Abstract
Let K be a totally real number field, π an irreducible cuspidal representation of \({{\rm GL}_{2}(K){\backslash}{\rm GL}_{2}(\mathbb{A}K)}\) with unitary central character, and χ a Hecke character of conductor \({\mathfrak{q}}\). Then \({L(1/2, \pi\oplus\chi) \ll (\mathcal{N}\mathfrak{q})^{\frac{1}{2}-\frac{1}{8}(1-2\theta)+\epsilon}}\), where 0 ≤ θ ≤ 1/2 is any exponent towards the Ramanujan–Petersson conjecture (θ = 1/9 is admissible). The proof is based on a spectral decomposition of shifted convolution sums and a generalized Kuznetsov formula.
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The first author was in part supported by an NSERC grant 311664-05 and a Sloan Research Fellowship. The second author was supported by European Community grants EIF 040371 and ERG 239277 within the 6th and 7th Framework Programmes and by OTKA grants K 72731 and PD 75126.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Blomer, V., Harcos, G. Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem. Geom. Funct. Anal. 20, 1–52 (2010). https://doi.org/10.1007/s00039-010-0063-x
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DOI: https://doi.org/10.1007/s00039-010-0063-x
Keywords and phrases
- Subconvexity
- ternary quadratic forms
- shifted convolution sums
- spectral decomposition
- Hilbert modular forms
- Kirillov model