. Introduction. This paper is motivated by the conjecture of Birch and Swinnerton-Dyer relating the rank of the Mordell-Weil group of an abelian variety defined over a number field with (in its crudest form) the order of vanishing of its Hasse-Weil L-function at the central critical point. Mestre [Mes] began the study of the implications of this conjecture towards providing upper bounds for the rank. He used " explicit formulae " similar to that of Riemann-Weil and assumed the analytic continuation and (perhaps more significantly) the Riemann hypothesis for those L-functions. Brumer [Br1] first studied the special case of the Jacobian variety J 0 (q) of the modular curve X 0 (q). This is an abelian variety defined over Q of dimension about q/12. Here analytic continuation is known, by the work of Eichler and Shimura [Sh1]. Assuming only the Riemann hypothesis for the L-functions of automorphic forms (of weight 2 and level q), Brumer proved rank a J 0 (q) 3 2 + o(1) dim J 0 (q) and conjectured that rank J 0 (q) = rank a J 0 (q) ∼ 1 2 dim J 0 (q) (based on the fact that the sign of the functional equation for the automorphic L-functions of weight 2 and level q is approximately half the time +1 and half the time −1). Other authors, notably Murty [Mur] (who first applied the Petersson formula in this context), considered the same problem. Most recently, Luo, Iwaniec, and Sar-nak [LIS], using the same assumptions, proved an estimate rank a J 0 (q) c + o(1) dim J 0 (q) for some (explicit) constant c < 1. This turns out to be quite significant in light of the general conjectures of Katz and Sarnak [KaS] on the distribution of zeros of families of L-functions. This paper approaches the same problem with a different emphasis: we wish to avoid all assumptions about the L-functions involved and obtain a bound of the correct order of magnitude. Indeed, we prove the following theorem. Theorem 1. There exists an absolute and effective constant C > 0 such that for any prime number q, rank a J 0 (q) C dim J 0 (q). If the Birch and Swinnerton-Dyer conjecture holds for J 0 (q), then rank J 0 (q) C dim J 0 (q). This theorem provides the first known unconditional bound for the analytic rank of a family …

The analytic rank of J0(q) and zeros of automorphic L-functions