Compactness of the Modular Curve $X_0(N)$ Number Theory Compatibility of Upper Numbering with Quotients Local Fields Integral Primitive Solutions Induce Primitive Congruence Solutions Diophantine Equations Hadamard–de la Vallée Poussin Nonvanishing Theorem Analytic Number Theory Completeness Criterion for Parametric Classifications of Integer Solutions Diophantine Equations Eichler-Shimura Construction of Newform Quotients Number Theory Identity Component of a Real Elliptic Curve Is a Subgroup Number Theory Dirichlet-Quality Approximation Criterion Elementary Number Theory Number Theory Area Analytic Number Theory Subarea