# Upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$

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.

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.

• What about $N$ squarefree ? Dec 15, 2022 at 13:39
• Even the case of $J_1(p)$ for $p$ prime is hard. Dec 15, 2022 at 16:38