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- W4377027424 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 $X_0(N)$." @default.
- W4377027424 created "2023-05-19" @default.
- W4377027424 creator A5015057168 @default.
- W4377027424 date "2023-05-16" @default.
- W4377027424 modified "2023-09-29" @default.
- W4377027424 title "The Recursive Structures of Manin Symbols over $mathbb{Q}$, Cusps and Elliptic Points on $X_0(N)$" @default.
- W4377027424 doi "https://doi.org/10.20944/preprints202305.1118.v1" @default.
- W4377027424 hasPublicationYear "2023" @default.
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