Cycles and Harmonic Forms on Locally Symmetric Spaces

1985 ◽  
Vol 28 (1) ◽  
pp. 3-38 ◽  
Author(s):  
John J. Millson
Keyword(s):  
Quadratic Forms ◽  
Symmetric Spaces ◽  
Number Fields ◽  
The Other ◽  
Weil Representation ◽  
Harmonic Forms ◽  
Congruence Subgroups ◽  
Locally Symmetric Spaces ◽  
Background Material ◽  
Totally Real

AbstractTwo constructions of cohomology classes for congruence subgroups of unit groups of quadratic forms over totally real number fields are given and shown to coincide. One is geometric, using cycles, and the other is analytic, using the oscillator (Weil) representation. Considerable background material on this representation is given.

scholarly journals There are no universal ternary quadratic forms over biquadratic fields

2020 ◽  
Vol 63 (3) ◽  
pp. 861-912 ◽  
Author(s):  
Jakub Krásenský ◽  
Magdaléna Tinková ◽  
Kristýna Zemková
Keyword(s):  
Quadratic Forms ◽  
Number Fields ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Mild Conditions ◽  
Positive Unit ◽  
Positive Elements ◽  
Totally Positive ◽  
Biquadratic Fields ◽  
Totally Real

AbstractWe study totally positive definite quadratic forms over the ring of integers $\mathcal {O}_K$ of a totally real biquadratic field $K=\mathbb {Q}(\sqrt {m}, \sqrt {s})$. We restrict our attention to classic forms (i.e. those with all non-diagonal coefficients in $2\mathcal {O}_K$) and prove that no such forms in three variables are universal (i.e. represent all totally positive elements of $\mathcal {O}_K$). Moreover, we show the same result for totally real number fields containing at least one non-square totally positive unit and satisfying some other mild conditions. These results provide further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal {O}_K$; we prove several new results about their properties.

scholarly journals On the image of complex conjugation in certain Galois representations

2016 ◽  
Vol 152 (7) ◽  
pp. 1476-1488 ◽  
Author(s):  
Ana Caraiani ◽  
Bao V. Le Hung
Keyword(s):  
Symmetric Spaces ◽  
Galois Representations ◽  
Real Field ◽  
Complex Conjugation ◽  
Automorphic Representations ◽  
Locally Symmetric Spaces ◽  
Cuspidal Automorphic Representations ◽  
Totally Real ◽  
Totally Real Field

We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $\text{GL}_{n}$ over a totally real field $F$.

scholarly journals Brauer equivalent number fields and the geometry of quaternionic Shimura varieties

2018 ◽  
Vol 70 (2) ◽  
pp. 675-687
Author(s):  
Benjamin Linowitz
Keyword(s):  
Symmetric Spaces ◽  
Number Fields ◽  
Shimura Varieties ◽  
Brauer Groups ◽  
Locally Symmetric Spaces ◽  
Totally Geodesic ◽  
Equivalent Number ◽  
Geodesic Surfaces

Abstract Two number fields are said to be Brauer equivalent if there is an isomorphism between their Brauer groups that commutes with restriction. In this paper, we prove a variety of number theoretic results about Brauer equivalent number fields (for example, they must have the same signature). These results are then applied to the geometry of certain arithmetic locally symmetric spaces. As an example, we construct incommensurable arithmetic locally symmetric spaces containing exactly the same set of proper immersed totally geodesic surfaces.

Tubes, Cohomology with Growth Conditions and an Application to the Theta Correspondence

1988 ◽  
Vol 40 (1) ◽  
pp. 1-37 ◽  
Author(s):  
Stephen S. Kudla ◽  
John J. Millson
Keyword(s):  
Finite Volume ◽  
Symmetric Spaces ◽  
Integral Formula ◽  
Growth Conditions ◽  
Harmonic Forms ◽  
Theta Correspondence ◽  
Locally Symmetric Spaces ◽  
Spinor Norm ◽  
Integer Coefficients ◽  
Locally Symmetric Space

In this paper we continue our effort [11], [12], [13], [14] to interpret geometrically the harmonic forms on certain locally symmetric spaces constructed by using the theta correspondence. The point of this paper is to prove an integral formula, Theorem 2.1, which will allow us to generalize the results obtained in the above papers to the finite volume case (the previous papers treated only the compact case). We then apply our integral formula to certain finite volume quotients of symmetric spaces of orthogonal groups. The main result obtained is Theorem 4.2 which is described below. We let (,) denote the bilinear form associated to a quadratic form with integer coefficients of signature (p, q). We assume that the fundamental group Γ ⊂ SO(p, q) of our locally symmetric space is the subgroup of the integral isometries of (,) congruent to the identity matrix modulo some integer N. We assume that N is chosen large enough so that Γ is neat (the multiplicative subgroup of C* generated by the eigenvalues of the elements of Γ has no torsion), Borel [2], 17.1 and that every element in Γ has spinor norm 1, Millson-Raghunathan [15], Proposition 4.1. These conditions are needed to ensure that our cycles Cx (see below) are orientable. The methods we will use apply also to unitary and quaternion unitary locally symmetric spaces, see [13].

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