masur_veech_volumes

masur_veech_volumes#

Masur-Veech volumes of Abelian strata and their connected components

surface_dynamics.flat_surfaces.masur_veech_volumes.masur_veech_volume(C, rational=False, method=None)#

Return the Masur-Veech volume of the stratum or component of stratum C.

INPUT:

  • C – a stratum or a connected component of stratum

  • rational (boolean) - if False (default) return the Masur-Veech volume and if True return the Masur-Veech volume divided by zeta(2g).

  • method - the method to use to compute the volume either

    • "table" - for a table lookup (all strata up to dimension 9 and some strata up to dimension 11)

    • "CMSZ" - the Chen-Möller-Sauvaget-Zagier recursion (currently only implemented for the principal stratum)

TESTS:

sage: from surface_dynamics import Stratum
sage: from surface_dynamics.flat_surfaces.masur_veech_volumes import masur_veech_volume

sage: H4 = Stratum([4], k=1)
sage: masur_veech_volume(H4, False, 'table')
61/108864*pi^6
sage: masur_veech_volume(H4, False, 'CMSZ')
61/108864*pi^6
sage: masur_veech_volume(H4.hyperelliptic_component(), False, 'table')
1/6720*pi^6
sage: masur_veech_volume(H4.hyperelliptic_component(), False, 'CMSZ')
1/6720*pi^6
sage: masur_veech_volume(H4.odd_component(), False, 'table')
1/2430*pi^6
sage: masur_veech_volume(H4.odd_component(), False, 'CMSZ')
1/2430*pi^6

sage: H6 = Stratum([6], k=1)
sage: all(masur_veech_volume(C, True, 'table') == masur_veech_volume(C, True, 'CMSZ') for C in H6.components())
True

TESTS:

sage: masur_veech_volume(Stratum([1,1,-2], k=1))
Traceback (most recent call last):
...
ValueError: meromorphic differentials with higher order poles
sage: masur_veech_volume(Stratum([1]*6, k=3))
Traceback (most recent call last):
...
NotImplementedError: higher order differentials
surface_dynamics.flat_surfaces.masur_veech_volumes.minimal_strata_CMSZ(gmax, rational=False)#

Return the volumes of cH(2g-2) for the genus g going from 1 up to gmax-1.

The algorithm is the one from Sauvaget [Sau18] involving an implicit equation. As explained in [CheMoeSauZag20], one could go through Lagrange inversion. Note that they miss factor 2 in their theorem 4.1.

EXAMPLES:

sage: from surface_dynamics.flat_surfaces.masur_veech_volumes import minimal_strata_CMSZ
sage: minimal_strata_CMSZ(6, True)
[2, 3/4, 305/576, 87983/207360, 1019547/2867200]
sage: minimal_strata_CMSZ(6, False)
[1/3*pi^2,
 1/120*pi^4,
 61/108864*pi^6,
 12569/279936000*pi^8,
 12587/3311616000*pi^10]

sage: from surface_dynamics import Stratum
sage: from surface_dynamics.flat_surfaces.masur_veech_volumes import masur_veech_volume
sage: for rat in [True, False]:
....:     V0, V2, V4, V6 = minimal_strata_CMSZ(5, rational=rat)
....:     MV0 = masur_veech_volume(Stratum([0], k=1), rat, 'table')
....:     assert V0 == MV0, (V0, MV0, rat)
....:     MV2 = masur_veech_volume(Stratum([2], k=1), rat, 'table')
....:     assert V2 == MV2, (V2, MV2, rat)
....:     MV4 = masur_veech_volume(Stratum([4], k=1), rat, 'table')
....:     assert V4 == MV4, (V4, MV4, rat)
....:     MV6 = masur_veech_volume(Stratum([6], k=1), rat, 'table')
....:     assert V6 == MV6, (V6, MV6, rat)
surface_dynamics.flat_surfaces.masur_veech_volumes.minimal_strata_hyp(g, rational=False)#

Return the volume of the hyperelliptic component H^{hyp}(2g-2).

The explicit formula appears in section 6.5 of [CheMoeSauZag20].

EXAMPLES:

sage: from surface_dynamics.flat_surfaces.masur_veech_volumes import minimal_strata_hyp
sage: minimal_strata_hyp(2)
1/120*pi^4
sage: minimal_strata_hyp(4)
1/580608*pi^8
sage: minimal_strata_hyp(10)
1/137733277917118464000*pi^20
sage: minimal_strata_hyp(10, rational=True)
668525/10499279483305984
surface_dynamics.flat_surfaces.masur_veech_volumes.minimal_strata_spin_diff(gmax, rational=False)#

Return the differences of volumes between even and odd components in H(2g-2) for the genus g going from 1 up to gmax-1.

If there are no even/odd components, the corresponding total volume is 0. Formulas are from [CheMoeSauZag20].

EXAMPLES:

sage: from surface_dynamics.flat_surfaces.masur_veech_volumes import minimal_strata_spin_diff
sage: minimal_strata_spin_diff(5)
[-1/3*pi^2, -1/120*pi^4, -143/544320*pi^6, -15697/1959552000*pi^8]
sage: minimal_strata_spin_diff(5, rational=True)
[-2, -3/4, -143/576, -15697/207360]
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