algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety \(L\)-function; Shimura variety; Hilbert-Blumenthal surface; Tate's conjectures for abelian fields; group of algebraic cycles; intersection cohomology; canonical intermediate-perversity extension; Hirzebruch- Zagier cycles; Tate class 