On L-functions of modular elliptic curves and certain K3 surfaces

Abstract

Inspired by Lehmer’s conjecture on the non-vanishing of the Ramanujan tau -function, one may ask whether an odd integer alpha can be equal to tau (n) or any coefficient of a newform f(z). Balakrishnan, Craig, Ono and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight kge 4. We use these methods for weight 2 and 3 newforms and apply our results to L-functions of modular elliptic curves and certain K3 surfaces with Picard number ge 19. In particular, for the complete list of weight 3 newforms f_lambda (z)=sum a_lambda (n)q^n that are eta -products, and for N_lambda the conductor of some elliptic curve E_lambda , we show that if |a_lambda (n)|<100 is odd with n>1 and (n,2N_lambda )=1, then aλ(n)∈{-5,9,±11,25,±41,±43,-45,±47,49,±53,55,±59,±61,±67,-69,±71,±73,75,±79,±81,±83,±89,±93±97,99}.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} a_\\lambda (n) \\in&\\{-5,9,\\pm 11,25, \\pm 41, \\pm 43, -45,\\pm 47,49, \\pm 53,55, \\pm 59, \\pm 61,\\\\&\\pm 67, -69,\\pm 71,\\pm 73,75, \\pm 79,\\pm 81, \\pm 83, \\pm 89,\\pm 93 \\pm 97, 99\\}. \\end{aligned}$$\\end{document}Assuming the Generalized Riemann Hypothesis, we can rule out a few more possibilities leaving aλ(n)∈{-5,9,±11,25,-45,49,55,-69,75,±81,±93,99}.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} a_\\lambda (n) \\in \\{-5,9,\\pm 11,25,-45,49,55,-69,75,\\pm 81,\\pm 93, 99\\}. \\end{aligned}$$\\end{document}