Infinitely many curves with isogenous Jacobians Let $g\geq 4$. Are there infinitely many compact genus $g$ Riemann surfaces with (mutually) isogenous Jacobians?
Does the situation change in positive characteristic?
 A: 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.
A: (Edit: this doesn't answer the question - I misread it to mean that we were looking for infinitely many isogenies of distinct Jacobians in arbitrary genus.)
There are explicit constructions that provide infinitely many examples of pairs of non-isomorphic curves with isogenous Jacobians in any genus.
Jean-François Mestre has a Richelot-like construction that, for each $g \ge 2$, produces a $g+1$-dimensional family of pairs of hyperelliptic curves with $(2,\ldots,2)$-isogenous Jacobians.  For the details, see https://www.ams.org/journals/jag/2013-22-03/S1056-3911-2012-00589-X/ (if you don't have access to that, then there's an earlier version in French at https://arxiv.org/abs/0902.3470 , and a translation of that version into English at http://www.lix.polytechnique.fr/~smith/Mestre---pairs.pdf ).
In characteristic $p$, you can produce examples over finite fields almost trivially using Frobenius.  Let $q$ be a power of $p$.  For each curve $C$ over $\mathbb{F}_{q^2}$, there is a Galois-conjugate curve $C'$ over $\mathbb{F}_{q^2}$ defined by raising the coefficients of the defining equations of $C$ to the $q$-th power, and then $q$-th powering defines a degree-$q$ morphism $\pi: C \to C'$, which (in genus $g>0$) induces an inseparable isogeny between $J_C$ and $J_{C'}$.  If $C$ and $C'$ are not isomorphic (and in general they are not), then you have a pair of distinct isogenous Jacobians.
