Non-positivity of certain functions associated with analysis on elliptic surfaces

Abstract

In this paper, we study some basic analytic properties of the boundary term of Fesenko's two-dimensional zeta integrals. In the case of the rational number field, we show that this term is the Laplace transform of certain infinite series consisting of $K$-Bessel functions. It is known that the non-positivity property of the fourth log derivative of such series is a sufficient condition for the Riemann hypothesis of the Hasse-Weil $L$-function attached to an elliptic curve. We show that such non-positivity is a necessary condition under some technical assumption.