Sur le rang de J_0(q)

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 $J^n_0(q)$, 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.