On the Calegari-Venkatesh conjecture connecting modular forms spaces and algebraic K-Theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914108487499776 |
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| author | Rahm, Alexander D. Emiliano, Torti |
| author_facet | Rahm, Alexander D. Emiliano, Torti |
| contents | Calegari and Venkatesh did construct, modulo small torsion, a surjection from the degree 2 homology of the rank 2 projective general linear group over a ring of algebraic integers (of odd class number, and with enough embeddings) to the 2nd algebraic K-group of that ring. They asked whether this surjection becomes an isomorphism when passing to the quotient modulo the Eisenstein ideal on the left hand side. We provide a new method (together with numerical examples) to lift elements in the opposite direction, enabled by a theorem in a more general setting, where we exploit a connection between the algebraic K-groups and the Steinberg homology groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00687 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Calegari-Venkatesh conjecture connecting modular forms spaces and algebraic K-Theory Rahm, Alexander D. Emiliano, Torti K-Theory and Homology Calegari and Venkatesh did construct, modulo small torsion, a surjection from the degree 2 homology of the rank 2 projective general linear group over a ring of algebraic integers (of odd class number, and with enough embeddings) to the 2nd algebraic K-group of that ring. They asked whether this surjection becomes an isomorphism when passing to the quotient modulo the Eisenstein ideal on the left hand side. We provide a new method (together with numerical examples) to lift elements in the opposite direction, enabled by a theorem in a more general setting, where we exploit a connection between the algebraic K-groups and the Steinberg homology groups. |
| title | On the Calegari-Venkatesh conjecture connecting modular forms spaces and algebraic K-Theory |
| topic | K-Theory and Homology |
| url | https://arxiv.org/abs/2504.00687 |