Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem

By V. Blomer, Gergely Harcos. 2009

70 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/335a7f77c97ae22918c1ae2e3532cdaaabdf7374

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Citations in the corpus

The horocycle flow at prime times (2011, methodology): “1 gives a slight improvement over [4] and also [29], in this q aspect, an it is optimal under The Ramanujan/Selberg Conjectures.”
On the Ramanujan conjecture over number fields (2010, methodology): “As an immediate application we obtain the following numerical improvement for subconvexity bounds of twisted L-functions over number fields [1]: Let K be a totally real number field, let π be a cuspidal automorphic representation of GL2(A) with unitary central character, and let χ be a Hecke charact”
The circle method and bounds for L-functions---II: Subconvexity for twists of GL(3) L-functions (2012, methodology): “At present we have the Weyl-type bound in the t-aspect, and the Burgess-type bound in the M aspect [2] over general number fields, under the Ramanujan conjecture.”