Quadratic Chabauty on \( X_0(N)^* \), Part~II: Quaternion Algebras

The first part explained how one can choose a correspondence so that all local height contributions away from \( p \) vanish. We now show that this is possible for all but finitely many squarefree levels. A sufficient condition is that the genus of \( X_0(N)^* \) is greater than one more than the total number of cycles (of length greater than \( 1 \)) in the dual graphs at all the bad primes. The number of cycles can be bounded by the number of fixed points of Atkin--Lehner involutions \( W_d \). These fixed points are related to embeddings of quadratic orders into Eichler orders in the definite quaternion algebra \( B_{p,\infty} \). This leads to explicit formulas for the number of \( W_d \)-fixed points in the supersingular locus. We compare a lower bound for the genus with an upper bound for the total number of cycles in order to prove the desired inequality.