/hilbertmodularsurfacesdata

hilbertmodularsurfacesdata

Repository for the data associated to

The data was computed with the Magma package hilbertmodularforms

Labels

In a given file, each row contains data pertaining to a particular Hilbert modular surface. The first entry in each such row is a label identifying the surface. The label is of the form field_label-level_label-component_label-ambient_type-gamma_typewhere

  • field_label is the LMFDB label of the real quadratic field $F$ associated to the surface;
  • level_label is the LMFDB label of the level $\mathfrak{N}$;
  • component_label is the LMFDB label of the ideal $\mathfrak{b}$ specifying a component of the surface;
  • ambient_type is either gl or sl depending on whether the associated Hilbert modular group is defined as a subset of $\operatorname{GL}_2^+(F)$ or $\operatorname{SL}_2(F)$
  • gamma_type is either 0, 1, or f according as the Hilbert modular group is $\Gamma_0(\mathfrak{N})$, $\Gamma_1(\mathfrak{N})$, or $\Gamma(\mathfrak{N})$ (or the $\operatorname{SL}$ variants of these).

File formats

  • Files with names ending in inv.txt contain data of geometric invariants of the surfaces. Each row is of the form label:[h[0,2],h[1,1]]:K^2:chi, where

    • label is the label of the surface;
    • h[0,2] and h[1,1] are the Hodge numbers $h^{0,2}$ and $h^{1,1}$;
    • K^2 is the self-intersection number of the canonical divisor; and
    • chiis the holomorphic Euler characteristic (what van der Geer calls the arithmetic genus)

  • Files with names ending in kposs.txt contain data of the possible Kodaira dimensions of each surface. Each row is of the form label:ks, where

    • label is the label of the surface; and
    • ks are the possible values of the Kodaira dimension $\kappa$ of the surface.

  • Files with names ending in cusps.txt contain data about the cusps on the surfaces. Each row contains data on one cusp and is of the form label:component-label:M-label:[[a1,a2],[b1,b2]]:bs:v, where

    • label is the label of the surface;
    • component-label is the LMFDB label of the ideal $\mathfrak{b}$ specifying a component of the surface;
    • M-label is the LMFDB label of the ideal $\mathfrak{M}$ associated to the cusp; see equation 3.1.2 in the section Cusps;
    • [[a1,a2],[b1,b2]] indicate the coordinates of the cusp $(a_1 + a_2 w : b_1 + b_2 w)$ as points in $\mathbb{P}^1(F)$, where $F = \mathbb{Q}(w)$;
    • bs gives the periodic part of the associated Hirzebruch-Jung continued fraction; equivalently, the self-intersection numbers of the curves in the resolution of the cusp; and
    • v is the number of times bs must be repeated to cover every curve in the resolution.

  • Files with names ending in hs.txt contain data about the Hilbert series of each surface. Here Hilbert series are represented as rational functions in x. Each row is of the form label:num:den, where

    • label is the label of the surface;
    • num is the numerator of the Hilbert series; and
    • den is the denominator of the Hilbert series.