Resonance for Maass forms in the spectral aspect
Let ƒ be a Maass cusp form for Γ0(N) with Fourier coefficients λƒ(n) and Laplace eigenvalue ¼+k2. For real α≠0 and β>0 consider the sum: ∑nλƒ(n)e(αnβ)Φ(n/X), where Φ is a smooth function of compact support. We prove bounds on the second spectral moment of this sum, with the eigenvalue tending tow...
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| Format: | Others |
| Language: | English |
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University of Iowa
2016
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| Online Access: | https://ir.uiowa.edu/etd/3179 https://ir.uiowa.edu/cgi/viewcontent.cgi?article=6509&context=etd |
| Summary: | Let ƒ be a Maass cusp form for Γ0(N) with Fourier coefficients λƒ(n) and Laplace eigenvalue ¼+k2. For real α≠0 and β>0 consider the sum:
∑nλƒ(n)e(αnβ)Φ(n/X), where Φ is a smooth function of compact support. We prove bounds on the second spectral moment of this sum, with the eigenvalue tending toward infinity. When the eigenvalue is sufficiently large we obtain an average bound for this sum in terms of X. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL(2)×GL(2). It contains in particular the Kuznetsov trace formula and 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 the same way for holomorphic cusp forms.
Furthermore, we prove similar bounds for
∑nλf(n)λg(n)e(αnβ)Φ(n/X),
where g is a holomorphic cusp form. As a corollary, we obtain a subconvexity bound for the L-function L(s, f ×g). This bound has the significant property of breaking convexity even for the trivial bound toward the Generalized Ramanujan Conjecture. |
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