The Recursive Structures of Manin Symbols over <named-content content-type="font-family:bold-italic"><math id="mm222222"><semantics><mi mathvariant="double-struck">Q</mi></semantics></math></named-content>, Cusps and Elliptic Points on <i>X</i>₀ (<i>N</i>)

Abstract

Firstly, we present a more explicit formulation of the complete system D(N) of representatives of Manin’s symbols over Q, which was initially given by Shimura. Then, we establish a bijection between D(M)×D(N) and D(MN) for (M,N)=1, which reveals a recursive structure between Manin’s symbols of different levels. Based on Manin’s complete system Π(N) of representatives of cusps on X0(N) and Cremona’s characterization of the equivalence between cusps, we establish a bijection between a subset C(N) of D(N) and Π(N), and then establish a bijection between C(M)×C(N) and C(MN) for (M,N)=1. We also provide a recursive structure for elliptical points on X0(N). Based on these recursive structures, we obtain recursive algorithms for constructing Manin symbols over Q, cusps, and elliptical points on X0(N). This may give rise to more efficient algorithms for modular elliptic curves. As direct corollaries of these recursive structures, we present a recursive version of the genus formula and prove constructively formulas of the numbers of D(N), cusps, and elliptic points on X0(N).

Keywords

• modular curve
• elliptic curve
• recursive structure
• Q %22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> Manin’s symbols over <named-content content-type="font-family:bold-italic"> <mml:math id="mm2222"> <mml:semantics> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:semantics> </mml:math> </named-content>
• cusps
• elliptic points