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, this term is the Laplace transform of certain infinite series consisting of KBessel functions. The positivity property of the fourth log derivative of such series is closely related to the Riemann hypothesis for the Hasse-Weil L-funct...

(1.1) E : y + a1xy + a3y = x + a2x + a4x+ a6 where a1, a2, a3, a4, a6 ∈ Z. Let N(E) denote the conductor of E, j(E) the j-invariant of E, and L(E, s) = ∑∞ n=1 a(n)n −s the Hasse-Weil L-function of E. If E is modular, then let FE(z) = ∑∞ n=1 aE(n)q n ∈ S2(N(E), χ1) be the associated weight 2 cusp form. Here χ1 denotes the trivial Dirichlet character. Throughout, D will denote a square-free integ...

Hilbert’s law of quadratic reciprocity, as reformulated by Hasse, states that for a quaternion algebra D over a number field K, the number of ramified places of D is even. In general a number field has three kinds of places: finite, real, and complex. The complex places are never ramified. If D is assumed unramified at all finite places, it follows that D is ramified at an even number of real p...

We are now ready to prove Hasse's theorem.

Hasse constants and their basic properties are introduced to facilitate the connection between the lattice of subalgebras of an algebra C and the natural action of the automorphism group Aut(C) on C. These constants are then used to describe the lattice of subloops of the smallest nonassociative simple Moufang loop.

In this paper we establish a Hasse principle concerning the linear dependence over Z of nontorsion points in the Mordell-Weil group of an abelian variety over a number field.

In a previous work we introduced slice graphs as a way to specify both infinite languages of directed acyclic graphs (DAGs) and infinite languages of partial orders. Therein we focused on the study of Hasse diagram generators, i.e., slice graphs that generate only transitive reduced DAGs. In the present work we show that any slice graph can be transitive reduced into a Hasse diagram generator r...

Del Pezzo surfaces are smooth projective surfaces, isomorphic over the algebraic closure of the base ,eld to P P or the blow-up of P in up to eight points in general position. In the latter case the del Pezzo surface has degree equal to 9 minus the number of points in the blow-up. The arithmetic of del Pezzo surfaces over number ,elds is an active area of investigation. It is known that the Has...

Two dimensional adelic objects were introduced by I. Fesenko in his study of the Hasse zeta function associated to a regular model E of the elliptic curve E. The Hasse-Weil L-function L(E, s) of E appears in the denominator of the Hasse zeta function of E . The two dimensional adelic analysis predicts that the integrand h of the boundary term of the two dimensional zeta integral attached to E i...