Automorphic Forms and Sums of Squares over Function Fields
نویسندگان
چکیده
منابع مشابه
Automorphic Forms and Sums of Squares over Function Fields
We develop some of the theory of automorphic forms in the function field setting. As an application, we find formulas for the number of ways a polynomial over a finite field can be written as a sum of k squares, k ≥ 2. Given a finite field Fq with q odd, we want to determine how many ways a polynomial in Fq[T ] can be written as a sum of k squares. For k ≥ 3 (or k = 2, −1 not a square in Fq), t...
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Let K be a totally real number field with Galois closure L. We prove that if f ∈ Q[x1, . . . , xn] is a sum of m squares in K[x1, . . . , xn], then f is a sum of 4m · 2[L:Q]+1 ([L : Q] + 1 2 ) squares in Q[x1, . . . , xn]. Moreover, our argument is constructive and generalizes to the case of commutative K-algebras. This result gives a partial resolution to a question of Sturmfels on the algebra...
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For a quadric Q over a real field k, we investigate whether finiteness of the Pythagoras number of the function field k(Q) implies the existence of a uniform bound on the Pythagoras numbers of all finite extensions of k. We give a positive answer if the quadratic form that defines Q is weakly isotropic. In the case where Q is a conic, we show that the Pythagoras number of k(Q) is 2 only if k is...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 1999
ISSN: 0022-314X
DOI: 10.1006/jnth.1999.2454