Local Invariants

  • Gerard van der Geer
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 16)


This chapter deals with some local invariants and properties of the cusps and quotient singularities. Since a cusp is associated to a purely arithmetic object, one naturally expects that the geometric invariants of the cusp singularity and its resolution correspond to arithmetic invariants. We shall see that the contribution χ (Mσ, Vσ) of a cusp σ to the arithmetic genus is essentially equal to the value at s = 0 of an L-series \( {L_{{{M_{\sigma }},\;{V_{\sigma }}}}}(s) \) associated to (Mσ, Vσ). In the quadratic case that can be checked directly since Meyer expressed such special L-values in terms of Dedekind sums, but for higher degree fields it turned out to be quite difficult to prove such identities. The relation between χ(Mσ, Vσ) and \( {L_{{{M_{\sigma }},\;{V_{\sigma }}}}}(0) \) stimulated Zagier and Shintani to compute the values at negative integers of L-series of totally real algebraic number fields directly using the cone decompositions occurring in the resolution of cusp singularities (the first in the quadratic case, the latter in the general case). It remains an interesting question to express the values at negative integers as geometric invariants.


Euler Number Local Invariant Geometric Invariant Arithmetic Genus Quotient Singularity 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Gerard van der Geer
    • 1
  1. 1.Mathematisch InstituutUniversiteit van AmsterdamAmsterdamThe Netherlands

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