On sums of three squares. (Q1424568)

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 \({\mathbb Q}\).