Quaternions , polarizations and class numbers
نویسنده
چکیده
We study abelian varieties A with multiplication by a totally indefinite quaternion algebra over a totally real number field and give a criterion for the existence of principal polarizations on them in pure arithmetic terms. Moreover, we give an expression for the number π 0 (A) of isomorphism classes of principal polarizations on A in terms of relative class numbers of CM fields by means of Eichler's theory of optimal embeddings. As a consequence, we exhibit simple abelian varieties of any even dimension admitting arbitrarily many non-isomorphic principal polarizations. On the other hand, we prove that π 0 (A) is uniformly bounded for simple abelian varieties of odd square-free dimension.
منابع مشابه
On Quaternion Maps with Memory
Quaternions 1.1 Quaternions are a class of hypercomplex numbers with four real components [1]. By analogy with the complex numbers being representable as a sum of real and imaginary parts (z a + bi), quaternions can also be written as a linear combination: q a + bi + cj + dk, (1) where 1, i, j, k make a group and satisfy the noncommutative rules: i2 j2 k2 -1, ij ji k, jk -kj ...
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1. A. F. Horadam. "Basic Properties of a Certain Generalized Sequence of Numbers." The Fibonacci Quarterly 3 (1965):161-75. 2. A. F. Horadam. "Complex Fibonacci Numbers and Fibonacci Quaternions." Amer. Math. Monthly 70 (1963):289-91. 3. A. L. Iakin. "Generalized Quaternions with Quaternion Components." The Fibonacci Quarterly 15 (1977):35Q-52. 4. A. L. Iakin. "Generalized Quaternions of Higher...
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