On the Hodge and Tate conjectures for moduli spaces of curves

Technical Summary: On the Hodge and Tate Conjectures for Moduli Spaces of Curves

Problem Statement
The Hodge and Tate conjectures are central, yet largely open, problems relating the cohomology of algebraic varieties to algebraic cycles. While progress has historically relied on varieties moving in nontrivial families (e.g., abelian varieties, K3 surfaces) or possessing special endomorphisms, the moduli spaces of stable curves, $\mathcal{M}_{g,n}$, present a distinct testing ground. These spaces do not deform in families but form an infinite discrete collection with rich, complicated Hodge structures and Galois representations. The paper addresses the verification of the Hodge and Tate conjectures, particularly in their generalized coniveau forms, for the cohomology of $\mathcal{M}_{g,n}$. The generalized forms ask whether specific substructures in cohomology (sub-Hodge structures or Galois subrepresentations) are supported on algebraic subsets of a predicted codimension, rather than merely asking if even-degree classes arise from cycles.

Methodology
The paper synthesizes recent cohomological computations with the inductive framework established by Arbarello and Cornalba. The methodology relies on three primary pillars:

1. Inductive Boundary Stratification: The proof strategy utilizes the stratification of $\mathcal{M}_{g,n}$ by topological types of stable curves. The boundary divisor $\bar{D}_{g,n}$ is a disjoint union of products of smaller moduli spaces. Using the localization sequence and weight arguments, cohomology classes on $\mathcal{M}_{g,n}$ are related to classes on smaller moduli spaces and the pure weight part of the interior cohomology.
2. Tautological Generation: A core technique is determining whether cohomology groups are generated by tautological classes (generated by $\kappa$-, $\psi$-, and boundary classes). If $H^{2k}(\mathcal{M}_{g,n})$ is tautological, the Hodge and Tate conjectures hold for codimension $k$ cycles. The paper employs "semi-tautological extensions" to control high-degree cohomology, reducing questions to base cases.
3. Arithmetic Inspiration and Classification: The results are guided by the classification of automorphic representations of motivic weight $\le 22$ by Chenevier and Lannes. This classification predicts the specific motivic structures (e.g., $S_{12}$, $S_{16}$) appearing in the cohomology of varieties smooth and proper over $\mathbb{Z}$. The paper uses these predictions to identify the expected support loci for non-tautological classes, particularly in odd degrees.
4. Base Case Verification: The inductive arguments require explicit verification of base cases for small $g$ and $n$. These are established using:
   • Point counting over finite fields and the Behrend–Grothendieck–Lefschetz trace formula.
   • The Chow–Künneth generation property.
   • Explicit geometric constructions (e.g., Mukai's description of canonical curves).
   • Topological vanishing theorems (e.g., Harer's bounds on virtual cohomological dimension).

Key Contributions and Results

• Verification of Conjectures in Specific Ranges: The paper proves that the generalized Hodge and Tate conjectures hold for $\mathcal{M}_{g,n}$ in a surprisingly wide range of degrees:

  • Codimension 1 and 2: Holds for all $g, n$.
  • Codimension 3: Holds for all $n$ and $g \notin \{7, 8, 9\}$.
  • Dimension $\le 7$: Holds for all $g, n$.
  • Generalized Hodge Conjecture: Holds for $H^k(\mathcal{M}_{g,n})$ for even $k \le 4$ (all $g,n$), $k=6$ ($g \notin \{7,8,9\}$), odd $k \le 13$ (all $g,n$), and $k = 2d-j$ for $0 \le j \le 15$.
  • Generalized Tate Conjecture: Holds over finitely generated fields $L$ in the same ranges, subject to ordinarity conditions on the characteristic $p$ of $L$ (specifically, $p$ must not divide the Ramanujan $\tau$-function for $S_{12}$ or the Fourier coefficients of the weight 16 cusp form for $S_{16}$).

• Structure of Cohomology Groups:

  • Even Degrees: For low even degrees ($k \le 6$, with exceptions for $g \in \{7,8,9\}$ at $k=6$), cohomology is generated by tautological classes.
  • Odd Degrees: $H^k(\mathcal{M}_{g,n}) = 0$ for odd $k < 11$. For $k=11$, the cohomology is a direct sum of Tate twists of $S_{12} := W_{11}H^{11}(\mathcal{M}_{1,11})$, which is related to the Ramanujan cusp form $\Delta$. For $k=13$, the cohomology is supported on divisors (boundary strata) and consists of Tate twists of $S_{12}$.
  • High Degrees: By Poincaré duality and Hard Lefschetz, results for low degrees imply results for high degrees ($2d-k$).

• Motivic Structures: The paper confirms that the motivic structures appearing in $H^*(\mathcal{M}_{g,n})$ up to weight 15 align with the Chenevier–Lannes classification. Specifically, odd cohomology vanishes below degree 11, $H^{11}$ arises from $S_{12}$, and $H^{13}$ arises from boundary-supported Tate twists of $S_{12}$.

• Conjectures on Chow Groups: The paper proposes the Chow 10 Conjecture, asserting that the rational Chow group $A^k(\mathcal{M}_{g,n})_\mathbb{Q}$ is generated by tautological cycles for $k \le 10$. This implies the cycle class map is an isomorphism for $k \le 10$.

Significance and Claims
The paper claims that the inductive structure of the boundary stratification, combined with modern arithmetic predictions and explicit low-genus calculations, verifies the generalized Hodge and Tate conjectures in a range previously inaccessible.

• Novelty of Approach: Unlike traditional approaches relying on variation in families, this work demonstrates that the "tautological gluing and forgetful morphisms" and boundary stratification of $\mathcal{M}_{g,n}$ provide sufficient structure to verify these deep conjectures.
• Bridging Arithmetic and Geometry: The paper highlights how arithmetic predictions (Chenevier–Lannes) regarding motives over $\mathbb{Z}$ serve as a guide for geometric proofs. The geometric results confirm these predictions unconditionally for moduli spaces of curves.
• Open Problems: The paper identifies specific remaining challenges, including:
  • Proving tautological generation for $H^6(\mathcal{M}_{g,n})$ when $g \in \{7, 8, 9\}$.
  • Verifying the generalized Hodge conjecture for $H^{15}(\mathcal{M}_{g,n})$.
  • Investigating the support of cohomology at non-ordinary primes (where $p | \tau(p)$), where the predicted coniveau jumps.
  • Constructing algebraic cycles for isomorphic motivic structures appearing in different moduli spaces (e.g., $S_{18}$ in $\mathcal{M}_{1,17}$ and $\mathcal{M}_{2,14}$) without a known geometric explanation.

The work establishes $\mathcal{M}_{g,n}$ as a robust testing ground for motivic conjectures, providing a "surprisingly wide range" of verified cases while delineating the precise boundaries where new phenomena (such as non-tautological classes and non-ordinary primes) emerge.