Ideal class groups of cyclotomic fields and modular forms of level 1 (Q1310830)

Let \(p\) be an odd prime, and let \(A\) denote the \(p\)-Sylow subgroup of the ideal class group of the cyclotomic field \(\mathbb{Q} (\mu_ p)\). The author studies the following conjecture, which would follow from Vandiver's Conjecture: For each odd character \(\psi\) of \(\text{Gal} (\mathbb{Q} (\mu_ p)/ \mathbb{Q})\), which is not the Teichmüller character, the \(\psi\)- eigenspace of \(A\) is cyclic of order \(L(0,\psi^{-1})\). This conjecture is related to the structure of the Hecke algebra of level 1 modular forms.