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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.

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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.

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  • $\begingroup$ What about $N$ squarefree ? $\endgroup$
    – Henri Cohen
    Commented Dec 15, 2022 at 13:39
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    $\begingroup$ Even the case of $J_1(p)$ for $p$ prime is hard. $\endgroup$
    – David Loeffler
    Commented Dec 15, 2022 at 16:38

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