Gauss made two conjectures about average values of class numbers of orders in quadratic number fields, later on proven by Lipschitz and Siegel. A version for function fields of odd characteristic was established by Hoffstein and Rosen. In this paper, we extend their results to the case of even characteristic. More precisely, we obtain formulas of average values of L-functions associated to orders in quadratic function fields over a constant field of characteristic two, and then derive formulas of average class numbers of these orders.

Abstract. A theorem of F. Hirzebruch relates continued fractions to class numbers of quadratic number fields. A version for function fields of odd characteristic was established by D. R. Hayes and C. D. González. We present here a complete treatment of the even charateristic theory, in particular, two class number relations involving continued fractions are derived, one of which is an analogue of the Hirzebruch relation in characteristic 2.

Let E be an elliptic curve defined over Q and P ∈ E(Q) a rational point of infinite order.Suppose that E has complex multiplication by an order in the imaginary quadratic field k. Denote by M E,P the set of rational primes such that splits in k, E has good reduction at , and P is a primitive point modulo . Under the generalized Riemann hypothesis, we can determine the positivity of the density of the set M E,P explicitly.

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