2023/10/13 by Chung-Hang Kwan, Chung‐Hang Kwan, Kwan, Chung-Hang
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Finite Group Theory Research
paper · pdf · doi:10.1017/s1474748026101844
Abstract This work is the second in a series, following Part I [Kwa24] ( Algebra Number Theory 18 .10 (2024)) and preceding Part III [Kwa25] ( Math. Ann. 391 .1 (2025)). We continue our investigation of spectral moments of <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mtext>GL</mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>×</mml:mo> <mml:mtext>GL</mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>L</mml:mi> </mml:math> \text GL(3)× \text GL(2) L GL left parenthesis 3 right parenthesis times GL left parenthesis 2 right parenthesis upper L -functions from the perspective of period integrals. Using an identity between two distinct periods for the <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mtext>GL</mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> \text GL(3) GL left parenthesis 3 right parenthesis Eisenstein series, we establish an exact Motohashi-type identity linking the shifted cubic moment of <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mtext>GL</mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>L</mml:mi> </mml:math> \text GL(2) L GL left parenthesis 2 right parenthesis upper L -functions to the shifted fourth moment of <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mtext>GL</mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mi>L</mml:mi> </mml:math> \text GL(1) L GL left parenthesis 1 right parenthesis upper L -functions. In addition, we offer a novel, intrinsic and automorphic account for the sources and symmetries of the full set of main terms for both moments, in agreement with the CFKRS Moment Conjectures ( Proc. Lond. Math. Soc. (3) 91 (2005)).