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On a theorem of Mestre and Schoof

John E. Cremona, Andrew V. Sutherland (2010)

Journal de Théorie des Nombres de Bordeaux

A well known theorem of Mestre and Schoof implies that the order of an elliptic curve E over a prime field 𝔽 q can be uniquely determined by computing the orders of a few points on E and its quadratic twist, provided that q > 229 . We extend this result to all finite fields with q > 49 , and all prime fields with q > 29 .

On a two-variable zeta function for number fields

Jeffrey C. Lagarias, Eric Rains (2003)

Annales de l’institut Fourier

This paper studies a two-variable zeta function Z K ( w , s ) attached to an algebraic number field K , 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 w = 1 this function becomes the completed Dedekind zeta function ζ ^ K ( s ) of the field K . The function is a meromorphic function of two complex variables with polar divisor s ( w - s ) , and it satisfies the functional equation Z K ( w , s ) = Z K ( w , w - s ) . We consider the special case K = , where for w = 1 this function...

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