On the \(\mathbb{Q}\)-rational cuspidal subgroup and the component group of \(J_0(p^r)\)
For a prime number \(p\) and a positive integer \(r\), let \(C_0(p^r)\) be the \({\mathbb Q}\)-rational cuspidal subgroup of the Jacobian \(J_0(p^r)\) of the classical modular curve \(X_0(p^r)\) over \({\mathbb Q}\). In Math. USSR, Izv. 6(1972), 19-64 (1973); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 36, 19-66 (1972; Zbl 0243.14008), \textit{Yu. I. Manin} has shown that \(C(p^r)\) is a finite abelian group, and one can state the question to find this group for any such \(p,r\). For \(r = 1\) and any prime \(p\) it is computed by \textit{A. P. Ogg} that the group \(C(p)\) is cyclic of order \((p-1)/(p-1,12)\) [see: Analytic Number Theory, St. Louis Univ. 1972, Proc. Symp. Pure Math. 24, 221-231 (1973; Zbl 0273.14008)]. For \(p \geq 5\) let \(a = a(p) = (p-1)/(p-1,12)\) and \(b = b(p) = (p+1)/(p+1,12)\). For such \(p\), \(a = a(p)\), \(b= b(p)\) it is shown by Lorenzini that the prime-to-2p part \(C(p^r)^{(2p)}\) of \(C(p^r)\) is isomorphic to the prime-to-2 part of \(({\mathbb Z}/a{\mathbb Z})^r \times ({\mathbb Z}/b{\mathbb Z})^{r-1}\), provided \(p \not\cong 11 \bmod 12\) [\textit{D. I. Lorenzini}, Compos. Math. 96, No. 2, 149-172 (1995; Zbl 0846.14017)]. In the paper under review the group \(C(p^r)\) is computed entirely for all primes \(p \geq 3\) and any positive integer \(r\). This result is applied to determine the component group \({\Phi}_{p^r}\) of the Néron model of \(J_0(p^r)\) over \({\mathbb Z}_p\) for \(p \geq 5\), this way extending a result of Lorenzini (loc. cit.). It is studied also the action of the Atkin-Lehner involution on the \(p\)-primary part of \(C(p^r)\), as well the effect of degeneracy maps on the component groups.
scientific article; zbMATH DE number 1062224