Quasi-elliptic integrals and periodic continued fractions

In this report we detail the following story. Several centuries ago, Abel noticed that the well-known elementary integral (latin small letter esh)dx/√x² + 2bx + c = log(x + b + √x² + 2bx + c) is just an augur of more surprising integrals of the shape (latin small letter esh)f(x)dx/√D(x) = log(p(x) + q(x) √D(x)). Here f is a polynomial of degree g and the D are certain polynomials of degree deg D(x) = 2g + 2. Specifically, f(x) = p′(x)/q(x) (so q divides p′). Note that, morally, one expects such integrals to produce inverse elliptic functions and worse, rather than an innocent logarithm of an algebraic function. Abel went on to study, well, abelian integrals, and it is Chebychev who explains - using continued fractions - what is going on with these 'quasi-elliptic' integrals. Recently, the second author computed all the polynomials D over the rationals of degree 4 that have an f as above. We will explain various contexts in which the present issues arise. Those contexts include symbolic integration of algebraic functions; the study of units in function fields; and, given a suitable polynomial g, the consideration of period length of the continued fraction expansion of the numbers √g(n) as n varies in the integers. But the major content of this survey is an introduction to period continued fractions in hyperelliptic - thus quadratic - function fields.