2019/03/15 by Jeanine Van Order, Van Order, Jeanine
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1903.06686
openalex publication_date 2019/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We obtain nonvanishing estimates for central values of certain self-dual\nRankin-Selberg L-functions on \GL2( bfAF) \×\n\GL2( bfAF), and more generally\n\GLr( bfAF) \× \GL2( bfAF) for r\n\≥ 2 an integer over F a totally real number field, contingent on the best\nknown approximations towards the generalized Lindel "of hypothesis for\n\GL2( bfAF)-automorphic forms in the level aspect, as\nwell as the best known approximations to the generalized Ramanujan conjecture\nhypothesis for \GL2( bfAF)-automorphic forms. We proceed\nby developing a spectral approach to the shifted convolution problem for\ncoefficients of \GL2( bfAF)-automorphic forms, accessing\nhe higher-rank case through the classical projection operator mathbb Pr1\nand the way it respects Fourier-Whittaker expansions. In the course of deriving\nour results, we supply the required nonvanishing hypothesis for recent work of\nDarmon-Rotger to bound Mordell-Weil ranks of elliptic curves in number fields\ncut out by tensor products of two odd, two-dimensional Artin representations\nwhose product of determinants is trivial. This in particular allows us to\ndeduce bounds (on average) for Mordell-Weil ranks of elliptic curves in ring\nclass extensions of real quadratic fields which had not been accessible\npreviously.\n