The Arithmetic of Quaternion Algebra

118CHAPTER 5. QUATERNION ARITHMETIC IN THE CASE WHERE THE EICHLER CONDITI
of Q( √ m) above a. For the different values of m we have
m 2 3 5 6 7 13 15 17 21 33
ζ Q( √ m)(−1) 1/12 1/6 1/30 1/2 2/3 1/6 2 1/3 1/3 1
h + m 1 2 1 2 2 1 4 1 2 2
Abelian cubic field: The Eichler order of level N without square factor,of discriminant
D, in a quaternion algebra totally defined over an abelian cubic field
of discriminant m 2 , and which has a class number equal to the class number h + m
of center, are the 19 maximal orders given by the following list:
m equation ζ(−1) D
7 x 3 − 7x − 7 −1/21 p2, p3, p (i)
13
9 x 3 − 3x + 1 −1/9 p3, p (i)
19
13 x 3 − x 2 − 4x − 1 −1/3 p13
Reference: Vigneras-Gueho [3].
Exercise
, p(i)
37
, p(i) 29 , p(i) 43
Euclidean orders. Prove it exists exactly 3 quaternion algebras totally defined
over Q, of which the maximal orders (over Z) are euclidean for the norm. Their
reduced discriminants are 2,3,5 respectively.
(I translate this book just for those who want read it but up to now still has
a little difficulty for reading French. I am not an expert both in quaternion
algebra and French language, so definitely there are many mistakes both in
mathematics and in language. When you read it you must be more careful than
usual. Correct them please.—translator,27 Sept. 2006. Beijing)