Sur le rang de J_0(q), by Emmanuel Kowalski and Philippe Michel

In this paper, we prove an unconditionnal bound for the analytic rank (i.e the order of vanishing at the critical point of the \$L\$ function) of the new part \$J^n_0(q)\$, of the jacobian of the modular curve \$X_0(q)\$. Our main resultis the following upper bound: for \$q\$ prime, one has \$\$rank_a(J_0^n(q))\ll \dim J_0^n(q)\$\$ where the implied constant is absolute. All previously known non trivials bounds of \$rank_a(J_0^n(q))\$ assumed the generalized Riemann hypothesis; here, our proof is unconditionnal, and is based firstly on the construction by Perelli and Pomykala of a new test function in the context of Riemann-Weil explicit formulas, and secondly on a density theorem for the zeros of \$L\$ functions attached to new forms.

Emmanuel Kowalski <ekowalsk@math.rutgers.edu>
Philippe Michel <michel@math.u-psud.fr>