The Second Moment of Twisted Modular L-Functions

By V. Blomer, Djordje Milićević. 2014

93 citations recorded by the source corpus.

AbstractWe prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment $$\underset{\chi \bmod {q}}{\left. \sum \right.^{\ast}} L(1/2, f_1\otimes \chi) \overline{L(1/2, f_2 \otimes \chi)}$$∑*χmodqL(1/2,f1⊗χ)L(1/2,f2⊗χ)¯of central values of L-functions of any two (possibly equal) fixed cusp forms f1, f2 twisted by all primitive characters modulo q, valid for all sufficiently factorable q including 99.9 % of all admissible moduli. The two key ingredients are a careful spectral analysis of a potentially highly unbalanced shifted convolution problem in Hecke eigenvalues and a new large sieve type bound for Kloosterman sums where the summation lengths can be below the square-root threshold of the modulus. Applications are given to simultaneous non-vanishing and lower bounds on higher moments of twisted L-functions.

Subfields: Algebra and Number Theory, Artificial Intelligence, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e8ce32da7557fc3a4bfd87a0fbc13fb64b5b4687

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Citations in the corpus

Twisted moments of L-functions and spectral reciprocity (2017, methodology): “cently obtained by Kıral and Young [KY] in an impressive 80-page tour de force argument. Our version for Maaß forms comes as a special case of a more general framework. Since L(1/2,f) is non-negative [BM], Theorem 3 implies immediately the strong subconvexity bound (1.13) L(1/2,f) ≪ε q(1+ϑ+ε)/5. Thi”
Bilinear forms with Kloosterman sums and applications (2015, methodology): “This will be taken up in the forthcoming paper [4] jointly with Blomer, Fouvry and Milićević.”
On moments of twisted L-functions (2014, methodology): “We proceed by Mellin inversion (as in [3, 29]): using the definition (2.”