Cusp forms of weight 1 associated to Fermat curves
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| Název: | Cusp forms of weight 1 associated to Fermat curves |
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| Autoři: | Yang, Tonghai |
| Zdroj: | Duke Math. J. 83, no. 1 (1996), 141-156 |
| Informace o vydavateli: | Duke University Press, 1996. |
| Rok vydání: | 1996 |
| Témata: | modular curve, Fermat curve, 11F11, 01 natural sciences, Modular and Shimura varieties, Plane and space curves, principal congruence subgroup, dimension formulas, canonical basis for the space of cusp forms of weight 1, 0103 physical sciences, canonical basis for the space of modular forms, Riemann-Roch theorems, 0101 mathematics, 11G18, Holomorphic modular forms of integral weight |
| Popis: | Let \(\Delta\) be the free subgroup of the principal congruence subgroup \(\Gamma (2)\) generated by \(A= \left(\begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix} \right)\) and \(B = \left (\begin{smallmatrix} 1 & 0 \\ 2 & 1 \end{smallmatrix} \right)\) (one has \(\Gamma(2) = \{\pm I\} \Delta)\), and for \(N\in\mathbb{N}\) let \(\Phi(N)\) be the subgroup of \(\Delta\) generated by \(A^N\) and \(B^N\) and the commutator \([\Delta,\Delta]\). Then the associated modular curve \(X(\Phi (N))\) is isomorphic to the Fermat curve \(X^N + Y^N=Z^N\). As is well-known, \(\Phi(N)\) is not a congruence subgroup unless \(N=1,2,4,8\). Using the above isomorphism the author constructs a canonical basis for the space of modular forms and cusp forms of weight 1 for \(\Phi(N)\), and, as a consequence, obtains explicit dimension formulas for these spaces. One should note that the classical Riemann-Roch theorem in general only gives explicit dimension formulas for spaces of cusp forms of weight \(k\geq 2\). |
| Druh dokumentu: | Article Other literature type |
| Popis souboru: | application/xml; application/pdf |
| ISSN: | 0012-7094 |
| DOI: | 10.1215/s0012-7094-96-08306-4 |
| Přístupová URL adresa: | https://zbmath.org/912123 https://doi.org/10.1215/s0012-7094-96-08306-4 https://projecteuclid.org/journals/duke-mathematical-journal/volume-83/issue-1/Cusp-forms-of-weight-1-associated-to-Fermat-curves/10.1215/S0012-7094-96-08306-4.full http://projecteuclid.org/euclid.dmj/1077244250 |
| Přístupové číslo: | edsair.doi.dedup.....1078e7ef6732f805c9a3d261d25e157d |
| Databáze: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstrakt: | Let \(\Delta\) be the free subgroup of the principal congruence subgroup \(\Gamma (2)\) generated by \(A= \left(\begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix} \right)\) and \(B = \left (\begin{smallmatrix} 1 & 0 \\ 2 & 1 \end{smallmatrix} \right)\) (one has \(\Gamma(2) = \{\pm I\} \Delta)\), and for \(N\in\mathbb{N}\) let \(\Phi(N)\) be the subgroup of \(\Delta\) generated by \(A^N\) and \(B^N\) and the commutator \([\Delta,\Delta]\). Then the associated modular curve \(X(\Phi (N))\) is isomorphic to the Fermat curve \(X^N + Y^N=Z^N\). As is well-known, \(\Phi(N)\) is not a congruence subgroup unless \(N=1,2,4,8\). Using the above isomorphism the author constructs a canonical basis for the space of modular forms and cusp forms of weight 1 for \(\Phi(N)\), and, as a consequence, obtains explicit dimension formulas for these spaces. One should note that the classical Riemann-Roch theorem in general only gives explicit dimension formulas for spaces of cusp forms of weight \(k\geq 2\). |
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| ISSN: | 00127094 |
| DOI: | 10.1215/s0012-7094-96-08306-4 |