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The analytic rank of $$J_ 0 (N) (\mathbb{Q})$$. (English) Zbl 0851.11036
Dilcher, Karl (ed.), Number theory. Fourth conference of the Canadian Number Theory Association, July 2-8, 1994, Dalhousie University, Halifax, Nova Scotia, Canada. Providence, RI: American Mathematical Society. CMS Conf. Proc. 15, 263-277 (1995).
Let $$N$$ be prime and let $$f$$ run over all weight 2 new forms for $$\Gamma_0 (N)$$. Assume the Riemann Hypothesis for $$L(f,s)$$ and write $$r_f$$ for the order of $$L(f,s)$$ at $$s = 1$$. Then it is shown that $\sum_f {r_f \over (f,f)} \leq {14 \pi \over 3} + \varepsilon$ for $$N \geq N (\varepsilon)$$. Assuming also the Lindelöf Hypothesis for $$L (\text{sym}^2 (f),s)$$ one gets $\sum_f r_f \leq \left( {3 \over 2} + \varepsilon \right) \dim S_2 (N) + o(N)$ as $$N$$ grows, $$S_2 (N)$$ being the space of weight 2 cusp forms. As a corollary one finds under the same hypotheses that the analytic rank of $$J_0 (N)^{new}$$ is at most $$({3 \over 2} + \varepsilon) \dim S_2 (N)^{new} + o(N)$$, a result established by A. Brumer [Astérisque 228, 41-68 (1995; see preceding review)] without using the symmetric square. The method employed differs from Brumer’s in that Poincaré series are used instead of the Eichler-Selberg trace formula.
An unconditional result $\sum_f (f,f) = {\pi \over 24} \bigl( \dim S_2 (N) \bigr)^2 + O(N^{13/10} \log^2N)$ is also established.
For the entire collection see [Zbl 0827.00036].

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14H40 Jacobians, Prym varieties 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11G18 Arithmetic aspects of modular and Shimura varieties