Author

I. V. Picinini

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Preprint Sep 2026

The Schur multiplier of $\rm{SL}_2$ and Dedekind zeta-functions over $S$-integers

In this paper, we obtain an exact sequence connecting $H_2(\rm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$ to $H_2(\rm{SL}_2(\mathcal{O}_{K,T}), \mathbb{Z})$, where $\mathcal{O}_{K,S}$ is a ring of $S$-integers and $T$ is a set of primes containing $S$. We apply this sequence to establish a relation between $H_2(\rm{SL}_2(\...

P. Amorim, I. V. Picinini, B. R. Ramos et al. · 0 citations

Preprint Aug 2026

The integral homology of $\text{SL}_2(\mathbb{Z}[1/n])$

Let $\mathbb{Z}[1/n] := \{a/n^r: a \in \mathbb{Z},r \in \mathbb{Z} \geq 0\}$ be the euclidean domain obtained from the ring of integers $\mathbb{Z}$ by localizing at $n$. Moreover, let $\text{SL}_2(\mathbb{Z}[1/n])$ be the group of all invertible 2-by-2 matrices of determinant one with entries in $\mathbb{Z}[1/n]$. The...

I. V. Picinini · 0 citations

Preprint Aug 2026

On the abelianization of congruence subgroups of $\mathrm{SL}_2$ over $S$-integers

In this work, we compute the first integral homology, or abelianization, of the congruence subgroups $\Gamma(A, \mathfrak{m}_A), \Gamma_1(A, \mathfrak{m}_A)$, and $\Gamma_0(A, \mathfrak{m}_A)$ for a local ring $A$ with maximal ideal $\mathfrak{m}_A$, showing that $H_1(\Gamma(A, \mathfrak{m}_A), \mathbb{Z})$ is isomorph...

P. Amorim, I. V. Picinini, B. R. Ramos et al. · 0 citations