Without pretending to understand Ben Wieland's argument, let me take the Shimura curve suggestion and run with it. In particular, modulo a few details I think the following should be true:

>The Shimura curve $X^{218}$ defines a complete curve in $M_2$

My first reaction upon seeing Jordan's Shimura curve suggestion was "No way". The reason is that a priori Shimura curves $X^D$ don't parametrize curves of genus 2, but abelian surfaces which admit the action of a maximal order in the unique $\mathbf{Q}$-quaternion algebra of discriminant $D$ (and the fixed choice of a polarization compatible with the action of the choice of maximal order). Indeed, infinitely many of these abelian surfaces are products of elliptic curves with complex multiplication, and these of course do not need to necessarily be Jacobians.

Thankfully we have at our disposal [the work of Hayashida and Nishi][1] who asked the question: For what pairs of isogenous elliptic curves $E,E'$ is there a genus 2 curve inside $E\times E'$?

In particular they show the following:

> If $E$ does not have CM, $E\times E$ is not a Jacobian

> If $E,E'$ have CM by a maximal order in $\mathbf{Q}(\sqrt{-m})$ then $E\times E'$ contains a genus 2 curve if and only if $m \ne 1,3,7,15$

by reducing the above question to one about a certain real valued 4-variable quadratic form over the integers. A priori this doesn't answer the question, because we have to concern ourselves with nonmaximal orders in $\mathbf{Q}(\sqrt{-m})$. However, it seems (at least in the cursory read of their paper I've been able to give this afternoon) that their methods can be generalized to nonmaximal orders with a little bit of care. If the apparent qualms people have about Ben Wieland's argument can be resolved, I'll do exactly that in a few days.

But assuming that all works out, life is great! In particular we can come up with a Shimura curve $X^D$ which avoids those particular products of CM elliptic curves if and only if there is a prime $p|D$ such that $\left(\dfrac{-m}{p}\right) = 1$ for $m= 1,3,7,15$. We can compute that the minimal such prime is $109$, which is $1 \bmod 12$, $4  \bmod 5$ and $4\bmod 7$.

We recall momentarily that $X^D$ is a complete curve if and only if the unique rational quaternion algebra of discriminant $D$ is a division algebra if and only if $D$ is the squarefree product of an even number of primes. For convenience we take $D = 2p$.

Therefore (again assuming the bit about nonmaximal orders) we have found $X^{218}$ to be a curve lying in the image of the Torelli map in $\mathcal{A}_2$, and thus in $M_2$.

Comments are very welcome!

  [1]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jmsj/1260975570