Christian Friesen

The statistics of continued fractions for polynomials over a finite field

Given a finite field F of order q and polynomials a, b ∈ F [X] of degrees m < n respectively, there is the continued fraction representation b/a = a1 + 1/(a2 + 1/(a3 + · · · + 1/ar)). Let CF (n, k, q) denote the number of such pairs for which deg b = n, deg a < n, and for 1 ≤ j ≤ r, deg aj ≤ k. We give both an exact recurrence relation, and an asymptotic analysis, for CF (n, k, q). The polynomial associated with the recurrence relation turns out to be of P-V type. We also study the distribution of r. Averaged over all a and b as above, this presents no difficulties. The average value of r is n(1− 1/q), and there is full information about the distribution. When b is fixed and only a is allowed to vary, we show that this is still the average. Moreover, few pairs give a value of r that differs from this average by more than O( √ n/q).

Class group frequencies of real quadratic function fields: the degree 4 case

The distribution of ideal class groups of F q (T, √M(T)) is examined for degree-four monic polynomials M E F q [T] when F q is a finite field of characteristic greater than 3 with q E [20000,100000] or q E [1020000,1100000] and M is irreducible or has an irreducible cubic factor. Particular attention is paid to the distribution of the p-Sylow part of the class group, and these results agree with those predicted using the Cohen-Lenstra heuristics to within about 1 part in 10000. An alternative set of conjectures specific to the cases under investigation is in even sharper agreement.

Cyclotomy of order 15 over GF(p2), p=4, 11(mod15)

For primes p≡4, 11(mod15) explicit formulae are obtained for the cyclotomic numbers of order 15 over GF(p2).

Remark on the Class Number of Q(√2p) Modulo 8 for p ≡5 (mod 8) a Prime

An explicit congruence modulo 8 is given for the class number of the real quadratic field Q(dr2p), wherep is a prime congruent to 5 modulo 8. Let Q denote the rational number field. Let Q(4d) denote the quadratic extension of Q having discriminant d. The class number of Q(Jd) is denoted by h(d). \fd>0 the fundamental unit (> 1) of Q(-/d) is denoted by ed. If d = p, where p = 5 (mod 8) is a prime, it is a classical result of Gauss that h(p)= 1 (mod2) (see for example [4, §3]) and the author [10, Theorem 1] has given an explicit congruence for h(p) modulo 4, namely;