Explicit resolutions of cubic cusp singularities. (English) Zbl 0999.11023
From the text: Resolutions of cusp singularities are crucial to many techniques in computational number theory, and therefore finding explicit resolutions of these singularities has been the focus of a great deal of research. This paper presents an implementation of a sequence of algorithms leading to explicit resolutions of cusp singularities (of Hilbert varieties) arising from totally real cubic fields. As an example, the implementation is used to compute values of partial zeta functions associated to these cusps following Shintani’s approach.
The resolutions derived here are based on the work of F. Ehlers [Math. Ann. 127-156 (1975; Zbl 0301.14003)] and E. Thomas and A. T. Vasquez [Math. Ann. 247, 1-20 (1980; Zbl 0403.14005)].
The resolutions derived here are based on the work of F. Ehlers [Math. Ann. 127-156 (1975; Zbl 0301.14003)] and E. Thomas and A. T. Vasquez [Math. Ann. 247, 1-20 (1980; Zbl 0403.14005)].
MSC:
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 11Y40 | Algebraic number theory computations |
| 14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |
| 14G35 | Modular and Shimura varieties |
| 11R16 | Cubic and quartic extensions |
| 11R42 | Zeta functions and \(L\)-functions of number fields |
Keywords:
fundamental domain; resolutions of cusp singularities; algorithms; totally real cubic fields; values of partial zeta-functions; Hilbert varietiesSoftware:
LiDIAReferences:
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