  Computation of Gauss's arithmetic-geometric mean involves iteration of a
simple step, whose algebro-geometric interpretation is the construction of an
elliptic curve isogenous to a given one, specifically one whose period is
double the original period. A higher genus analogue should involve the explicit
construction of a curve whose jacobian is isogenous to the jacobian of a given
curve. The doubling of the period matrix means that the kernel of the isogeny
should be a lagrangian subgroup of the group of points of order 2 in the
jacobian. In genus 2 such a construction was given classically by Humbert and
was studied more recently by Bost and Mestre. In this article we give such a
construction for general curves of genus 3. We also give a similar but simpler
construction for hyperelliptic curves of genus 3. We show that the
hyperelliptic construction is a degeneration of the general one, and we prove
that the kernel of the induced isogeny on jacobians is a lagrangian subgroup of
the points of order 2. We show that for g at least 4 no similar construction
exists, and we also reinterpret the genus 2 case in our setup. Our construction
of these correspondences uses the bigonal and the trigonal constructions,
familiar in the theory of Prym varieties.
