On a theorem of Landau. (Sur un théorème de Landau.)
A well known theorem of Mestre and Schoof implies that the order of an elliptic curve over a prime field can be uniquely determined by computing the orders of a few points on and its quadratic twist, provided that . We extend this result to all finite fields with , and all prime fields with .
This paper studies a two-variable zeta function attached to an algebraic number field , introduced by van der Geer and Schoof, which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov divisors. When this function becomes the completed Dedekind zeta function of the field . The function is a meromorphic function of two complex variables with polar divisor , and it satisfies the functional equation . We consider the special case , where for this function...