Hecke Algebra Isomorphisms and Adelic Points on Algebraic Groups

  • Gunther Cornelissen

    Mathematisch Instituut, Universiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, Nederland
  • Valentijn Karemaker

    Department of Mathematics, University of Pennsylvania, David Rittenhouse Labs, 209 South 33rd Street, Philadelphia, PA 19104-6395, U.S.A.
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Let GG denote a linear algebraic group over Q\bold{Q} and KK and LL two number fields. We establish conditions on the group GG, related to the structure of its Borel groups, under which the existence of a group isomorphism G(AK,f)G(AL,f)G(\bold{A}_{K,f}) \cong G(\bold{A}_{L,f}) over the finite adeles implies that KK and LL have isomorphic adele rings. Furthermore, if GG satisfies these conditions, KK or LL is a Galois extension of Q\bold{Q}, and G(AK,f)G(AL,f)G(\bold{A}_{K,f}) \cong G(\bold{A}_{L,f}), then KK and LL are isomorphic as fields. We use this result to show that if for two number fields KK and LL that are Galois over Q\bold{Q}, the finite Hecke algebras for GL(n)\mathrm{GL}(n) (for fixed n2n \geq 2) are isomorphic by an isometry for the L1L^1-norm, then the fields KK and LL are isomorphic. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field, if it is Galois over Q\bold{Q}.

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Gunther Cornelissen, Valentijn Karemaker, Hecke Algebra Isomorphisms and Adelic Points on Algebraic Groups. Doc. Math. 22 (2017), pp. 851–871

DOI 10.4171/DM/580