Motivic Modularity of CM K3 Surfaces

Technical Summary: Motivic Modularity of CM K3 Surfaces

Problem Statement
The paper addresses the explicit description of algebraic Hecke characters associated with K3 surfaces defined over a number field $k \subset \mathbb{C}$ that possess complex multiplication (CM). While the Shimura-Taniyama theorem establishes the existence of algebraic Hecke characters $\chi_1, \dots, \chi_{2n}$ such that the Hasse-Weil L-function of an abelian variety $A$ decomposes as $L(s, A) = \prod L(s, \chi_i)$, the explicit determination of these characters for K3 surfaces has been limited to specific cases. Previous results covered K3 surfaces with non-symplectic group actions, Kummer surfaces associated with self-products or products of CM elliptic curves, and certain K3 surfaces with Picard rank 16. A significant gap remained in providing a general formula for the algebraic Hecke characters of CM K3 surfaces, particularly those with Picard ranks 8 and 14, which were not covered by the aforementioned specific constructions.

Methodology
The author employs the theory of complex multiplication for K3 surfaces, specifically relying on the main theorem of complex multiplication for K3 surfaces established by Rizov. The methodology proceeds through the following steps:

1. Galois Representations and Transcendental Lattices: The paper considers the $l$-adic Galois representation $\rho_l$ acting on the transcendental part of the second cohomology, $T_l(X_k) \otimes \mathbb{Q}_l$, where $X$ is a K3 surface. The dimension of this representation is $22 - \rho(X_k)$, where $\rho(X_k)$ is the Picard rank.
2. Mumford-Tate Groups and CM Fields: Utilizing the definition of CM K3 surfaces, the author identifies the reflex field $E$ (a CM field) and the associated Mumford-Tate group $G$. The group $G$ is shown to be isomorphic to the kernel of the norm map $Nm: \text{Res}_{E/\mathbb{Q}} \mathbb{G}_{m,E} \to \text{Res}_{F/\mathbb{Q}} \mathbb{G}_{m,F}$, where $F$ is the maximal totally real subfield of $E$.
3. Construction of Hecke Characters: By applying Rizov's main theorem, the action of the Galois group on the transcendental lattice is shown to factor through a continuous homomorphism into $G(\mathbb{A}_f)$. The author constructs a unique locally constant homomorphism $u: \mathbb{A}_k^\times \to E^\times$ that satisfies a specific functional equation involving the norm map $N_{k/E}$ and complex conjugation.
4. Derivation of Characters: The algebraic Hecke characters $\chi_\tau$ are explicitly defined by projecting the homomorphism $u(t) \cdot N_{k/E}(t) / \overline{N_{k/E}(t)}$ via embeddings $\tau: E \to \mathbb{C}$.
5. Verification of Properties: The paper verifies that these constructed characters are indeed algebraic Hecke characters and that they diagonalize the $l$-adic Galois representation. The unramifiedness of the representation at a prime $p$ is shown to be equivalent to the unramifiedness of the Hecke character at $p$.

Key Contributions and Results
The primary result is Theorem 1.3, which provides an explicit formula for the algebraic Hecke characters of any K3 surface $X$ defined over a number field $k$ with CM by a CM field $E$. The theorem establishes:

• The existence of a unique locally constant homomorphism $u: \mathbb{A}_k^\times \to E^\times$ satisfying specific norm and conjugation properties.
• The construction of algebraic Hecke characters $\chi_\tau$ for each embedding $\tau \in \text{Hom}(E, \mathbb{C})$.
• The decomposition of the $l$-adic Galois representation $\rho_l$ on the transcendental lattice as a direct sum of the representations associated with these characters: $\rho_l \cong \bigoplus_{\tau} \rho_{l, \chi_\tau}$.
• The equality of L-functions: $L(s, T_l(X_k) \otimes \mathbb{Q}_l) = \prod_{\tau} L(s, \chi_\tau)$, termed the "Motivic modularity" of $X$.

Furthermore, the paper applies this general theorem to a new class of examples in Section 4: Kummer surfaces associated with simple CM abelian surfaces.

• Proposition 4.7 demonstrates that the Hecke characters derived from the general construction in Theorem 1.3 for such Kummer surfaces coincide exactly with those obtained by applying the main theorem of complex multiplication directly to the underlying simple abelian surface. This result is noted as independent of previous work by Shioda-Inose and Rizov, which focused on products of elliptic curves.

Significance and Claims
The paper claims to provide a unified and explicit formula for the algebraic Hecke characters of CM K3 surfaces over number fields, extending the scope of known results to include surfaces with Picard ranks 8 and 14. The author emphasizes that this construction recovers and generalizes previous specific cases (such as those involving non-symplectic actions or products of elliptic curves).

The significance of the work lies in:

1. Explicit Determination: Moving beyond the existence of such characters to their explicit construction via the reflex field and norm maps.
2. Generality: Covering cases previously excluded from explicit formulas, specifically those with Picard ranks 8 and 14, which are known to exist for CM fields of degree up to 20.
3. Consistency: Verifying that the general construction aligns with classical CM theory for abelian varieties when applied to Kummer surfaces associated with simple abelian surfaces.

The author modestly notes that while the paper does not prove that specific K3 surfaces mentioned in prior literature (e.g., those in Costa et al.) are of CM type, it demonstrates that if they are of CM type, their transcendental Galois representations are governed by the algebraic Hecke characters constructed in this paper, thereby recovering their results under that assumption.