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\begin{document}

\title{Dense Algebraic Loci of Hodge Classes}

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

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

\date{October 17, 2026}

\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, and that the real multiplications by $\sqrt5$, $\sqrt6$, $\sqrt7$ 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. 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. Both statements remain open.
\par\medskip\upshape\noindent\textit{2020 Mathematics Subject Classification.} 14C30 (transcendental methods, Hodge theory); 14K20 (analytic theory of abelian varieties); 14J28 (K3 surfaces); 11E12 (quadratic forms over global rings and fields).
\end{abstract}

\keywords{Hodge conjecture; abelian varieties; Weil classes; K3 surfaces}

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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
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\input{appA_checks}

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\end{document}