\section{Introduction}\label{sec:01-introduction}
The Hodge conjecture and Tate conjecture are two central conjectures connecting algebraic topology of algebraic varieties, properties of algebraic cycles and categories of motives. These conjectures are still wide open in general, and only known in some special cases. 

Let $X=\Mbar_{g,n}$ be the smooth proper moduli stack of stable curves
of genus $g$ with $n$ ordered marked points, where $g,n\ge0$ and
$2g-2+n>0$. For $r\ge0$, let $N^rH^{k}(X)$ be the geometric coniveau
filtration (classes supported on closed algebraic subsets of
codimension at least $r$) and $N_H^rH^{k}(X)$ the Hodge coniveau
filtration (the largest rational sub-Hodge structure of Hodge types
$(p,q)$ with $p,q\ge r$); the precise definitions are recalled at the
beginning of Section~\ref{sec:17-geometric-coniveau}. The generalized
Hodge conjecture asserts that these two filtrations agree.

Payne~\cite{payne2026} surveys recent progress on these conjectures on $\Mbar_{g,n}$. In particular, it is known that the generalized Hodge conjecture holds for $H^{k}(\Mbar_{g,n})$ for all odd $k\leq 13$. In this paper, we prove that the generalized Hodge conjecture holds for $H^{15}(\Mbar_{g,n})$, answering Payne's question (Problem 6.2 of that survey).

\begin{theorem}
\label{thm:introduction-ghc}
The generalized Hodge conjecture holds for $H^{15}(\Mbar_{g,n})$. 
\end{theorem}

Theorem~\ref{thm:introduction-ghc} is proved as
Corollary~\ref{cor:degree-fifteen-ghc} in
Section~\ref{sec:17-geometric-coniveau}.

Put
\[
 \Lmot=H^2(\mathbb P^1),\qquad
 \Soneone=H^{11}(\Mbar_{1,11}),\qquad
 \Pmot=\Lmot^2\otimes\Soneone,
\]
and let $\Sonesix$ be the rank-two Hodge structure attached to the
normalized level-one cusp form of weight sixteen. The Hodge types of
$\Pmot$ are $(13,2)$ and $(2,13)$, whereas those of $\Sonesix$ are
$(15,0)$ and $(0,15)$. Previous work identifies these as the only
possible simple constituents of $H^{15}(\Mbar_{g,n})$, with $\Sonesix$ occurring only in genus
one~\cite{canning2024a,canning2024b}; we recall the precise statements
in Theorem~\ref{thm:motivic-types}. We use the same symbols for their
compatible $\ell$-adic realizations, and refer to these compatible
realizations as motives.

Write $H^{15}(\Mbar_{g,n})_{\Pmot}$ for the canonical $\Pmot$-isotypic
sub-Hodge structure, namely the sum of all sub-Hodge structures
isomorphic to $\Pmot$.

\begin{theorem}[The geometric coniveau filtration]
\label{thm:introduction-coniveau}
We have that
\[
 N^1H^{15}(\Mbar_{g,n})=N^2H^{15}(\Mbar_{g,n})=H^{15}(\Mbar_{g,n})_{\Pmot},
 \qquad N^rH^{15}(\Mbar_{g,n})=0\quad(r\ge3).
\]
The subspace $H^{15}(\Mbar_{g,n})_{\Pmot}$ is supported on a single closed
algebraic subset of the boundary of codimension at least two in $\Mbar_{g,n}$.
For $g\ge2$, this subspace is all of $H^{15}(\Mbar_{g,n})$.
If $g\ge2$ and $H^{15}(\Mbar_{g,n})\ne0$, its geometric coniveau is exactly two.
\end{theorem}

Theorem~\ref{thm:introduction-coniveau} is proved as
Theorem~\ref{thm:degree-fifteen-coniveau} in
Section~\ref{sec:17-geometric-coniveau}.
The geometric input is the identification of $H^{15}(\Mbar_{g,n})_{\Pmot}$ with the sum of the boundary Gysin images.
The Hodge types of $\Pmot$ and $\Sonesix$ give
$N_H^1H^{15}(X)=N_H^2H^{15}(X)=H^{15}(X)_{\Pmot}$ and
$N_H^rH^{15}(X)=0$ for $r\ge3$; comparing with
Theorem~\ref{thm:introduction-coniveau} proves
Theorem~\ref{thm:introduction-ghc}.

We also determine the $\Sn_n$-equivariant multiplicity spaces of
these Hodge constituents. The action of $\Sn_n$ is induced by
relabelling the marked points. For a partition $\lambda\vdash n$, let $V_\lambda$
denote the rational Specht module. Set $\Khook_n=V_{(n-14,1^{14})}$ for
$n\ge15$ and $\Khook_n=0$ for $n<15$, and let $\delta_{g,1}$ be one for
$g=1$ and zero otherwise.

\begin{theorem}
\label{thm:introduction-degree-fifteen}
There is a rational $\Sn_n$-module
$\En_{g,n}$, unique up to isomorphism, and an $\Sn_n$-equivariant
isomorphism of pure Hodge structures
\[
 H^{15}(\Mbar_{g,n})\cong
 \En_{g,n}\otimes\Pmot
 \;\oplus\;\delta_{g,1}\Khook_n\otimes\Sonesix.
\]
The multiplicity modules carry Hodge type $(0,0)$. After scalar
extension of these modules to $\mathbb Q_\ell$, the same decomposition
holds for the $\ell$-adic Galois representations before
semisimplification, for every prime $\ell$. The exact nonvanishing
range is
\[
\begin{array}{c|ccccc}
 g&0&1&2&3&g\geq4\\ \hline
 H^{15}(\Mbar_{g,n})\ne0
 &\varnothing&n\geq13&n\geq11&n\geq10&n\geq9.
\end{array}
\]
\end{theorem}

Theorem~\ref{thm:introduction-degree-fifteen} is proved as
Theorem~\ref{thm:main} in Section~\ref{sec:16-main-proof}.
The module $\En_{g,n}$ records the $\Sn_n$-equivariant multiplicities
of $\Pmot$. We determine these modules by computing the series
\[
 F_g=\sum_{\substack{n\ge0\\2g-2+n>0}}\ch_n(\En_{g,n})
 \in\widehat\Lambda=\prod_{n\ge0}\Lambda_n,
\]
where $\Lambda_n$ is the space of rational symmetric functions
homogeneous of degree $n$, and $\ch_n$ is the Frobenius characteristic;
the conventions are given in Notations~\ref{not:symmetric-functions} and~\ref{not:characters}.

One has $F_0=0$. We give formulas for $F_1$ and $F_2$ in
Theorems~\ref{thm:genus-one} and~\ref{thm:genus-two}. In genus three,
Definition~\ref{def:genus-three-character} and
Theorem~\ref{thm:genus-three} give a finite rational intersection
algorithm for every coefficient of $F_3$. For $g\ge4$,
Theorem~\ref{thm:uniform-high-genus} gives a uniform Schur-positive
formula for $F_g$. These calculations establish the decomposition
and nonvanishing range in Theorem~\ref{thm:main}.

\begin{samepage}
Writing $e_{g,n}=\dim\En_{g,n}$, the decomposition also gives
\[
 h^{13,2}=h^{2,13}=e_{g,n},\qquad
 h^{15,0}=h^{0,15}=\delta_{g,1}\binom{n-1}{14},
\]
with all other degree-fifteen Hodge numbers zero and the binomial
coefficient taken to be zero for $n<15$; see
Corollary~\ref{cor:expert-questions}.
\end{samepage}

\subsection{Related work to the determination of $H^{15}(\Mbar_{g,n})$}\label{subsec:introduction-related-work}

The relation between modular forms and boundary geometry has a long
history. Keel's presentation identifies the Chow ring of $\Mbar_{0,n}$
with cohomology, so odd cohomology vanishes in genus
zero~\cite{keel1992}. In positive genus, Pikaart used elliptic cusp-form
classes and boundary strata to construct nonzero odd cohomology in
sufficiently large genus~\cite{pikaart1995}.
Graber--Pandharipande subsequently used odd K\"unneth components on
elliptic boundary products to detect non-tautological algebraic
cycles~\cite{graberpandharipande2003}.

Arbarello--Cornalba introduced boundary restriction as an inductive
method for studying the cohomology of $\Mbar_{g,n}$ and proved vanishing
in degrees one, three, and five~\cite{arbarello1998}.
Bergstr\"om--Faber--Payne established the additional base cases by
point counting and extended the vanishing to degrees seven and
nine~\cite{bergstrom2024}. Canning--Larson--Payne then determined
degree eleven: it is nonzero precisely for $g=1$, $n\geq11$, where
\[
 H^{11}(\Mbar_{1,n})\cong V_{(n-10,1^{10})}\otimes\Soneone
\]
equivariantly~\cite[Theorem 1.1 and Proposition 2.3]{canning2023}.
The genus-one calculations build on Getzler's equivariant Serre
series and the correction of
Consani--Faber~\cite{getzler1998,consani2006}, which also supply the
generating-series input for our genus-one formula. Consani--Faber
also realized level-one cusp-form motives as alternating summands in
the cohomology of $\Mbar_{1,n}$.

In degree thirteen, Canning--Larson--Payne--Willwacher prove that
every class is obtained by gluing an elliptic degree-eleven class to
a second component carrying the unit class in $H^0$.
For $g\geq2$, the direct sum of these Gysin maps is an isomorphism,
giving the complete equivariant
decomposition~\cite[Theorem 1.7]{canning2026}.
Earlier, Canning--Larson--Payne had attached components of distinct
positive genera to an elliptic component to construct classes in
higher odd degrees.
Their nonvanishing theorem already implies
$H^{15}(\Mbar_{g,n})\ne0$ for every $g\geq4$, $n\geq9$:
take the genera of the two attached components to be $1$ and
$g-2$~\cite[Theorem 1.5]{canning2023}.
Thus the positive direction of our high-genus nonvanishing range
is known independently of the multiplicity calculation.

The possible degree-fifteen constituents are also known.
Canning--Larson--Payne prove that, for $g\geq2$,
$H^{15}(\Mbar_{g,n})$ is a direct sum of copies of $\Pmot$ as a
Hodge structure, and has the same form after Galois
semisimplification~\cite[Theorem 1.1]{canning2024a}.
Their framework of semi-tautological extensions supplies general
finite-generation results and a generation theorem for
$H_{15}(\Mbar_{g,n})$~\cite[Theorems 1.3 and 1.7]{canning2024a}.
This generation statement does not give a multiplicity formula or a
presentation of $H^{15}$ by the boundary Gysin maps considered here.
They also isolated the primitive $\Sonesix$ in
$H^{15}(\Mbar_{1,15})$~\cite[Lemma 8.3]{canning2024a}.
In genus one, Canning--Larson--Payne--Willwacher identify the
$\Sonesix$ summand as $\Khook_n\otimes\Sonesix$, with a complement
whose semisimplification consists of copies of
$\Pmot$~\cite[Theorem 1.3]{canning2024b}.
Their study of these cusp structures in the weight-graded compactly
supported cohomology of the open spaces $\Mopen_{g,n}$ addresses a
related problem; the present calculation concerns the pure group
$H^{15}$ of the compactification.
Their subsequent work places holomorphic forms, including the
degree-fifteen contribution, in a uniform framework of functors on
finite marking sets, or $\mathbf{FA}$-modules~\cite{canning2025}.

Explicit multiplicity calculations are available in finite ranges.
Petersen computed the cohomology of symplectic local systems on the
moduli space of principally polarized abelian surfaces in terms of
Tate structures and elliptic and Siegel modular forms, up to
semisimplification~\cite{petersen2015}. These results provide inputs
for our genus-two calculation.
Bergstr\"om--Faber computed equivariant Euler characteristics in genus
three through $n=14$, conditionally for $n\geq9$ on the predicted
Galois constituents~\cite{bergstromfaber2023}.
Canning--Larson--Payne subsequently confirmed the cases $n=9,10,11$
unconditionally~\cite[Theorem 1.9]{canning2024a}.

Payne asks for the generalized Hodge conjecture for
$H^{15}(\Mbar_{g,n})$~\cite[Problem 6.2]{payne2026}.
Theorem~\ref{thm:introduction-coniveau} answers it.
The semisimplicity statement in~\cite[Theorem C]{payne2026}, applied
in degree $2d-15$ with $d=3g-3+n$, also gives Galois semisimplicity
in degree fifteen by Poincar\'e duality.

% \subsection*{Roadmap}
% Section~\ref{sec:03-motivic-types} identifies the simple constituents
% (Theorem~\ref{thm:motivic-types}) and the ordinary one-edge boundary
% source with its Gysin map (Proposition~\ref{prop:h13-input}).
% Section~\ref{sec:04-low-genus} sets up the weight-fifteen boundary row,
% reduces boundary generation to finitely many symplectic coefficients
% (Proposition~\ref{prop:primitive-degree-fifteen-quotient}), and proves
% the genus-one and genus-two formulas.
% Section~\ref{sec:05-genus-three-generation} proves boundary generation in
% genus three, and Section~\ref{sec:06-genus-three-formula} derives the
% genus-three formula from a rational Gram matrix.
% Section~\ref{sec:07-ordinary-image} determines the ordinary Gysin image
% for $g\ge4$ as a direct sum of four families
% (Theorem~\ref{thm:ordinary-image}).
% Section~\ref{sec:08-geometric-bases} proves the base cases in genera four
% and five through Hurwitz spaces, and
% Section~\ref{sec:09-marking-supports} reduces the remaining genus-five
% cases to seven symplectic coefficients, which
% Sections~\ref{sec:10-genus-two-detectors}--\ref{sec:13-eight-row-coefficients}
% show to vanish by genus-two test classes; Section~\ref{sec:11-top-coefficients}
% also proves compact nonseparating injectivity
% (Theorem~\ref{thm:compact-injectivity}).
% Section~\ref{sec:15-ordinary-lifting} proves ordinary lifting
% (Theorem~\ref{thm:ordinary-lifting}), Section~\ref{sec:16-main-proof}
% completes the induction in the genus and assembles
% Theorem~\ref{thm:introduction-degree-fifteen}, and
% Section~\ref{sec:17-geometric-coniveau} determines geometric coniveau
% and proves the generalized Hodge conjecture in degree fifteen.

We have become aware of recent work by Bradford, Iribar L\'opez,
Kretschmer, and Larson~\cite{BIKL26}, whose results overlap in part
with those of this paper. To clarify the chronology, Pharos completed
the mathematical body of this manuscript on 23 September 2026, before
their preprint appeared on arXiv on 9 October 2026. The intervening
period was devoted to the author's study and checking of the arguments.

More precisely, they study the semi-tautological extension generated by
$H^{11}(\Mbar_{1,11})$ and give explicit generators, complete relations,
and bases for its degree-fifteen part in every genus $g\geq2$. They also
show that this part equals the full cohomology in genus two, and in
genus three under a conjectural dimension assumption
\cite[Theorems 1.1--1.3]{BIKL26}. This gives a concrete and systematic
description of the relevant boundary classes and their relations.

We use a similar approach based on boundary classes, gluing maps, and
restrictions to boundary strata. Our calculations also address generation
of the full cohomology in higher genus and its symmetric-group
multiplicities. In low genus, some technical arguments differ: we use
local-system cohomology to study generation, including an unconditional
argument in genus three, and intersection pairings to compute the
genus-three multiplicities.

The overlap provides useful opportunities to check the calculations in
both papers. In particular, our genus-two dimension formula agrees with
theirs for all numbers of markings, and the higher-genus dimension checks
carried out so far are also consistent with their presentation. Their
explicit bases and relations make these comparisons particularly
accessible, and we find their treatment a valuable complement to the
calculations presented here.

\subsection*{Acknowledgements}

The main result of this paper was obtained using the Pharos system. Pharos is an automated mathematical reasoning system that inherits selected design principles from Rethlas~\cite{Ju26} and Danus~\cite{Liu26}, especially the fact graph and multi-agent architecture, while substantially redesigning and extending the remaining system components. The author would like to express their sincere gratitude to Bin Dong, Guoxiong Gao, Jiedong Jiang, Shurui Liu, and Bin Wu for their substantial contributions to the development and refinement of Pharos. The details of the use of generative AI are documented in
Appendix~\ref{app:ai} for transparency.