A Proof of a Conjecture of C.L. Siegel Original Research Article

Let Mn be the Minkowski fundamental domain for the space of n × n real, symmetric, and positive definite matrices under the action of the unimodular group SLn(Z). C. L. Siegel conjectured that d(A, B) − f(A, B) ≤ C(n), for A, B set membership, variant Mn, where d and f are the geodesic and the reduced geodesic distances, respectively, and C(n) is a constant depending only on n. This conjecture appears in his book ("Zur Reduktionstheorie Quadratischer Formen," The Mathematical Society of Japan, 1959). By reducing the problem to the diagonal matrices in Mn, we obtain a proof.