A Burgess-like subconvex bound for twisted L-functions

Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, X a primitive character of conductor q, and s a point on the critical line Rs = 1/2. It is proved that L(g circle times chi, s) 0 is arbitrary and theta = 7/64 is the current known approximation towards the RamannJan-Petersson conjecture (which would allow theta = 0); moreover, the dependence on s and all the parameters of g is polynomial. This result is an analog of Burgess' classical subconvex bound for Dirichlet L-functions. In Appendix 2 the above result is combined with a theorem of Waldspurger and the adelic calculations of Baruch-Mao to yield an improved uniform upper bound for the Fourier coefficients of holomorphic half-integral weight cusp forms

Hybrid bounds for twisted L-functions

The aim of this paper is to derive bounds on the critical line Rs 1/2 for L- functions attached to twists f circle times chi of a primitive cusp form f of level N and a primitive character modulo q that break convexity simultaneously in the s and q aspects. If f has trivial nebentypus, it is shown that L(f circle times chi, s) << (N vertical bar s vertical bar q)(epsilon) N-4/5(vertical bar s vertical bar q)(1/2-1/40), where the implied constant depends only on epsilon > 0 and the archimedean parameter of f. To this end, two independent methods are employed to show L(f circle times chi, s) << (N vertical bar s vertical bar q)(epsilon) N-1/2 vertical bar S vertical bar(1/2)q(3/8) and L(g,s) << D-2/3 vertical bar S vertical bar(5/12) for any primitive cusp form g of level D and arbitrary nebentypus (not necessarily a twist f circle times chi of level D vertical bar Nq(2))

Bounding sup-norms of cusp forms of large level

Let f be an $L^2$-normalized weight zero Hecke-Maass cusp form of square-free level N, character $\chi$ and Laplacian eigenvalue $\lambda\geq 1/4$. It is shown that $\| f \|_{\infty} \ll_{\lambda} N^{-1/37}$, from which the hybrid bound $\|f \|_{\infty} \ll \lambda^{1/4} (N\lambda)^{-\delta}$ (for some $\delta > 0$) is derived. The first bound holds also for $f = y^{k/2}F$ where F is a holomorphic cusp form of weight k with the implied constant now depending on k.Comment: version 3: substantially revised versio

Equidistribution of Heegner Points and Ternary Quadratic Forms

We prove new equidistribution results for Galois orbits of Heegner points with respect to reduction maps at inert primes. The arguments are based on two different techniques: primitive representations of integers by quadratic forms and distribution relations for Heegner points. Our results generalize one of the equidistribution theorems established by Cornut and Vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow. Moreover, for fixed fundamental discriminant and variable conductor, we deduce an effective surjectivity theorem for the reduction map from Heegner points to supersingular points at a fixed inert prime. Our results are applicable to the setting considered by Kolyvagin in the construction of the Heegner points Euler system

Nodal domains of Maass forms I

This paper deals with some questions that have received a lot of attention since they were raised by Hejhal and Rackner in their 1992 numerical computations of Maass forms. We establish sharp upper and lower bounds for the $L^2$-restrictions of these forms to certain curves on the modular surface. These results, together with the Lindelof Hypothesis and known subconvex $L^\infty$-bounds are applied to prove that locally the number of nodal domains of such a form goes to infinity with its eigenvalue.Comment: To appear in GAF

The subconvexity problem for \GL_{2}

Generalizing and unifying prior results, we solve the subconvexity problem for the $L$-functions of \GL_{1} and \GL_{2} automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino--Ikeda.Comment: Almost final version to appear in Publ. Math IHES. References updated