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\begin{document}

\title[Dense Algebraic Loci of Hodge Classes]{Dense Algebraic Loci of Hodge Classes}

\author{Deep Bhattacharjee\,\orcidlink{0000-0003-0466-750X}}
\address{Formerly, Electro-Gravitational Space Propulsion Laboratory (EGSPL), Bhubaneswar, Odisha 751030, India; formerly, University of Calcutta, 87/1 College Street, Kolkata 700073, India}
\email{itsdeep@live.com}
\thanks{Corresponding author: Deep Bhattacharjee. Second address: \nolinkurl{d.bhattacharjee@erl-forschung.de}.}

\author{Ushashi Bhattacharya\,\orcidlink{0009-0002-2254-3914}}
\address{Formerly, National Taiwan University, Taipei 106319, Taiwan}
\email{rai.bhattacharya.in@gmail.com}

\date{October 21, 2026}

\subjclass[2020]{14C30, 14K20, 14J28, 11E12}
\keywords{Hodge conjecture; abelian varieties; Weil classes; K3 surfaces}

\begin{abstract}
\itshape
We prove unconditionally that the Hodge conjecture is equivalent to two statements about abelian varieties, that the locus on which either statement holds is dense in every family and is either the whole family or a meagre set, that the real multiplications by $\sqrt6$ and by the cubic fields $\QQ(\zeta_{7})^{+}$ and $\QQ(\zeta_{9})^{+}$ are algebraic on explicit families of K3 surfaces, the cubic ones on the surfaces of Mumford type whose quaternion algebra is ramified at two real places, and that the Hodge conjecture holds on every power and every Hilbert scheme of points of the K3 surfaces of the one-dimensional Fricke families of levels $5$ and $7$, which carry complex multiplication by $\QQ(\sqrt{-5})$ and $\QQ(\sqrt{-7})$. For each of the two statements we build the object it needs next, a complex on a product of two Weil fourfolds and a class on an abelian fourteenfold, prove what it does, and prove at which step it stops; among the steps, every convolution of the Weil line bundles on $E_{0}^{8}$ whose shifts lie in seven consecutive integers fails the polarised criterion in eighteen of the twenty-eight arrangements of the shifts, an explicit curve of genus four makes the Prym map of cyclic triple covers \'etale onto the Weil domain of $\QQ(\sqrt{-3})$, and the conjecture for a class is equivalent to the vanishing of the mass gap of a multiple of it plus a complete intersection, which Koll\'ar's hypersurfaces show cannot be dropped for integral classes. Both statements remain open.
\end{abstract}

\maketitle
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\input{sec1_intro}
\input{sec2_conventions}
\input{sec3_reduction}
\input{sec4_loci}
\input{sec5_f3prime}
\input{sec6_f2}
\input{sec7_earlier}
\input{sec8_status}

\appendix
\input{appB_derivations}
\input{appA_checks}

\input{declarations}

\input{bibliography}

\end{document}