Does there exist an upper bound of the analytic rank of the modular Jacobian varieties $J\_1(N)$?
(Or more generally of $J\_\Gamma$ for a congruence subgroup $\Gamma\_0 \subseteq \Gamma \subseteq \Gamma\_1$.)
I want one like $ rank J\_1(N) < C \dim J\_1(N)$, for some nice small constant $C$.
($N$ is an arbitrary positive integer, or it’s ok to assume that it is a prime number)


I know there’s such upper bound for $J\_0(p)$ and for $C$ smaller than $1.2$.


(See Kowalski, E., Michel, P., The analytic rank of J 0 ( q ) and zeros of automorphic L -functions, theorem 1,
and Kowalski, E., Michel, P., VanderKam, J. M., Nonvanishing of higher derivatives of
automorphic L-function.)


Can we generalize them?
I am not familiar with the analytic number theory, the symmetric square of modular forms, and every analytic things mentioned in the papers at all.



Upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$
I remember discussing this with Emmanuel Kowalski not long ago. The short answer is that generalising the result to $J\_1(N)$ is an open problem, and seems to be very difficult.


