Quasi-elliptic integrals and periodic continued fractions

In this report we detail the following story. Several centuries ago, Abel n oticed that the well-known elementary integral integral dx/rootx(2) + 2bx + c = log(x + b + rootx(2) + 2bx + c) is just an augur of more surprising integrals of the shape integralf(x)dx/rootD(x) = log(p(x) + q(x)rootD(x)). Here f is a polynomial of degree g and the D are certain polynomials of deg ree deg D(x) = 2g + 2. Specifically, f(x) = p'(x)/q(x) (so q divides p'). N ote that, morally, one expects such integrals to produce inverse elliptic f unctions and worse, rather than an innocent logarithm of an algebraic funct ion. Abel went on to study, well, abelian integrals, and it is Chebychev who exp lains - using continued fractions - what is going on with these 'quasi-elli ptic' integrals. Recently, the second author computed all the polynomials D over the rationals of degree 4 that have anf as above. We will explain var ious contexts in which the present issues arise. Those contexts include sym bolic integration of algebraic functions; the study of units in function fi elds; and, given a suitable polynomial g, the consideration of period lengt h of the continued fraction expansion of the numbers rootg(n) as n varies i n the integers. But the major content of this survey is an introduction to period continued fractions in hyperelliptic - thus quadratic - function fie lds. 1991 Mathematics Subject Classification: 11J70, 11A65, 11J68.