Sparse equidistribution problems, period bounds and subconvexity

Abstract

Abstract. We introduce a "geometric" method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the level aspect over number fields, which implies subconvexity for certain standard and Rankin-Selberg L-functions; and to bounding Fourier coefficients of automorphic forms.

Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

“…Both of these ideas would suffice to prove effective equidistribution of orbits of horospherical subgroups in any rank, as would the work of Burger [4]. More detailed analysis of the quantitative equidistribution of the horocycle flow for SL 2 (R) may be found in [23,60]; quantitative mixing rates are discussed in [26], and analysis of equidistribution of closed horospheres and its relevance to the theory of automorphic forms may be found in [61].…”

Symmetries in algebra and number theory (SANT)

“…also [40,44]) that generalizes work of Duke [18]: Under the assumption of a subconvex bound as above it is proved that a certain family of Heegner points and certain d -dimensional subvarieties are equidistributed on the Hilbert modular variety PSL 2 Finally we note that the subconvex bound in Theorem 1 (in particular for K = Q) is a crucial input for certain subconvex bounds of higher degree L-functions, which in turn have other arithmetic applications. We refer the reader to [24,8] for more details.…”

“…In an unpublished manuscript [13] (see also [14]), Cogdell, PiatetskiiShapiro and Sarnak obtained δ = 1/18 for holomorphic Hilbert cusp forms using deep bounds for triple products [35]. As an application of an ingenious and very flexible geometric method, Venkatesh [40] (see also [34]) proved recently -among other things -Theorem 1 with δ = 1/24. Our method is quite different from all of these works and will yield in particular as a by-product a solution of a problem of Selberg, see Theorem 2 below.…”

Equidistribution, <em>L</em>-functions and ergodic theory: on some problems of Yu. Linnik

“…The first such result (in the eigenvalue aspect) was given by Bernstein-Reznikov [BR04], [BR05], and a little later a result in the level aspect was given by Venkatesh [Ven05]. These two methods seem to be quite distinct.…”

“…corresponds to a holomorphic Hilbert modular form). Recently, the second author developed a new method which we discuss in section 4 below and established, amongst other things, the bounds (3.1), (3.2), (3.3) and (3.4) for F an arbitrary number field, π2 fixed but arbitrary and π1 with a trivial central character [Ven05]. Eventually the authors combined their respective methods from [Mic04] and [Ven05] to obtain (3.3) and (3.4) for π1 with an arbitrary central character.…”