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Motohashi’s fourth moment identity for non-archimedean test functions and applications
- Part of
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- Journal:
- Compositio Mathematica / Volume 156 / Issue 5 / May 2020
- Published online by Cambridge University Press:
- 17 April 2020, pp. 1004-1038
- Print publication:
- May 2020
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- Article
- Export citation
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Motohashi established an explicit identity between the fourth moment of the Riemann zeta function weighted by some test function and a spectral cubic moment of automorphic
$L$-functions. By an entirely different method, we prove a generalization of this formula to a fourth moment of Dirichlet 
$L$-functions modulo 
$q$ weighted by a non-archimedean test function. This establishes a new reciprocity formula. As an application, we obtain sharp upper bounds for the fourth moment twisted by the square of a Dirichlet polynomial of length 
$q^{1/4}$. An auxiliary result of independent interest is a sharp upper bound for a certain sixth moment for automorphic 
$L$-functions, which we also use to improve the best known subconvexity bounds for automorphic 
$L$-functions in the level aspect.
