Using a cubic moment, we prove a Weyl-type subconvexity bound for the quadratic twists of a holomorphic newform of square-free level, trivial nebentypus, and arbitrary even weight. This generalizes work of Conrey and Iwaniec in that the newform that is being twisted may have arbitrary square-free level, and also that the quadratic character may have even conductor. One of the new tools developed in this paper is a more general Petersson formula for newforms of square-free level.

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Abstract. We prove formulas for power moments for point counts of elliptic curves over a finite field k such that the groups of k-points of the curves contain a chosen subgroup. These formulas express the moments in terms of traces of Hecke operators for certain congruence subgroups of SL 2 (Z). As our main technical input we prove an Eichler-Selberg trace formula for a family of congruence subgroups of SL 2 (Z) which include as special cases the groups Γ 1 (N ) and Γ(N ). Our formulas generalize results of Birch and Ihara (the case of the trivial subgroup, and the full modular group), and previous work of the authors (the subgroups Z/2Z and (Z/2Z) 2 and congruence subgroups Γ 0 (2), Γ 0 (4)). We use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field, generalizing results of Vlǎduţ, Gekeler, Howe, and others.

We prove a Lindelöf-on-average upper bound for the fourth moment of Dirichlet L-functions of conductor q along a coset of the subgroup of characters modulo d when q * |d, where q * is the least positive integer such that q 2 |(q * ) 3 . As a consequence, we finish the previous work of the authors and establish a Weyl-strength subconvex bound for all Dirichlet L-functions with no restrictions on the conductor.