On sums of three squares.
type: Item
instance of: scholarly article
title: On sums of three squares.
zbMATH Open document ID: 1050.11043
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2004-03-16T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93890
review text: This article is a summary of upcoming joint work of the author with I. Piatetski-Shapiro and P. Sarnak. The key result is a subconvexity estimate for the central value \(L(1/2,\pi_f \otimes \chi_1)\) of the twist by a ray class character \(\chi_1\) of the \(L\)-series attached to a Hilbert modular Hecke eigenform \(f\) of weight \((k,\ldots,k)\) (or the corresponding automorphic representation \(\pi_f\)) with the conductor of \(\chi_1\) varying. The subconvexity estimate is obtained using the amplification technique and averaging over a suitable family. By generalization of Waldspurger's formula, the above central values are related to the Fourier coefficients of the half integral weight modular form corresponding to \(f\) under Shimura's correspondence. Using this relation, an asymptotic formula for representation numbers of (squarefree) numbers by ternary quadratic forms is derived, generalizing known results from the case of the ground field \({\mathbb Q}\).
reviewed by: Rainer Schulze-Pillot
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 11F41
zbMATH DE Number: 2058852
zbMATH Keywords: sums of squares
zbMATH Keywords: integral quadratic forms
zbMATH Keywords: theta series
zbMATH Keywords: Waldspurger's theorem
zbMATH Keywords: subconvexity estimates
author: James Cogdell
MaRDI profile type: Publication
OpenAlex ID: W2071750534
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DOI: 10.5802/JTNB.385
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description: scientific article; zbMATH DE number 2058852