Zeta-functions and Ideal Classes of Quaternion Orders
نویسندگان
چکیده
Inspired by Stark’s analytic proof of the finiteness of the class number of a ring of integers in an algebraic number field, we give a new proof of the finiteness of the number of classes of ideals in a maximal order of a quaternion division algebra over a totally real number field. Previous proofs of this well-known result have used adeles or geometry of numbers, while our proof uses the classical analytic theory of zeta functions. We also note that this approach leads to alternative proofs of Eichler’s mass formula and the even parity of the number of ramified primes in the quaternion algebra.
منابع مشابه
Twisted Zeta Functions of Quaternion Orders
Given an abelian Galois extension K/F of number fields, a quaternion algebra A over F that is ramified at all infinite primes, and a character χ of the Galois group of K over F , we consider the twist of the zeta function of A by the character χ. We show that such twisted zeta functions provide a factorization of the zeta function of A(K) = A ⊗F K. Also, the quotient of the zeta function for A(...
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