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2018•arXiv (Cornell University)Open access

Rankin-Selberg Trace Formula for $\mathrm{GL}_2$: Geometric Side

Han Wu

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Abstract

In the 80s, Zagier and Jacquet-Zagier tried to derive the Selberg trace formula by applying the Rankin-Selberg method to the automorphic kernel function. Their derivation was incomplete due to a puzzle in a special case. We solve this puzzle and complete the derivation in the first part of the paper. The main input is an extension of the theory of regularized integrals invented by Zagier, which is of independent interest. In the second part, we also give another form of the geometric side of the formula, which explicitly relates the trace formula to some relative trace formulas.

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What this paper is about

In the 80s, Zagier and Jacquet-Zagier tried to derive the Selberg trace formula by applying the Rankin-Selberg method to the automorphic kernel function. Their derivation was incomplete due to a puzzle in a special case. We solve this puzzle and complete the derivation in the first part of the paper. The main input is an extension of the theory of regularized integrals invented by Zagier, which is of independent interest. In the second part, we also give another form of the geometric side of the formula, which explicitly relates the trace formula to some relative trace formulas.

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Available abstract

In the 80s, Zagier and Jacquet-Zagier tried to derive the Selberg trace formula by applying the Rankin-Selberg method to the automorphic kernel function. Their derivation was incomplete due to a puzzle in a special case. We solve this puzzle and complete the derivation in the first part of the paper. The main input is an extension of the theory of regularized integrals invented by Zagier, which is of independent interest. In the second part, we also give another form of the geometric side of the formula, which explicitly relates the trace formula to some relative trace formulas.

Key concepts: Selberg trace formula, TRACE (psycholinguistics), Mathematics, Extension (predicate logic), Pure mathematics, Kernel (algebra), Automorphic form, Algebra over a field

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