Inspired by the question 


[Does the moduli space of smooth curves of genus g contain an elliptic curve](https://mathoverflow.net/questions/103120/does-the-moduli-space-of-smooth-curves-of-genus-g-contain-an-elliptic-curve)


and its amazing answers, I ask (pure out of curiosity) whether the moduli space $M\_3$ of (smooth projective connected) curves of genus $3$ contains a (smooth projective connected) curve of genus $2$. 


The existence of such a genus two curve is (Edit: **stronger**) than the existence of a surface $S$, a genus two curve $C$ and a smooth projective non-isotrivial morphism $S\to C$ whose fibres are genus three curves. 


If the answer is positive, how explicit can our answer be made? I'm already aware of the fact that $M\_g$ contains a complete curve for all $g\geq 3$. For instance, in the paper by Chris Zaal 


<http://dare.uva.nl/document/38546>


many curves of some genus (I think 513) are shown to embed into $M\_3$.


Of course, by Shafarevich' conjecture, if $K(C)$ denotes the function field of $C$, there are only finitely many non-isotrivial $K(C)$-isomorphism classes of genus three curves over $K(C)$ with good reduction over $C$. I'm asking whether there exists some genus two curve $C$ such that there exists a genus three curve over $K(C)$ with good reduction over $C$. 


**Edit:** the arithmetic analogue also has a negative answer.


The latter (**weaker**) phrasing of my question allows us to formulate an arithmetic analogue of the above question. (I know that I'm considering function fields over $\mathbf{C}$ and that some of you might argue function fields over $\mathbf{F}\_p$ are a better analogue of number fields.) This arithmetic analogue reads as follows. There exists a number field of "genus two" such that there exist a genus three curve over $K$ with good reduction over the ring of integers of $K$. Here a number field of "genus two" should be a number field of absolute discriminant $e^2$. I'll take this to mean discriminant at most $8$.


**Arithmetic analogue. (Abrashkin-Fontaine)** There do not exist non-zero smooth abelian schemes over the ring of integers of a number field of absolute discriminant at most 8.


There are many related questions I'd also like to ask. For example, what is the minimal $g$ such that $M\_g$ contains a genus two curve? Or, what is the minimal $g$ such that $M\_3$ contains a genus $g$ curve? And, finally, is there an example of a complete curve in $M\_g$ which is defined over $\overline{\mathbf{Q}}$? (Edit: The answer to the last question is positive. This is explained in the comments below.)



Does the moduli space of genus three curves contain a complete genus two curve
There does not exist a map of a smooth complete genus 2 curve to $M\_3$. 


Such a map would give rise to a surface $S$ (of general type) which violates the Bogomolov-Miyaoka-Yau inequality $c\_1(S)^2 \leq 3c\_2(S)$. This inequality is equivalent to $3\sigma (S) \leq e(S)$ where $\sigma$ and $e$ are the signature and topological euler characteristic of the surface. The euler characteristic of this surface is 8 (since it is multiplicative for fiber bundles) and by the index theorem, the signature is given by 4 times the integral of $\lambda\_1$ over the curve in $M\_3$. Since $\lambda\_1$ is ample on $M\_3$, $\sigma$ must be positive and divisible by 4. 


This argument is due to Dieter Kotschick in his paper "Signatures, Monopoles, and Mapping Class Groups" (MRL vol 5, 1998). 


