Cancellation in additively twisted sums on GL(n)

By Stephen D. Miller. 2004

105 citations recorded by the source corpus.

In a previous paper with Schmid we considered the regularity of automorphic distributions for GL (2,R), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting, establishing the nontrivial bound Σ n ≤ T a n e 2 π i n α = O e ( T 3/4+e ), uniformly in α ∈ R, for a n the coefficients of the L -function of a cusp form on GL (3, Z)\ GL (3,R). We also derive an equivalence (Theorem 7.1) between analogous cancellation statements for cusp forms on GL ( n ,R), and the sizes of certain period integrals. These in turn imply estimates for the second moment of cusp form L -functions.

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

View Paper: https://www.semanticscholar.org/paper/1b4fe8762455179daed5f33e2f8921b94b74c66e

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In Algebra and Number Theory

1. Nicolas Templier
2. Atsushi Ichino
3. R. Munshi

In Discrete Mathematics and Combinatorics

1. Nicolas Templier
2. Atsushi Ichino
3. R. Munshi

In Mathematical Physics

1. Nicolas Templier
2. Atsushi Ichino
3. R. Munshi

Citations in the corpus

• The circle method and bounds for L-functions---II: Subconvexity for twists of GL(3) L-functions (2012, methodology): “One may apply the Voronoi summation formula on the sum over n or directly apply Miller’s bound from [10] to the”
• On the Voronoĭ formula for GL(n) (2013, methodology): “Recent applications of the formula for GL(3) may be found in Miller-Schmid [29,31], Sarnak-Watson [32] and Li [26,27].”
• Bounds for GL(3)×GL(2) L -functions and GL(3) L -functions (2008, unknown): “For other applications, see [Mil06] and [Li09].”