Abstract:DISCLAIMER Mention of trade names or commercial products does not constitute endorsement or recommendation for use.
u FOREWORDThe formation of the U.S. Environmental Protection Agency in 1970 ushered in the first decade of environmental awareness as a total national phenomenon. It was a decade punctuated by major Congressional mandates to restore the nation's waters, to reduce air pollution, and to find a comprehensive approach to other environmental problems-those associated with pesticide use, hazardous wast… Show more
“…for primitive Dirichlet characters χ of conductor M. Burgess employed an ingenious technique to bound short character sums by higher moments of complete character sums, for which one has strong bounds coming from the Riemann hypothesis for curves on finite fields (Weil's theorem). Similar subconvex bound in the level aspect for GL(2) L-functions was first obtained by Duke, Friedlander and Iwaniec [3] using the amplification technique. For f a newform of level M and trivial nebentypus their result gives the subconvex bound…”
Let f be a holomorphic modular form of prime level p and trivial nebentypus. We show that there exists a computable δ > 0, such that L 1 2 , Sym 2 f ≪ p 1 2 −δ , with the implied constant depending only on δ and the weight of f .
“…for primitive Dirichlet characters χ of conductor M. Burgess employed an ingenious technique to bound short character sums by higher moments of complete character sums, for which one has strong bounds coming from the Riemann hypothesis for curves on finite fields (Weil's theorem). Similar subconvex bound in the level aspect for GL(2) L-functions was first obtained by Duke, Friedlander and Iwaniec [3] using the amplification technique. For f a newform of level M and trivial nebentypus their result gives the subconvex bound…”
Let f be a holomorphic modular form of prime level p and trivial nebentypus. We show that there exists a computable δ > 0, such that L 1 2 , Sym 2 f ≪ p 1 2 −δ , with the implied constant depending only on δ and the weight of f .
“…Our job now is to get a nontrivial estimate for (13), beyond square-root cancellation in the character sum S(m 1 , m 2 , n, h; q). For h = 0, the zero shift, the character sum S(m 1 , m 2 , n, 0; q) can be evaluated precisely, and then one can use the large sieve inequality of Duke, Friedlander and Iwaniec [8] for Kloosterman fractions to get extra cancellation on the sum over n and m. Alternatively one can use reciprocity and then Voronoi yet again on the sum over m 2 , to get a much better result. However for non-zero shift the character sum S(m 1 , m 2 , n, h; q) can not be computed explicitly, and hence it is not clear how to obtain extra cancellation.…”
Section: Now We Apply Cauchy and Lemma 4 To Concludementioning
confidence: 99%
“…Non-trivial bound of this sum often has deep implications, e.g. subconvexity and equidistribution (QUE) (see [2], [6], [7], [11], [12], [13], [14], [16], [17], [18], [20], [24]). In this paper we will consider a higher rank analogue -…”
For the shifted convolution sumwhere λ 1 (1, m) are the Fourier coefficients of a SL(3, Z) Maass form π 1 , and λ 2 (m) are those of a SL(2, Z) Maass or holomorphic form π 2 , and 1 ≤ |h| ≪ X 1+ε , we establish the bound D h (X) ≪π 1 ,π 2 ,ε X 1− 1 20 +ε . The bound is uniform with respect to the shift h.1991 Mathematics Subject Classification. 11F66, 11M41.
“…For number fields, t-aspect subconvexity was achieved by Cogdell, Piatetski-Shapiro and Sarnak [5], Petridis and Sarnak [19] and Diaconu and Garrett [6]. The first subconvex bound in the M aspect (for t = 0) was obtained by Duke, Friedlander and Iwaniec [7] (for holomorphic forms of full level), Bykovskii [4] (for general holomorphic forms) and later by Harcos [9], Michel [15] and Blomer, Harcos and Michel [3] (for Maass forms). Over number fields this was established by Venkatesh [21] and Blomer and Harcos [2].…”
Let f be a Hecke-Maass or holomorphic primitive cusp form of arbitrary level and nebentypus, and let χ be a primitive character of conductor M . For the twisted L-function L(s, f ⊗χ) we establish the hybrid subconvex boundfor t ∈ R. The implied constant depends only on the form f and ε.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.