Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

The Recursive Structures of Manin Symbols over $\mathbb{Q}$, Cusps and Elliptic Points on $X_0(N)$

Version 1 : Received: 15 May 2023 / Approved: 16 May 2023 / Online: 16 May 2023 (07:23:25 CEST)

A peer-reviewed article of this Preprint also exists.

Wang, S. The Recursive Structures of Manin Symbols over Q, Cusps and Elliptic Points on X0 (N). Axioms 2023, 12, 597. Wang, S. The Recursive Structures of Manin Symbols over Q, Cusps and Elliptic Points on X0 (N). Axioms 2023, 12, 597.

Abstract

Firstly, we present a more explicit formulation of the complete system $D(N)$ of representatives of Manin's symbols over $\mathbb{Q}$, which was initially given by Shimura. Then we establish a bijection between $D(M)\times D(N)$ 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 $\Pi (N)$ of representatives of cusps on $X_0(N)$ and Cremona's characterization of the equivalence between cusps, we establish a bijection between a subset $C(N)$ of $D(N)$ and $\Pi (N)$, and then establish a bijection between $C(M)\times C(N)$ and $C(MN)$ for $(M,N)=1$. We also provide a recursive structure for elliptical points on $X_0(N)$. Based on these recursive structures, we obtain recursive algorithms for constructing Manin symbols over $\mathbb{Q}$, cusps and elliptical points on $X_0(N)$. This gives rise to a more efficient algorithms for modular elliptic curve. As direct corollaries of these recursive structures, we present a recursive version of the genus formula and an elementary proof of formulas of the numbers of $D(N)$, cusps and elliptical points on $X_0(N)$.

Keywords

modular curve; elliptic curve; recursive structure; Manin's symbols over $\mathbb{Q}$; cusps; elliptical points; algorithmic number theory

Subject

Computer Science and Mathematics, Algebra and Number Theory

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