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Extra resources for Binary Quadratic Forms: An Algorithmic Approach (Algorithms and Computation in Mathematics)
Example text
1. 2). That is, we have f I2 = f and f (U V ) = (f U )V for any form f and all matrices U, V ∈ GL(2, R). 2 Equivalence Let U ∈ Z(2,2) with det U = 0, let g = f U , and let n ∈ R. If (x, y) is a representation of (det U )n by g then (det U )f U (x, y) = (det U )n. Hence, U (x, y) is a representation of n by f . 6) sends representations of (det U )n by g to representations of n by f . 1. 1 and set g = f U . The vector (1, 0) is a representation of −1 by g. We have det U = −1. Therefore, U (1, 0) = (−4, 3) is a representation of 1 by f .
Let m and n be integers. First, let m = 0. We have 0 n = 1 0 m n very if n = ±1, otherwise. Now let m = 0. 13 can be used to reduce the 0 for some n . Write determination of m n to the computation of n n = (−1)x 2y n with an odd positive n and x, y ≥ 0. Then the definition of the Kronecker symbol tells us that m n = (−1)x+y(m 2 −1)/8 m . n 44 3 Constructing Forms So it suffices to explain the computation m n with an odd and positive n. 13, we may replace m by m mod n. Hence, we assume that 0 ≤ m < n.
M 2. If m ≡ m (mod n), then m n = n . m m 3. m n n = nn . m m mm 4. n n = n . (n−1)/2 5. −1 . n = (−1) 2 (n2 −1)/8 6. n = (−1) . (n−1)(m−1)/4 n 7. If m, n are odd and positive, then m n = (−1) m . m 8. If m = 0, m ≡ 0, 1 (mod 4) and n ≡ n (mod |m|), then n = m n . Proof. 12. 13 we can evaluate the Legendre symbol ∆ p for a fixed discriminant ∆ and many primes p as follows. We ∆ compute the table of all values ∆ n for 0 ≤ n < ∆. If we want to compute p ∆ for a prime number p, then we can find the value ∆ p = p mod |∆| by a table look-up.