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].