algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety curve; function field; Jacobian; abelian variety; finite field; Mordell-Weil group; torsion; rank; \(L\)-function; Birch and Swinnerton-Dyer conjecture; Tate-Shafarevich group; Tamagawa number; endomorphism algebra; descent; height; Néron model; Kodaira-Spencer map; monodromy 