Points S-entiers des courbes elliptiques. (S-integral points of elliptic curves)
Let \(E\) be an elliptic curve defined over a field \(K\) with discriminant \(\Delta\) and Weierstrass equation \(y^ 2+a_ 1xy+a_ 3y=x^ 3+a_ 2x^ 2+a_ 4x+a_ 6\), with \(a_ i\in K\). Given a point \(M=(x,y)\in E(K)\) and a positive integer \(m\) we have \(mM=(\varphi_ m(M)/\psi^ 2_ m(M),\omega_ m(M)/\psi^ 3_ m(M))\), where \(\varphi_ m\), \(\psi_ m\), \(\omega\in\mathbb{Z}[a_ 1,\dots,a_ 6,x,y]\) are polynomials defined inductively [cf. \textit{J. Silverman}, ``The arithmetic of elliptic curves'' (1986; Zbl 0585.14026); chapter III, ex. 3.7]. The author takes \(K\) to be a valued field with valuation \(v\) and such that \(v(a_ i)\geq 0\), for \(i=1,2,3,4,6\). He shows that if \(M\in E(K)\) is not the point at infinity \(\infty\), the reduction of \(M\) modulo \(v\) is not \(\infty\) then the following conditions are equivalent: (a) \(v(\psi_ 2(M))>0\) and \(v(\psi_ 3(M))>0\). (b) For each \(m\geq 2\), \(v(\psi_ m(M))>0\). (c) There exists \(m_ 0\geq 2\) such that \(v_{m_ 0}(M))>0\) and \(v(\psi_{m_ 0+1}(M))>0\). (d) There exists \(n_ 0\geq 2\) such that \(v(\psi_{n_ 0}(M))>0\) and \(v(\varphi_{n_ 0}(M))>0\). (e) \(M\) modulo \(v\) is not a singular point of the reduced curve \(\tilde E_ v\). Further, taking \(K=\mathbb{Q}\), \(a_ i\in\mathbb{Z}\), \(S\) a finite set of prime numbers, \(M=(x,y)=(a/d^ 2,b/d^ 3)\), with \(\text{gcd}(a,d)=1\), and \(mM=(\hat\varphi_ m/d^ 2\hat\psi^ 2_ m,\hat\omega_ m/d^ 3\hat\psi^ 3_ m)\), where \(\hat\varphi_ m(M)=d^{2m^ 2}\varphi_ m(M)\), \(\hat\psi_ m(M)=d^{m^ 2-1}\psi_ m(M)\) and \(\hat\omega_ m(M)=d^{3m^ 2}\omega_ m(M)\) are integers, he obtains the following characterization of \(S\)-integral points. If \(M\) is a rational point of infinity order and \(mM\) is \(S\)-integral, then \(M\) itself is \(S\)-integral. Hence as a corollary of the previous equivalent conditions the author proves that if \(M\) is an \(S\)-integral point of infinity order and \(T\subset S\) is the set of primes in \(S\) for which \(M\) mod \(p\) is singular, then \(mM\) is \(S\)-integral if and only if \(\hat\psi_ m=\pm\prod_{p\in S}p^{e_ p}\), where \(e_ p\geq 1\), if \(p\in T\) and \(e_ p\geq 0\), if \(p\in S-T\). Moreover, he also shows that if \(M\) is a rational point of infinity order and \(L=(u,v)\) is a \(\mathbb{Q}\)-rational 2- torsion point of \(E\), then \(L+mM\) is \(S\)-integral if and only if \(\hat\varphi_ m-ud^ 2\hat\psi^ 2_ m=\pm\prod_{p\in S}p^{e_ p}\), if \(u\in\mathbb{Z}\), \(4(\hat\varphi_ m-ud^ 2\hat\psi^ 2_ m)=\pm\prod_{p\in S}p^{e_ p}\), if \(u\notin\mathbb{Z}\), where \(e_ p\geq 1\), if \(p\in T\), and \(e_ p\geq 0\), if \(p\in S-T\).
scientific article; zbMATH DE number 219905