Sur le rang de J_0(q)

Emmanuel Kowalski; Philippe Michel

paper

Sur le rang de J_0(q)

arXiv:math/9707237

Abstract

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 $J_{0}^{n} (q)$ , of the jacobian of the modular curve $X_{0} (q)$ . Our main resultis the following upper bound: for $q$ prime, one has $r an k_{a} (J_{0}^{n} (q)) ≪ dim J_{0}^{n} (q)$ where the implied constant is absolute. All previously known non trivials bounds of $r an k_{a} (J_{0}^{n} (q))$ 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.