Relative trace formulas and subconvexity estimates for \(L\)-functions of Hilbert modular forms (Q2833606)

Averages of central values base change \(L\)-functions of the form NEWLINE\[NEWLINE\sum_{f \in B_k(N)} \frac{L(1/2, f) L(1/2, f \times \chi)}{L(1, \pi, \text{Ad})},NEWLINE\]NEWLINE where the sum is taken over a basis of cuspidal holomorphic (new)forms of weight \(k\) and level \(N\) and \(\chi\) is a quadratic character, belong to the most fascinating mean values for \(L\)-functions. The asymptotic formula features \(L(1, \chi)\) in the main term which motivated \textit{H. Iwaniec} and \textit{P. Sarnak} [Isr. J. Math. 120, Part A, 155--177 (2000; Zbl 0992.11037)] to investigate this in the context of possible Siegel zeros. In fact, as a consequence of Gross-Zagier type formulae, the average can be expressed in closed form (if \(\chi\) is odd, i.e., corresponds to an imaginary quadratic field, and \(N\) is prime), various applications of which were given in papers by \textit{P. Michel} and \textit{D. Ramakrishnan} [in: Number theory, analysis and geometry. In memory of Serge Lang. Berlin: Springer. 437--459 (2012; Zbl 1276.11057)].NEWLINENEWLINEFollowing earlier work of \textit{D. Ramakrishnan} and \textit{J. Rogawski} [Pure Appl. Math. Q. 1, No. 4, 701--735 (2005; Zbl 1143.11029)] and the first author [J. Number Theory 156, 195--246 (2015; Zbl 1395.11080)], the present paper uses a relative trace formula to evaluate asymptotically a version of the above mean value twisted by Hecke eigenvalues in the context of Hilbert modular forms for arbitrary level (not necessarily squarefree) and under relaxed conditions on the character \(\chi\).NEWLINENEWLINEAs applications the authors provide a subconvexity estimate of the product of \(L\)-functions in the numerator in the weight aspect, a non-vanishing result (cf.\ also [the authors, Can. J. Math. 68, No. 4, 908--960 (2016; Zbl 1404.11057)] and a (``vertical'') equidistribution statement for Satake parameters.