• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site
Menu

Working paper

The Borel-Moore homology of an arithmetic quotient of the Bruhat-Tits building of PGL of a non-archimedean local field in positive characteristic and modular symbols

Kondo S., Yasuda S.
We study the homology and the Borel-Moore homology with coefficients in Q of a quotient (called arithmetic quotient) of the Bruhat-Tits building of PGL of a nonarchimedean local field of positive characteristic by an arithmetic subgroup (a special case of the general definition in Harder's article (Invent.\ Math.\ 42, 135-175 (1977)). We define an analogue of modular symbols in this context and show that the image of the canonical map from homology to Borel-Moore homology is contained in the sub Q-vector space generated by the modular symbols. By definition, the limit of the Borel-Moore homology as the arithmetic group becomes small is isomorphic to the space of Q-valued automorphic forms that satisfy certain conditions at a distinguished (fixed) place (namely that it is fixed by the Iwahori subgroup and the center at the place). We show that the limit of the homology with C-coefficients is identified with the subspace consisting of cusp forms. We also describe an irreducible subquotient of the limit of Borel-Moore homology as an induced representation in a precise manner and give a multiplicity one type result.