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Lifting of supersingular points on \(X_0(p^r)\) and lower bound of ramification index. (English) Zbl 1032.11024

Let \(\mathbb{Q}_p\) be the \(p\)-adic number field, \(\mathbb{Q}_p^{ur}\) the maximal unramified extension of \(\mathbb{Q}_p\) and \(K\) a finite extension of \(\mathbb{Q}_p^{ur}\). We denote by \(O\) the ring of integers of \(K\), by \({\mathfrak m}\) the maximal ideal of \(O\) and by \(e_K\) the degree of \(K\) over \(\mathbb{Q}_p^{ur}\). In this paper the authors prove that if there exists an elliptic curve \(E\) with a cyclic subgroup of order \(p^r\) defined over \(O\) whose reduction modulo \({\mathfrak m}\) is a supersingular elliptic curve, then \[ e_K\geq\begin{cases} 2p^l, &\text{if }r=2l+1,\\ p^l+ p^{l-1}, &\text{if }r=2l. \end{cases} \] Two proofs of this result are given. One is obtained by using the formal groups associated to elliptic curves, and the other relies on the crossing theorem of modular curves.
Let \(X_0(p)\) be the modular curve corresponding to the group \[ \Gamma_0(p)= \left\{ \left( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix}\right)\in \text{SL}_2 (\mathbb{Z})/c\equiv 0\bmod p\right\} \] and \(J_0(p)\) the Jacobian variety of \(X_0(p)\). Let \(X_0^+(p^r)= X_0(p^r)/ w_{p^r}\), where \(w_{p^r}\) is the fundamental involution of \(X_0(p^r)\). We denote by \(n(p,r)\) the number of \(\mathbb{Q}\)-rational points on \(X_0^+(p^r)\) which are neither cusps nor CM points. Further, we set \(J_0^-(p)= J_0(p)/ (1+w_{p^r})\). Using the above result the authors obtain that if \(r\geq 2\) and the genus of \(X_0^+(p^r)\) is greater than 0, then \(n(p,r)= 0\) for \(p=2,3,7,11\), \(p=5\) with \(r\geq 4\), \(p=13\) with \(r\geq 3\) and \(p\geq 17\) with \(\#J_0^- (p)(\mathbb{Q})< \infty\).

MSC:

11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
11G07 Elliptic curves over local fields
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