Weierstrass points on \(X_0(p)\) and supersingular \(j\)-invariants
The authors study the arithmetic of the Weierstrass points on modular curves. Such curves are an important subject in arithmetic, and is well known that \(X_0(N)\) is the moduli space of elliptic curves with a prescribed cyclic subgroup of order \(N\). More specifically, the authors consider Weierstrass points in \(X_0(p)\), where \(p\) is a prime number greater than 4. In [Ill. J. Math. 22, 31--35 (1978; Zbl 0374.14005)], \textit{A.~Ogg} proved, among other results, that if \(Q\) is a \(\mathbb{Q}\)-rational Weierstrass point of \(X_0(p)\), then the reduction of \(Q\) modulo \(p\) is supersingular, meaning that the underlying elliptic curve is supersingular. In the present work the authors describe the relationship between the supersingular \(j\)-invariants and the set of \(j(Q)\) for Weierstrass points \(Q \in X_0(p)\). They define two polynomials, namely, \(F_p(x) := \prod_{Q \in X_0(p)} (x - j(Q))^{wt(Q)}\), where \(wt(Q)\) is the Weierstrass weight of \(Q\), and \(S_p(x):= \prod_{E/ \overline{\mathbb{F}}_p \text{\;supersingular}} (x - j(E)) \in \mathbb{F}_p[x]\), where the product is taken over \(\overline{\mathbb{F}}_p\)-isomorphism classes of elliptic curves, and denoting by \(g_p\) the genus of \(X_0(p)\), their main result states that \(F_p(x)\) has \(p\)-integral rational coefficients and satisfies \(F_p(x) \equiv S_p(x)^{g_p(g_p - 1)} \pmod{p}\).
scientific article; zbMATH DE number 1922397
scientific article