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arXiv · 2512.12041

Generalized Jacobians of graphs

Bruce W. Jordan·Kenneth A. Ribet·Anthony J. Scholl
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Abstract

We define a generalized Jacobian $\mathrm{J}_\mathfrak{m}(\mathit{Gr})$ and a generalized Picard group $\mathrm{P}_\mathfrak{m}(\mathit{Gr})$ of a graph $\mathit{Gr}$ with respect to a modulus $ \mathfrak{m}=\sum_{i=1}^s m_iw_i$ with $w_i$ vertices of $\mathit{Gr}$ and $m_i\geq 1$. These groups occur as the component groups of N\'{e}ron models of generalized Jacobians. We prove a universal mapping property for $\mathrm{J}_\mathfrak{m}(\mathit{Gr})$ and show that an Abel-Jacobi map in this context induces an isomorphism from $\mathrm{P}_\frak{m}(\mathit{Gr})$ to $\mathrm{J}_\mathfrak{m}(\mathit{Gr})$. We also reinterpret $\mathrm{P}_\mathfrak{m}(\mathit{Gr})$ in terms of sheaves on the geometric realization $\left| \mathit{Gr}\right|$ of $\mathit{Gr}$, making a connection with tropical geometry.

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Bruce W. Jordan, Kenneth A. Ribet, Anthony J. Scholl. 2025-12-12. Generalized Jacobians of graphs. https://arxiv.org/abs/2512.12041

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