Supersingular Drinfeld modules, Brandt matrices, and rank-metric codes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914486673211392 |
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| author | Micheli, Giacomo Papikian, Mihran |
| author_facet | Micheli, Giacomo Papikian, Mihran |
| contents | We prove a stabilization result for the $\mathbb{F}_q$-dimension of spaces of morphisms between supersingular Drinfeld modules, filtered by degree: for any two supersingular rank-$2$ Drinfeld $\mathbb{F}_q[T]$-modules in characteristic $\frak{p}$ of degree $d$, the dimension $m_s$ of the space of morphisms of $τ$-degree at most $s$ satisfies $m_s = 2(s+1)-(d-1)$ for all $s\geq d-2$. This is proved using the theory of Brandt matrices and properties of $L$-functions of automorphic forms for $\mathrm{GL}_2$ over function fields. The stabilization formula, combined with an analysis of zero entries in Brandt matrices and a hyperplane-avoidance argument, yields semifield rank-metric codes. We also describe an efficient algorithm for computing the relevant Brandt matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_17080 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Supersingular Drinfeld modules, Brandt matrices, and rank-metric codes Micheli, Giacomo Papikian, Mihran Number Theory 11G09, 94B60, 11F41, 11R52, 16S36 We prove a stabilization result for the $\mathbb{F}_q$-dimension of spaces of morphisms between supersingular Drinfeld modules, filtered by degree: for any two supersingular rank-$2$ Drinfeld $\mathbb{F}_q[T]$-modules in characteristic $\frak{p}$ of degree $d$, the dimension $m_s$ of the space of morphisms of $τ$-degree at most $s$ satisfies $m_s = 2(s+1)-(d-1)$ for all $s\geq d-2$. This is proved using the theory of Brandt matrices and properties of $L$-functions of automorphic forms for $\mathrm{GL}_2$ over function fields. The stabilization formula, combined with an analysis of zero entries in Brandt matrices and a hyperplane-avoidance argument, yields semifield rank-metric codes. We also describe an efficient algorithm for computing the relevant Brandt matrices. |
| title | Supersingular Drinfeld modules, Brandt matrices, and rank-metric codes |
| topic | Number Theory 11G09, 94B60, 11F41, 11R52, 16S36 |
| url | https://arxiv.org/abs/2604.17080 |