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Let $K$ be a field of characteristic zero, $X/K$ a hyperbolic $(g,n)$-curve. There is a famous short exact sequence

$$1\longrightarrow\pi_1(X\otimes \overline{K})\longrightarrow\pi_1(X)\longrightarrow \operatorname{Gal}_K\longrightarrow 1.$$

Let $\Pi_{g,n}$ be the profinite completion of the fundamental group $\pi_1(g,n)$ of a genus $g$ compact Riemann surface of with $n$ points punctured. After fixing an isomorphism $\pi_1(X\otimes \overline{K})\overset{\sim}{\to}\Pi_{g,n}$, one gets two exact sequences

$$1\longrightarrow\Pi_{g,n}\longrightarrow\pi_1(X)\longrightarrow \operatorname{Gal}_K\longrightarrow 1$$

$$1\longrightarrow \operatorname{Inn}(\Pi_{g,n }) \longrightarrow \operatorname{Aut}(\Pi_{g,n}) \longrightarrow \operatorname{Out}(\Pi_{g,n}) \longrightarrow 1$$

and the outer Galois representation

$$\rho_X:\operatorname{Gal}_K\longrightarrow \operatorname{Out}(\Pi_{g,n}).$$

Note that the kernel of $\rho_X$ is independent of the choice of the isomorphism. Voevodsky made a conjecture that $\rho_X$ is injective when $X$ is an affine hyperbolic curve and $K$ is a number field. The conjecture was proved by Matsumoto then extended to the proper case by Hoshi–Mochizuki.

In his letter to Faltings, Grothendieck conjectured that any hyperbolic curve (actually for any algebraic curve) over a number field should be anabelian i.e. the geometry of any hyperbolic curve $X$ over a number field is determined by $\rho_X$. Following the earlier work of Nakamura and Tamagawa, this was proved by Mochizuki. Meanwhile, Grothendieck conjectured that the moduli space of hyperbolic curves would be also anabelian. So it’s natural to ask whether Voevodsky’s conjecture holds after replacing $\Pi_{g,n}$ with $\mathscr{M}_{g,n}$, the moduli stack over $K$ of smooth geometrically connected proper curves of genus $g$ with $n$ marked points.

Just like $\Pi_{g,n}$, one has the profinite completion $\mathscr G_{g,n}$ of the oriented mapping class group $\operatorname{MCG}_{g,n} $ of a genus $g$ topological surface with $n$ marked points, which is isomorphic to $\pi_1(\mathscr M_{g,n}\otimes \overline{K})$. Then one gets a Galois representation

$$\rho_{g,n}:\operatorname{Gal}_K\longrightarrow \operatorname{Out}(\mathscr G_{g,n})$$

again. By the work of Matsumoto–Tamagawa, one can study $\rho_{g,n}$ with the associated universal monodromy representation

$$\rho_{g,n}^{\text{um}}:\pi_1(\mathscr{M}_{g,n})\longrightarrow \operatorname{Out}(\Pi_{g,n})$$

given by the exact sequence

$$1\longrightarrow\Pi_{g,n}\longrightarrow \pi_1(\mathscr{M}_{g,n+1})\longrightarrow \pi_1(\mathscr{M}_{g,n})\longrightarrow 1.$$

Using their combinatorial anabelian geometry, Hoshi–Mochizuki showed that $\rho_{g,n}^{\text{um}}$ is injective if and only if $\rho_{g,n}^{\text{um}}|_{\mathscr{G}_{g,n}}$ is injective. Then finally, one just needs to note that the injectivity of $\rho_{g,n}^{\text{um}}|_{\mathscr{G}_{g,n}}$ is equivalent to the congruence subgroup problem for $\operatorname{MCG}_{g,n}$.

Let $K$ be a field of characteristic zero, $X/K$ a hyperbolic $(g,n)$-curve. There is a famous short exact sequence

$$1\longrightarrow\pi_1(X\otimes \overline{K})\longrightarrow\pi_1(X)\longrightarrow \operatorname{Gal}_K\longrightarrow 1.$$

Let $\Pi_{g,n}$ be the profinite completion of the fundamental group $\pi_1(g,n)$ of a genus $g$ compact Riemann surface of with $n$ points punctured. After fixing an isomorphism $\pi_1(X\otimes \overline{K})\overset{\sim}{\to}\Pi_{g,n}$, one gets two exact sequences

$$1\longrightarrow\Pi_{g,n}\longrightarrow\pi_1(X)\longrightarrow \operatorname{Gal}_K\longrightarrow 1$$

$$1\longrightarrow \operatorname{Inn}(\Pi_{g,n }) \longrightarrow \operatorname{Aut}(\Pi_{g,n}) \longrightarrow \operatorname{Out}(\Pi_{g,n}) \longrightarrow 1$$

and the outer Galois representation

$$\rho_X:\operatorname{Gal}_K\longrightarrow \operatorname{Out}(\Pi_{g,n}).$$

Note that the kernel of $\rho_X$ is independent of the choice of the isomorphism. In his letter to Faltings, Grothendieck conjectured that any hyperbolic curve (actually for any algebraic curve) over a number field should be anabelian i.e. the geometry of any hyperbolic curve $X$ over a number field is determined by $\rho_X$. In order to prove some special cases of Grothendieck’a conjecture, Voevodsky made a conjecture that $\rho_X$ is injective when $X$ is an affine hyperbolic curve and $K$ is a number field. Voevodsky’s conjecture was proved by Matsumoto then extended to the proper case by Hoshi–Mochizuki. 

Following the earlier work of Nakamura and Tamagawa, the anabelian conjecture was proved by Mochizuki. Meanwhile, Grothendieck conjectured that the moduli space of hyperbolic curves would be also anabelian. So it’s natural to ask whether Voevodsky’s conjecture holds after replacing $\Pi_{g,n}$ with $\mathscr{M}_{g,n}$, the moduli stack over $K$ of smooth geometrically connected proper curves of genus $g$ with $n$ marked points.

Just like $\Pi_{g,n}$, one has the profinite completion $\mathscr G_{g,n}$ of the oriented mapping class group $\operatorname{MCG}_{g,n} $ of a genus $g$ topological surface with $n$ marked points, which is isomorphic to $\pi_1(\mathscr M_{g,n}\otimes \overline{K})$. Then one gets a Galois representation

$$\rho_{g,n}:\operatorname{Gal}_K\longrightarrow \operatorname{Out}(\mathscr G_{g,n})$$

again. By the work of Matsumoto–Tamagawa, one can study $\rho_{g,n}$ with the associated universal monodromy representation

$$\rho_{g,n}^{\text{um}}:\pi_1(\mathscr{M}_{g,n})\longrightarrow \operatorname{Out}(\Pi_{g,n})$$

given by the exact sequence

$$1\longrightarrow\Pi_{g,n}\longrightarrow \pi_1(\mathscr{M}_{g,n+1})\longrightarrow \pi_1(\mathscr{M}_{g,n})\longrightarrow 1.$$

Using their combinatorial anabelian geometry, Hoshi–Mochizuki showed that $\rho_{g,n}^{\text{um}}$ is injective if and only if $\rho_{g,n}^{\text{um}}|_{\mathscr{G}_{g,n}}$ is injective. Then finally, one just needs to note that the injectivity of $\rho_{g,n}^{\text{um}}|_{\mathscr{G}_{g,n}}$ is equivalent to the congruence subgroup problem for $\operatorname{MCG}_{g,n}$.

Let $K$ be a field of characteristic zero, $X/K$ a hyperbolic $(g,n)$-curve. There is a famous short exact sequence

$$1\longrightarrow\pi_1(X\otimes \overline{K})\longrightarrow\pi_1(X)\longrightarrow \operatorname{Gal}_K\longrightarrow 1.$$

Let $\Pi_{g,n}$ be the profinite completion of the fundamental group $\pi_1(g,n)$ of a genus $g$ compact Riemann surface of with $n$ points punctured. After fixing an isomorphism $\pi_1(X\otimes \overline{K})\overset{\sim}{\to}\Pi_{g,n}$, one gets two exact sequences

$$1\longrightarrow\Pi_{g,n}\longrightarrow\pi_1(X)\longrightarrow \operatorname{Gal}_K\longrightarrow 1$$

$$1\longrightarrow \operatorname{Inn}(\Pi_{g,n }) \longrightarrow \operatorname{Aut}(\Pi_{g,n}) \longrightarrow \operatorname{Out}(\Pi_{g,n}) \longrightarrow 1$$

and the outer Galois representation

$$\rho_X:\operatorname{Gal}_K\longrightarrow \operatorname{Out}(\Pi_{g,n}).$$

Note that the kernel of $\rho_X$ is independent of the choice of the isomorphism. Voevodsky made a conjecture that $\rho_X$ is injective when $X$ is an affine hyperbolic curve and $K$ is a number field. The conjecture was proved by Matsumoto then extended to the proper case by Hoshi–Mochizuki.

In his letter to Faltings, Grothendieck conjectured that any hyperbolic curve (actually for any algebraic curve) over a number field should be anabelian i.e. the geometry of any hyperbolic curve $X$ over a number field is determined by $\rho_X$. Following the earlier work of Nakamura and Tamagawa, this was proved by Mochizuki. Meanwhile, Grothendieck conjectured that the moduli space of hyperbolic curves would be also anabelian. So it’s natural to ask whether Voevodsky’s conjecture holds after replacing $\Pi_{g,n}$ with $\mathscr{M}_{g,n}$, the moduli stack over $K$ of smooth geometrically connected proper curves of genus $g$ with $n$ marked points.

Just like $\Pi_{g,n}$, one has the profinite completion $\mathscr G_{g,n}$ of the oriented mapping class group $\operatorname{MCG}_{g,n} $ of a genus $g$ topological surface with $n$ marked points, which is isomorphic to $\pi_1(\mathscr M_{g,n}\otimes \overline{K})$. Then one gets a Galois representation

$$\rho_{g,n}:\operatorname{Gal}_K\longrightarrow \operatorname{Out}(\mathscr G_{g,n})$$

again. By the work of Matsumoto–Tamagawa, one can study $\rho_{g,n}$ with the associated universal monodromy representation

$$\rho_{g,n}^{\text{um}}:\pi_1(\mathscr{M}_{g,n})\longrightarrow \operatorname{Out}(\Pi_{g,n})$$

given by the exact sequence

$$1\longrightarrow\Pi_{g,n}\longrightarrow \pi_1(\mathscr{M}_{g,n+1})\longrightarrow \pi_1(\mathscr{M}_{g,n})\longrightarrow 1.$$

Using their combinatorial anabelian geometry, Hoshi–Mochizuki showed that $\rho_{g,n}^{\text{um}}$ is injective if and only if $\rho_{g,n}^{\text{um}}|_{\mathscr{G}_{g,n}}$ is injective. Then finally, one just needs to note that the injectivity of $\rho_{g,n}^{\text{um}}|_{\mathscr{G}_{g,n}}$ is equivalent to the congruence subgroup problem for $\operatorname{MCG}_{g,n}$.

Let $K$ be a field of characteristic zero, $X/K$ a hyperbolic $(g,n)$-curve. There is a famous short exact sequence

$$1\longrightarrow\pi_1(X\otimes \overline{K})\longrightarrow\pi_1(X)\longrightarrow \operatorname{Gal}_K\longrightarrow 1.$$

Let $\Pi_{g,n}$ be the profinite completion of the fundamental group $\pi_1(g,n)$ of a genus $g$ compact Riemann surface of with $n$ points punctured. After fixing an isomorphism $\pi_1(X\otimes \overline{K})\overset{\sim}{\to}\Pi_{g,n}$, one gets two exact sequences

$$1\longrightarrow\Pi_{g,n}\longrightarrow\pi_1(X)\longrightarrow \operatorname{Gal}_K\longrightarrow 1$$

$$1\longrightarrow \operatorname{Inn}(\Pi_{g,n }) \longrightarrow \operatorname{Aut}(\Pi_{g,n}) \longrightarrow \operatorname{Out}(\Pi_{g,n}) \longrightarrow 1$$

and the outer Galois representation

$$\rho_X:\operatorname{Gal}_K\longrightarrow \operatorname{Out}(\Pi_{g,n}).$$

Note that the kernel of $\rho_X$ is independent of the choice of the isomorphism. Voevodsky made a conjecture that $\rho_X$ is injective when $X$ is an affine hyperbolic curve and $K$ is a number field. The conjecture was proved by Matsumoto then extended to the proper case by Hoshi–Mochizuki.

In his letter to Faltings, Grothendieck conjectured that any hyperbolic curve (actually for any algebraic curve) over a number field should be anabelian i.e. the geometry of any hyperbolic curve $X$ over a number field is determined by $\rho_X$. Following the earlier work of Nakamura and Tamagawa, this was proved by Mochizuki. Meanwhile, Grothendieck conjectured that the moduli space of hyperbolic curves would be also anabelian. So it’s natural to ask whether Voevodsky’s conjecture holds after replacing $\Pi_{g,n}$ with $\mathscr{M}_{g,n}$, the moduli stack over $K$ of smooth geometrically connected proper curves of genus $g$ with $n$ marked points.

Just like $\Pi_{g,n}$, one has the profinite completion $\mathscr G_{g,n}$ of the oriented mapping class group $\operatorname{MCG}_{g,n} $ of a genus $g$ topological surface with $n$ marked points, which is isomorphic to $\pi_1(\mathscr M_{g,n}\otimes \overline{K})$. Then one gets a Galois representation

$$\rho_{g,n}:\operatorname{Gal}_K\longrightarrow \operatorname{Out}(\mathscr G_{g,n})$$

again. By the work of Matsumoto–Tamagawa, one can study $\rho_{g,n}$ with the associated universal monodromy representation

$$\rho_{g,n}^{\text{um}}:\pi_1(\mathscr{M}_{g,n})\longrightarrow \operatorname{Out}(\Pi_{g,n})$$

given by the exact sequence

$$1\longrightarrow\Pi_{g,n}\longrightarrow \pi_1(\mathscr{M}_{g,n+1})\longrightarrow \pi_1(\mathscr{M}_{g,n})\longrightarrow 1.$$

Using their combinatorial anabelian geometry, Hoshi–Mochizuki showed that $\rho_{g,n}^{\text{um}}$ is injective if and only if $\rho_{g,n}^{\text{um}}|_{\mathscr{G}_{g,n}}$ is injective. Then finally, one just needs to note that the injectivity of $\rho_{g,n}^{\text{um}}|_{\mathscr{G}_{g,n}}$ is equivalent to the congruence subgroup problem for $\operatorname{MCG}_{g,n}$.

Let $K$ be a field of characteristic zero, $X/K$ a hyperbolic $(g,n)$-curve. There is a famous short exact sequence

$$1\longrightarrow\pi_1(X\otimes \overline{K})\longrightarrow\pi_1(X)\longrightarrow \operatorname{Gal}_K\longrightarrow 1.$$

Let $\Pi_{g,n}$ be the profinite completion of the fundamental group $\pi_1(g,n)$ of a genus $g$ compact Riemann surface of with $n$ points punctured. After fixing an isomorphism $\pi_1(X\otimes \overline{K})\overset{\sim}{\to}\Pi_{g,n}$, one gets two exact sequences

$$1\longrightarrow\Pi_{g,n}\longrightarrow\pi_1(X)\longrightarrow \operatorname{Gal}_K\longrightarrow 1$$

$$1\longrightarrow \operatorname{Inn}(\Pi_{g,n }) \longrightarrow \operatorname{Aut}(\Pi_{g,n}) \longrightarrow \operatorname{Out}(\Pi_{g,n}) \longrightarrow 1$$

and the outer Galois representation

$$\rho_X:\operatorname{Gal}_K\longrightarrow \operatorname{Out}(\Pi_{g,n}).$$

Note that the kernel of $\rho_X$ is independent of the choice of the isomorphism. Voevodsky made a conjecture that $\rho_X$ is injective when $X$ is an affine hyperbolic curve and $K$ is a number field. The conjecture was proved by Matsumoto then extended to the proper case by Hoshi–Mochizuki.

In his letter to Faltings, Grothendieck conjectured that any hyperbolic curve (actually for any algebraic curve) over a number field should be anabelian i.e. the geometry of any hyperbolic curve $X$ over a number field is determined by $\rho_X$. Following the earlier work of Nakamura and Tamagawa, this was proved by Mochizuki. Meanwhile, Grothendieck conjectured that the moduli space of hyperbolic curves would be also anabelian. So it’s natural to ask whether Voevodsky’s conjecture holds after replacing $\Pi_{g,n}$ with $\mathscr{M}_{g,n}$, the moduli stack over $K$ of smooth geometrically connected proper curves of genus $g$ with $n$ marked points.

Just like $\Pi_{g,n}$, one has the profinite completion $\mathscr G_{g,n}$ of the oriented mapping class group $\operatorname{MCG}_{g,n} $ of a genus $g$ topological surface with $n$ marked points, which is isomorphic to $\pi_1(\mathscr M_{g,n}\otimes \overline{K})$. Then one gets a Galois representation again

$$\rho_{g,n}:\operatorname{Gal}_K\longrightarrow \operatorname{Out}(\mathscr G_{g,n}).$$

By the work of Matsumoto–Tamagawa, one can study $\rho_{g,n}$ with the associated universal monodromy representation

$$\rho_{g,n}^{\text{um}}:\pi_1(\mathscr{M}_{g,n})\longrightarrow \operatorname{Out}(\Pi_{g,n})$$

given by the exact sequence

$$1\longrightarrow\Pi_{g,n}\longrightarrow \pi_1(\mathscr{M}_{g,n+1})\longrightarrow \pi_1(\mathscr{M}_{g,n})\longrightarrow 1.$$

Using their combinatorial anabelian geometry, Hoshi–Mochizuki showed that $\rho_{g,n}^{\text{um}}$ is injective if and only if $\rho_{g,n}^{\text{um}}|_{\mathscr{G}_{g,n}}$ is injective. Then finally, one just needs to note that the injectivity of $\rho_{g,n}^{\text{um}}|_{\mathscr{G}_{g,n}}$ is equivalent to the congruence subgroup problem for $\operatorname{MCG}_{g,n}$.

Let $K$ be a field of characteristic zero, $X/K$ a hyperbolic $(g,n)$-curve. There is a famous short exact sequence

$$1\longrightarrow\pi_1(X\otimes \overline{K})\longrightarrow\pi_1(X)\longrightarrow \operatorname{Gal}_K\longrightarrow 1.$$

Let $\Pi_{g,n}$ be the profinite completion of the fundamental group $\pi_1(g,n)$ of a genus $g$ compact Riemann surface of with $n$ points punctured. After fixing an isomorphism $\pi_1(X\otimes \overline{K})\overset{\sim}{\to}\Pi_{g,n}$, one gets two exact sequences

$$1\longrightarrow\Pi_{g,n}\longrightarrow\pi_1(X)\longrightarrow \operatorname{Gal}_K\longrightarrow 1$$

$$1\longrightarrow \operatorname{Inn}(\Pi_{g,n }) \longrightarrow \operatorname{Aut}(\Pi_{g,n}) \longrightarrow \operatorname{Out}(\Pi_{g,n}) \longrightarrow 1$$

and the outer Galois representation

$$\rho_X:\operatorname{Gal}_K\longrightarrow \operatorname{Out}(\Pi_{g,n}).$$

Note that the kernel of $\rho_X$ is independent of the choice of the isomorphism. Voevodsky made a conjecture that $\rho_X$ is injective when $X$ is an affine hyperbolic curve and $K$ is a number field. The conjecture was proved by Matsumoto then extended to the proper case by Hoshi–Mochizuki.

In his letter to Faltings, Grothendieck conjectured that any hyperbolic curve (actually for any algebraic curve) over a number field should be anabelian i.e. the geometry of any hyperbolic curve $X$ over a number field is determined by $\rho_X$. Following the earlier work of Nakamura and Tamagawa, this was proved by Mochizuki. Meanwhile, Grothendieck conjectured that the moduli space of hyperbolic curves would be also anabelian. So it’s natural to ask whether Voevodsky’s conjecture holds after replacing $\Pi_{g,n}$ with $\mathscr{M}_{g,n}$, the moduli stack over $K$ of smooth geometrically connected proper curves of genus $g$ with $n$ marked points.

Just like $\Pi_{g,n}$, one has the profinite completion $\mathscr G_{g,n}$ of the oriented mapping class group $\operatorname{MCG}_{g,n} $ of a genus $g$ topological surface with $n$ marked points, which is isomorphic to $\pi_1(\mathscr M_{g,n}\otimes \overline{K})$. Then one gets a Galois representation

$$\rho_{g,n}:\operatorname{Gal}_K\longrightarrow \operatorname{Out}(\mathscr G_{g,n})$$

again. By the work of Matsumoto–Tamagawa, one can study $\rho_{g,n}$ with the associated universal monodromy representation

$$\rho_{g,n}^{\text{um}}:\pi_1(\mathscr{M}_{g,n})\longrightarrow \operatorname{Out}(\Pi_{g,n})$$

given by the exact sequence

$$1\longrightarrow\Pi_{g,n}\longrightarrow \pi_1(\mathscr{M}_{g,n+1})\longrightarrow \pi_1(\mathscr{M}_{g,n})\longrightarrow 1.$$

Using their combinatorial anabelian geometry, Hoshi–Mochizuki showed that $\rho_{g,n}^{\text{um}}$ is injective if and only if $\rho_{g,n}^{\text{um}}|_{\mathscr{G}_{g,n}}$ is injective. Then finally, one just needs to note that the injectivity of $\rho_{g,n}^{\text{um}}|_{\mathscr{G}_{g,n}}$ is equivalent to the congruence subgroup problem for $\operatorname{MCG}_{g,n}$.