# Input for the conjecture

• $$k \subset \mathbb{C}$$ finitely generated field

• $$X$$ smooth projective $$k$$-variety

• Several cohomology theories of $$X$$

# $$\mathrm{H}^{i}_{\textrm{B}}(X)$$ — Hodge theory

• $$\mathrm{H}_{\textrm{B}} := \mathrm{H}^{i}_{\textrm{sing}}(X(\mathbb{C}), \mathbb{Q})$$

• Singular cohomology

• Hodge structure: $$\mathbb{C}^{*} \to \mathrm{GL}(\mathrm{H}_{\mathrm{B}}) \otimes \mathbb{R}$$

• Mumford–Tate group: $G_{\textrm{B}}(\mathrm{H}_{\mathrm{B}}) \subset \mathrm{GL}(\mathrm{H}_{\mathrm{B}})$

# $$\mathrm{H}^{i}_{\ell}(X)$$ — Étale cohomology

• $$\mathrm{H}_{\ell} := \mathrm{H}^{i}_{\textrm{ét}}(X_{\bar{k}},\mathbb{Q}_{\ell})$$

• $$\ell$$-adic étale cohomology

• Galois representation: $$\rho \colon \Gamma_{k} \to \mathrm{GL}(\mathrm{H}_{\ell})$$

• Image of Galois: $G_{\ell}(\mathrm{H}_{\ell}) := \overline{\mathrm{Im}(\rho)} \subset \mathrm{GL}(\mathrm{H}_{\ell})^{\circ}$

# Mumford–Tate conjecture

• Comparison theorem (Artin): $\mathrm{H}_{\textrm{B}} \otimes \mathbb{Q}_{\ell} \cong \mathrm{H}_{\ell}$

• Consequently: $\mathrm{GL}(\mathrm{H}_{\textrm{B}}) \otimes \mathbb{Q}_{\ell} \cong \mathrm{GL}(\mathrm{H}_{\ell})$

• Conjecture: $G_{\mathrm{B}}(\mathrm{H}_{\mathrm{B}}) \otimes \mathbb{Q}_{\ell} \cong G_{\ell}(\mathrm{H}_{\ell})$

# Known cases of the Mumford–Tate conjecture

• Abelian varieties of dimension $$\le 3$$

• K3 surfaces

• Some other surfaces with $$p_{g} = 1$$

• A few other “special” cases

• The conjecture is not additive.

# Main theorem

• $$A$$: abelian surface

• $$X$$: K3 surface

• The Mumford–Tate conjecture is true for $$\mathrm{H}^{2}(A \times X)$$

• Künneth theorem gives $\mathrm{H}^{2}(A \times X) \cong \mathrm{H}^{2}(A) \oplus \mathrm{H}^{2}(X)$

• Write $$H = \mathrm{H}^{2}(A)$$; $$V = \mathrm{H}^{2}(X)$$; $$M = H \oplus V$$

• MTC is known for both summands $$H$$ and $$V$$

• $G_{\mathrm{B}}(M) \subset G_{\mathrm{B}}(H) \times G_{\mathrm{B}}(V)$ with surjective projections onto both factors

• Similar for $$G_{\ell}$$

# More remarks

• MTC is known for the centres of the Mumford–Tate group and the image of Galois: $Z_{B}(M) \otimes \mathbb{Q}_{\ell} \cong Z_{\ell}(M)$

• $$G_{\mathrm{B}}(M)$$ and $$G_{\ell}(M)$$ are reductive

• Conclusion: focus on the semisimple parts (or even the Lie algebras)

# Hodge theory of K3 surfaces

• Zarhin: there is a field $$E$$; CM or TR (totally real) $G_{\mathrm{B}}(V)^{\text{der}} = \begin{cases} \mathrm{SO}_{E}(n) & \text{if E is TR} \\ \mathrm{SU}_{E}(n) & \text{if E is CM} \\ \end{cases}$

• Condition: $$[E : \mathbb{Q}] \cdot n \le 21$$

• Also applies to $$H = \mathrm{H}^{2}(A)$$ with condition $$[E : \mathbb{Q}] \cdot n \le 5$$

• Since MTC is known for $$A$$ and $$X$$, we also have a description of $$G_{\ell}(H)$$ and $$G_{\ell}(V)$$

# Proof strategy

• Finite list of possible groups: 5 for $$H$$, 66 for $$V$$

• Deligne: $$G_{\ell}(M)^{\text{der}} \subset G_{\mathrm{B}}(M)^{\text{der}} \otimes \mathbb{Q}_{\ell}$$

• Try to prove that $G_{\ell}(M)^{\text{der}} = G_{\ell}(H)^{\text{der}} \times G_{\ell}(V)^{\text{der}}$

• This would imply MTC

# Proof: part 1

• Suppose that $$G_{\mathrm{B}}(H)^{\text{der}} = \mathrm{SO}_{\mathbb{Q}}(3)$$ and $$G_{\mathrm{B}}(V)^{\text{der}} = \mathrm{SO}_{E}(5)$$, for some field $$E$$

• Classification of semisimple Lie algebras:

• The only subgroup of $$G_{\ell}(H)^{\text{der}} \times G_{\ell}(V)^{\text{der}}$$ with surjective projections is the product itself
• Similarly $$5 \cdot 45$$ cases

# Proof: part 2

• Suppose that $$G_{\mathrm{B}}(H)^{\text{der}} = \mathrm{SO}_{\mathbb{Q}}(3)$$ and $$G_{\mathrm{B}}(V)^{\text{der}} = \mathrm{SO}_{E}(3)$$, for some field $$E \ne \mathbb{Q}$$

• $G_{\ell}(V)^{\text{der}} = \prod_{\lambda|\ell}\mathrm{Res}_{E_{\lambda}/\mathbb{Q}_{\ell}} \mathrm{SO}_{E_{\lambda}}(3)$

• Chebotaryov: there is a prime $$\ell$$, such that $$E_{\lambda} \ne \mathbb{Q}_{\ell}$$

• Hence $$G_{\ell}(M)^{\text{der}} = G_{\ell}(H)^{\text{der}} \times G_{\ell}(V)^{\text{der}}$$ for this particular $$\ell$$

• Larsen–Pink: If MTC is true for one $$\ell$$, then true for all $$\ell$$

• This technique proves another $$5 \cdot 13$$ cases

# Proof: part 3

• There are $$5 \cdot 8$$ remaining cases

• On the side of the K3 surface $$X$$, 5 of the 8 cases have $$[E : \mathbb{Q}] \cdot n \le 5$$, just like the abelian surface

• Reduce these to MTC for low-dimensional abelian varieties, using Kuga–Satake varieties, and apply techniques of D. Lombardo

• Remaining $$5 \cdot 3$$ cases are very involved; use similar techniques + more algebra + geometrical input (even from char $$p$$)

• We win!