Yangbo Ye

A new bound K2/3+ɛ for Rankin-Selberg ℒ-functions for Hecke congruence subgroups

Let f be a holomorphic Hecke eigenform for Γ0(N ) of weight k, or a Maass eigenform for Γ0(N ) with Laplace eigenvalue 1/4 + k. Let g be a fixed holomorphic or Maass cusp form for Γ0(N ). A subconvexity bound for central values of the Rankin-Selberg L-function L(s, f ⊗ g) is proved in the k-aspect: L(1/2+ it, f ⊗ g) N ,g,t,e k, while a convexity bound is only k. This new bound improves earlier subconvexity bounds for these Rankin-Selberg L-functions by Sarnak, the authors, and Blomer. Techniques used include a result of Good, spectral large sieve, meromorphic continuation of a shifted convolution sum to −1/2 passing through all Laplace eigenvalues, and a weighted stationary phase argument.

Weighted stationary phase of higher orders

The subject matter of this paper is an integral with exponential oscillation of phase f(x) weighted by g(x) on a finite interval [α β]: When the phase f(x) has a single stationary point in (α β), an nth-order asymptotic expansion of this integral is proved for n ≥ 2: This asymptotic expansion sharpens the classical result for n = 1 by M. N. Huxley. A similar asymptotic expansion was proved by V. Blomer, R. Khan and M. Young under the assumptions that f(x) and g(x) are smooth and g(x) is compactly supported on R: In the present paper, however, these functions are only assumed to be continuously differentiable on [α β] 2n + 3 and 2n + 1 times, respectively. Because there are no requirements on the vanishing of g(x) and its derivatives at the endpoints α and β, the present asymptotic expansion contains explicit boundary terms in the main and error terms. The asymptotic expansion in this paper is thus applicable to a wider class of problems in analysis, analytic number theory, and other fields.

Improved subconvexity bounds for GL(2)xGL(3) and GL(3) L-functions by weighted stationary phase

Let

Improved subconvexity bounds for (2)×(3) and (3) -functions by weighted stationary phase

f

Zero correlation with lower-order terms for automorphic L-functions

be a fixed self-contragradient Hecke-Maass form for

The Lifting of Kloosterman Sums

SL(3,\mathbb Z)

Zero density for automorphic L-functions

, and

Subconvexity bounds for Rankin-Selberg L-functions for congruence subgroups

u

The prime number theorem and Hypothesis H with lower-order terms

an even Hecke-Maass form for

Asymptotics for cuspidal representations by functoriality from GL(2)

SL(2,\mathbb Z)