Abstract
We study the arithmetic properties of Weierstrass points on the modular curves X_0^+(p) for primes p. In particular, we obtain a relationship between the Weierstrass points on X_0^+(p) and the j-invariants of supersingular elliptic curves in characteristic p.
Highlights
A Weierstrass point on a compact Riemann surface M of genus g is a point Q ∈ M at which some holomorphic differential ω vanishes to order at least g
Ogg [20] showed that for modular curves X0(pM) where p is a prime with p M and with the genus of X0(M) equal to 0, the Weierstrass points of X0(pM) occur at points whose underlying elliptic curve is supersingular when reduced modulo p
Ahlgren and Ono [3] showed for the M = 1 case that all supersingular elliptic curves modulo p correspond to Weierstrass points of X0(p), and they demonstrated a precise correspondence between the two sets
Summary
A Weierstrass point on a compact Riemann surface M of genus g is a point Q ∈ M at which some holomorphic differential ω vanishes to order at least g. Ogg [20] showed that for modular curves X0(pM) where p is a prime with p M and with the genus of X0(M) equal to 0, the Weierstrass points of X0(pM) occur at points whose underlying elliptic curve is supersingular when reduced modulo p. Ahlgren and Ono [3] showed for the M = 1 case that all supersingular elliptic curves modulo p correspond to Weierstrass points of X0(p), and they demonstrated a precise correspondence between the two sets. El-Guindy [8] generalized Theorem 1.1 by considering FpM where M is squarefree, showing that FpM(x) has p-integral rational coefficients and is divisible by Sp(x)μ(M)gpM(gpM−1), where μ(M) := [ : 0(M)] and gpM is the genus of X0(pM), and where He gave an explicit factorization of FpM(x) in most cases where M is prime.
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