The Mahler measure of exact polynomials and special $L$-values of $K3$ surfaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912445417652224 |
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| author | Trieu, Thu Ha |
| author_facet | Trieu, Thu Ha |
| contents | We express the Mahler measure of an exact polynomial in arbitrarily many variables in terms of Deligne-Beilinson cohomology. We then focus on the relationship between the Mahler measure of four-variable exact polynomials and the special value of the $L$-function of $K3$ surfaces at $s = 4$. This result extends the three-variable case studied in \cite{Tri23}. Finally, we prove, under Beilinson's conjecture, that the Mahler measure of the polynomial $(x+1)(y+1)(z+1) + t$ is expressed in terms of the Riemann zeta function and the $L$-function of the modular form of weight 3 and level 7. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00893 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Mahler measure of exact polynomials and special $L$-values of $K3$ surfaces Trieu, Thu Ha Number Theory Algebraic Geometry Primary: 19F27, Secondary: 11G55, 11R06, 14J28, 14J27 We express the Mahler measure of an exact polynomial in arbitrarily many variables in terms of Deligne-Beilinson cohomology. We then focus on the relationship between the Mahler measure of four-variable exact polynomials and the special value of the $L$-function of $K3$ surfaces at $s = 4$. This result extends the three-variable case studied in \cite{Tri23}. Finally, we prove, under Beilinson's conjecture, that the Mahler measure of the polynomial $(x+1)(y+1)(z+1) + t$ is expressed in terms of the Riemann zeta function and the $L$-function of the modular form of weight 3 and level 7. |
| title | The Mahler measure of exact polynomials and special $L$-values of $K3$ surfaces |
| topic | Number Theory Algebraic Geometry Primary: 19F27, Secondary: 11G55, 11R06, 14J28, 14J27 |
| url | https://arxiv.org/abs/2412.00893 |