Main theorem
\(A\): abelian surface
\(X\): K3 surface
- The Mumford–Tate conjecture is true for \(\mathrm{H}^{2}(A \times X)\)
\(k \subset \mathbb{C}\) finitely generated field
\(X\) smooth projective \(k\)-variety
Several cohomology theories of \(X\)
\(\mathrm{H}_{\textrm{B}} := \mathrm{H}^{i}_{\textrm{sing}}(X(\mathbb{C}), \mathbb{Q})\)
Singular cohomology
Hodge structure: \(\mathbb{C}^{*} \to \mathrm{GL}(\mathrm{H}_{\mathrm{B}}) \otimes \mathbb{R}\)
Mumford–Tate group: \[ G_{\textrm{B}}(\mathrm{H}_{\mathrm{B}}) \subset \mathrm{GL}(\mathrm{H}_{\mathrm{B}}) \]
\(\mathrm{H}_{\ell} := \mathrm{H}^{i}_{\textrm{ét}}(X_{\bar{k}},\mathbb{Q}_{\ell})\)
\(\ell\)-adic étale cohomology
Galois representation: \(\rho \colon \Gamma_{k} \to \mathrm{GL}(\mathrm{H}_{\ell})\)
Image of Galois: \[ G_{\ell}(\mathrm{H}_{\ell}) := \overline{\mathrm{Im}(\rho)} \subset \mathrm{GL}(\mathrm{H}_{\ell})^{\circ} \]
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: \[ G_{\mathrm{B}}(\mathrm{H}_{\mathrm{B}}) \otimes \mathbb{Q}_{\ell} \cong G_{\ell}(\mathrm{H}_{\ell}) \]
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.
\(A\): abelian surface
\(X\): K3 surface
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 \(G_{\ell}\)
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)
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 \(G_{\ell}(V)\)
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
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:
Similarly \(5 \cdot 45\) cases
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
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!