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The Recursive Structures of Manin Symbols over , Cusps and Elliptic Points on X0 (N)
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Journal Article
The Recursive Structures of Manin Symbols over , Cusps and Elliptic Points on X0 (N)
2023
Overview
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 ) .
Publisher
MDPI AG
Subject
