For N integer $\ge 1$, K. Murty and D. Ramakrishnan defined the Nth Heisenberg curve, as the compactified quotient $X^{\prime }_N$ of the upper half-plane by a certain non-congruence subgroup of the modular group. They ask whether the Manin–Drinfeld principle holds, namely, if the divisors supported on the cusps of those curves are torsion in the Jacobian. We give a model over $\mathbf {Z}[\mu _N,1/N]$ of the Nth Heisenberg curve as covering of the Nth Fermat curve. We show that the Manin–Drinfeld principle holds for $N=3$, but not for $N=5$. We show that the description by generator and relations due to Rohrlich of the cuspidal subgroup of the Fermat curve is explained by the Heisenberg covering, together with a higher covering of a similar nature. The curves $X_N$ and the classical modular curves $X(n)$, for n even integer, both dominate $X(2)$, which produces a morphism between Jacobians $J_N\rightarrow J(n)$. We prove that the latter has image $0$ or an elliptic curve of j-invariant $0$. In passing, we give a description of the homology of $X^{\prime }_{N}$.

Romyar Sharifi has constructed a map $\varpi _M$ from the first homology of the modular curve $X_1(M)$ to the K-group $K_2(\operatorname {\mathrm {\mathbf {Z}}}[\zeta _M+\zeta _M^{-1}, \frac {1}{M}]) \otimes \operatorname {\mathrm {\mathbf {Z}}}[1/2]$, where $\zeta _M$ is a primitive Mth root of unity. Sharifi conjectured that $\varpi _M$ is annihilated by a certain Eisenstein ideal. Fukaya and Kato proved this conjecture after tensoring with $\operatorname {\mathrm {\mathbf {Z}}}_p$ for a prime $p\geq 5$ dividing M. More recently, Sharifi and Venkatesh proved the conjecture for Hecke operators away from M. In this note, we prove two main results. First, we give a relation between $\varpi _M$ and $\varpi _{M'}$ when $M' \mid M$. Our method relies on the techniques developed by Sharifi and Venkatesh. We then use this result in combination with results of Fukaya and Kato in order to get the Eisenstein property of $\varpi _M$ for Hecke operators of index dividing M.

We compute the $p$-adic $L$-functions of evil Eisenstein series, showing that they factor as products of two Kubota–Leopoldt $p$-adic $L$-functions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens.

Two integral structures on the $\mathbb{Q}$-vector space of modular forms of weight two on $X_0(N)$ are compared at primes $p$ dividing $N$ at most once. When $p=2$ and $N$ is divisible by a prime that is $3$ mod $4$, this comparison leads to an algorithm for computing the space of weight one forms mod $2$ on $X_0(N/2)$. For $p$ arbitrary and $N>4$ prime to $p$, a way to compute the Hecke algebra of mod $p$ modular forms of weight one on $\varGamma_1(N)$ is presented, using forms of weight $p$, and, for $p=2$, parabolic group cohomology with mod $2$ coefficients. Appendix A is a letter of October 1987 from Mestre to Serre in which he reports on computations of weight one forms mod $2$ of prime level. Appendix B, by Wiese, reports on an implementation for $p=2$ in Magma, using Stein’s modular symbols package, with which Mestre’s computations are redone and slightly extended.