2019/10/17 by Rob de Jeu, de Jeu, Rob, Xavier-François Roblot +1
Mathematics · Arts and Humanities · #Algebraic Geometry and Number Theory #Historical Studies and Socio-cultural Analysis
paper · pdf · doi:10.48550/arxiv.1910.07739
We conjecture that the p-adic L-function of a non-trivial irreducible even\nArtin character over a totally real field is non-zero at all non-zero integers.\nThis implies that a conjecture formulated by Coates and Lichtenbaum at negative\nintegers extends in a suitable way to all positive integers. We also state a\nconjecture that for certain characters the Iwasawa series underlying the p-adic\nL-series have no multiple roots except for those corresponding to the zero at\ns=0 of the p-adic L-function. We provide some theoretical evidence for our\nfirst conjecture, and prove both conjectures by means of computer calculations\nfor a large set of characters (and integers where appropriate) over the\nrationals and over real quadratic fields, thus proving many instances of\nconjectures by Coates and Lichtenbaum and by Schneider. The calculations and\nthe theoretical evidence also prove that certain p-adic regulators\ncorresponding to 1-dimensional characters for the rational numbers are units in\nmany cases. We also verify Gross' conjecture for the order of the zero of the\np-adic L-function at s=0 in many cases. We gather substantial statistical data\non the constant term of the underlying Iwasawa series, and propose a model for\nits behaviour for certain characters.\n