In this note,we present a simple approach for bounding the shifted convolution sum involving the Fourier coefficients of half-integral weight holomorphic cusp forms and Maass cusp forms.
Let f be a Maass cusp form for Г0（N） with Fourier coefficients 1 k2. λf（n） and Laplace eigenvalue 1/4 ＋k2 For real α≠0 and β 〉 0, consider the sum Sx（f; α,β） = ∑n λf（n）e（αnβ）φ（n/X）, where ...Let f be a Maass cusp form for Г0（N） with Fourier coefficients 1 k2. λf（n） and Laplace eigenvalue 1/4 ＋k2 For real α≠0 and β 〉 0, consider the sum Sx（f; α,β） = ∑n λf（n）e（αnβ）φ（n/X）, where φ is a smooth function of compact support. We prove bounds for the second spectral moment of Sx （f;α, β）, with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X. This implies that if f has its eigenvalue beyond X1/2＋ε, the standard resonance main term for Sx（f; ±2√q 1/2）, q ∈Z＋, cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL（2） × GL（2）. It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of Kε≤ L≤ K1-ε. The same bounds can be proved in a similar way for holomorphie cusp forms.展开更多
基金supported by US National Science Foundation (Grant No.DMS-0855600)
文摘In this note,we present a simple approach for bounding the shifted convolution sum involving the Fourier coefficients of half-integral weight holomorphic cusp forms and Maass cusp forms.
文摘Let f be a Maass cusp form for Г0（N） with Fourier coefficients 1 k2. λf（n） and Laplace eigenvalue 1/4 ＋k2 For real α≠0 and β 〉 0, consider the sum Sx（f; α,β） = ∑n λf（n）e（αnβ）φ（n/X）, where φ is a smooth function of compact support. We prove bounds for the second spectral moment of Sx （f;α, β）, with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X. This implies that if f has its eigenvalue beyond X1/2＋ε, the standard resonance main term for Sx（f; ±2√q 1/2）, q ∈Z＋, cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL（2） × GL（2）. It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of Kε≤ L≤ K1-ε. The same bounds can be proved in a similar way for holomorphie cusp forms.
