Let $(\mathcal{M}_{1,1})_{\overline{\mathbb{Q}}}$ be the moduli stack of elliptic curves over $\overline{\mathbb{Q}}$. By Oda, we know that its etale fundamental group is $\widehat{SL_2(\mathbb{Z})}$.

Let $(\mathcal{M}_{1,2})^\circ_{\overline{\mathbb{Q}}}$ denote the universal family of punctured elliptic curves (with the identity section removed), and let $E^\circ_{\overline{\mathbb{Q}}}$ be a geometric fiber. Then it is often said that we have the following exact sequence:

$$1\rightarrow\pi_1(E^\circ_{\overline{\mathbb{Q}}})\rightarrow\pi_1((\mathcal{M}_{1,2})^\circ_{\overline{\mathbb{Q}}})\rightarrow\pi_1((\mathcal{M}_{1,1})_{\overline{\mathbb{Q}}})\rightarrow 1$$

We know that $\pi_1(E^\circ_{\overline{\mathbb{Q}}}) \cong \widehat{F_2}$, the profinite completion of the free group of rank 2, so this exact sequence induces an outer representation $$\pi_1((\mathcal{M}_{1,1})_{\overline{\mathbb{Q}}}) = \widehat{SL_2(\mathbb{Z})}\longrightarrow Out(\widehat{F_2})$$

It's often said, for example in page 376 of

Makoto Matsumoto, Arithmetic fundamental groups and moduli of curves


Makoto Matsumoto & Akio Tamagawa, Mapping-class-group action versus Galois action on profinite fundamental groups

that this action comes from the natural (outer) action of the mapping class group $\Gamma_{1,1}$ of $E^\circ_\mathbb{C}$ on $\pi_1(E^\circ_\mathbb{C})$ by isotopy classes of orientation preserving self-diffeomorphisms, but of course it is never cited.

While I have no doubt that this is true, can anyone tell me where this is actually proven? I'd like to use this result in a paper, but I feel bad about just stating it without reference.

Actually I've even proven it myself, though it took me around 3-4 pages to explain fully, and my proof is very specific to elliptic curves. (Essentially I take the "universal" elliptic surface over the $j$-line punctured at 0 and 1728, compute the outer representation associated to the fundamental group of the base (isomorphic to $F_2$) by computing the monodromy around the singular fibers using the homological invariant, and then prove that the induced map $\mathbb{A}^1_j\setminus\{0,1728\}\rightarrow(\mathcal{M}_{1,1})_{\mathbb{C}}$ is surjective on fundamental groups by showing that pullbacks of connected covers remain connected.)

  • $\begingroup$ In Joan Birman's book on braid groups, an action of the braid group $B_n$ on $n$ strands on the free group $F_n$ on $n$ generators is given; the mapping class group to which you refer seems to be the action of $B_3$ on three strands, and since the affine part of the elliptic curve is a two-sheeted cover of the plane minus the three roots of the cubic polynomial in question, the pure braid group $P_3$ acts on the $\pi _1$ of the punctured ellpitic curve; this must be the action, and this may be due to (Emil) Artin. $\endgroup$ – Venkataramana Oct 10 '15 at 2:12

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