Curves with infinitely many points of fixed degree
In an earlier paper [Compos. Math. 58, 133-134 (1986; Zbl 0596.14021)], the author had shown that, for a given prime \(l>240\), there are only finitely many elliptic curves \(E\) over quadratic fields \(L\) having a rational isogeny of degree \(l\) over \(L\). Applying a recent theorem of Faltings the author is now in a position to prove the analogous result for \(l\)-isogenies over number fields \(L\) of arbitrary degree \([L : \mathbb{Q}] \leq d\) provided the prime satisfies \(l>120d\). The theorem of Faltings, corroborating a conjecture of S. Lang, implies that if the \(d\)-th symmetric product \(C^{(d)}\) of a curve \(C\) over a number field \(K\) contains infinitely many \(K\)-rational points, then there exists a \(K\)-rational covering \(\pi : C \to \mathbb{P}^ 1_ K\) of degree \(\leq 2d\). This theorem is applied to the modular curves \(X_ 0 (l)\) over \(K = \mathbb{Q} (\sqrt 5)\). Then \(2 \cong {\mathfrak p}\) is inert in \(K\) and hence the residue field \(k_{\mathfrak p}\) has cardinality 4. Now, on the one hand, the reduced curve \(X_ 0^{({\mathfrak p})} (l)\) has \(\# X_ 0^{({\mathfrak p})} (l)(k_{\mathfrak p}) \geq [{l \over 12}] + 1\) rational points over \(k_{\mathfrak p}\). On the other hand, since the family of curves \(\{X_ 0 (l)\}_ l\) for odd primes \(l\) ``behaves asymptotically good at \({\mathfrak p}\)'', the theorem of Faltings implies the estimate \(\# X_ 0^{({\mathfrak p})} (l)(k_{\mathfrak p}) \leq 2d (\# k_{\mathfrak p} + 1) = 10d\) in case \(X_ 0(l)\) has infinitely many points of degree \(d\) over \(K\) or, equivalently, in case \(\# X_ 0 (l_ )^{(d)} (K)\) is infinite. Combining these two inequalities yields that for \(l>120d\), the curve \(X_ 0 (l)\) can have only finitely many points of degree \(d\) over \(K\). Since \(X_ 0 (l)(L)\) parametrizes elliptic curves over \(L\) with an \(L\)-rational isogeny of degree \(l\), where \([L:K]\leq d\), one concludes that, for \(l>120d\), there are only finitely many elliptic curves over \(L\) with an \(L\)-rational isogeny of degree \(l\). \{Reviewer's remark: The proof given yields the assertion actually only for primes \(l>240d\). However, an additional argument then leads to the stronger assertion with \(l>120d\}\).
number of elliptic curves with rational isogeny
scientific article; zbMATH DE number 553997