TITLE: Infinitely many curves with isogenous Jacobians
QUESTION [18 upvotes]: Let $g\geq 4$. Are there infinitely many compact genus $g$ Riemann surfaces with (mutually) isogenous Jacobians?
Does the situation change in positive characteristic?

REPLY [4 votes]: I believe I have an example in genus 5:
Humbert curves (see either Varley's "Weddle's Surfaces, Humbert's Curves, and a Certain 4-Dimensional Abelian Variety", or exercise batch F in chapter 6 of ACGH) are in 1-1 correspondence with 5-tuples lines in the plane -- the plane being $I_C(2)$ -- up to projective transformations of the plane. These 5 lines are the image of the double cover from $W^1_4(C)$ to rank < 4 quadrics enveloping the canonical image of $C$. Each of the 5 lines is marked with 4 points - its intersection points with the other four lines, and is the 2-quotient (under the map $W^1_4\to I_C(2)$ ) of an elliptic curve which lies inside the Jacobian of $C$, where four points are the ramification points. The Jacobian of C is isogenous to the product of these five elliptic curves.
Giving the plane the projective coordinates $x,y,z$, we now consider the lines: $$x=0,\quad y=0,\quad x=z,\quad y=z,\quad x+y=az.$$
The $y$ (resp $x$) coordinate of the intersection points on the first (resp second) line are $0,1,a,\infty$; whereas the $x$ (resp $y$) coordinate of the intersection points on the third (resp fourth) lines are $0,1, 1-a, \infty$. Since $x\mapsto 1-x$ takes $0\to1,1\to0,\infty\to\infty, a\to1-a$, the four elliptic curves associated with these four lines are isomorphic. Finally, The elliptic curve associated with fifth line has an extra involution, (flipping $x,y$), so it is the same curve regardless of $a$. Hence if we pick any $a_1,a_2$ which give isogenous elliptic curves, we get isogenous Humbert curves.