Relative Kloosterman Integrals For Gl(3): III

1993 ◽  
Vol 45 (6) ◽  
pp. 1211-1230 ◽  
Author(s):  
Zhengyu Mao
Keyword(s):  
Symmetric Space ◽  
Number Field ◽  
Trace Formula ◽  
Unitary Group ◽  
Quadratic Extension

AbstractLet E be a quadratic extension of a number field F with Galois conjugation σ, G’ the quasi-split unitary group in three variables, G the group GL(3, E). We let S be the space of the matrices s in G such that σ(s)s = e. One conjectures a comparison identity between the relative Kuznietsov trace formula for the symmetric space S and the ordinary Kuznietsov trace formula for the group G’ (See [10]). We prove the corresponding “fundamental lemmas”.

On a Generalization of a Result of Waldspurger

1996 ◽  
Vol 48 (1) ◽  
pp. 105-142 ◽  
Author(s):  
Jiandong Guo
Keyword(s):  
Number Field ◽  
Trace Formula ◽  
Strong Evidence ◽  
Riemann Hypothesis ◽  
Symmetric Spaces ◽  
Quadratic Extension ◽  
Automorphic Representation ◽  
Cuspidal Representation ◽  
Orbital Integrals ◽  
Langlands Correspondence

AbstractWe consider a generalization of a trace formula identity of Jacquet, in the context of the symmetric spaces GL(2n)/GL(/n) × GL(n) and G′/H′. Here G′ is an inner form of GL(2n) over F with a subgroup H′ isomorphic to GL(n, E) where E/F is a quadratic extension of number field attached to a quadratic idele class character η of F. A consequence of this identity would be the following conjecture: Let π be an automorphic cuspidal representation of GL(2n). If there exists an automorphic representation π′ of G′ which is related to π by the Jacquet-Langlands correspondence, and a vector ø in the space of π′ whose integral over H′ is nonzero, then both L(1/2, π) and L(1/2,π ⊗ η) are nonvanishing. Moreover, we have L(1/2, π)L(1/2, π ⊗ η) > 0. Here the nonvanishing part of the conjecture is a generalization of a result of Waldspurger for GL(2) and the nonnegativity of the product is predicted from the generalized Riemann Hypothesis. In this article, we study the corresponding local orbital integrals for the symmetric spaces. We prove the "fundamental lemma for the unit Hecke functions" which says that unit Hecke functions have "matching" orbital integrals. This serves as the first step toward establishing the trace formula identity and in the same time it provides strong evidence for what we proposed.

scholarly journals Higher genus theory

2020 ◽  
Author(s):  
Peter Koymans ◽  
Carlo Pagano
Keyword(s):  
Number Field ◽  
Quadratic Forms ◽  
Class Group ◽  
Quadratic Extension ◽  
Number Fields ◽  
Explicit Description ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Genus Theory ◽  
Narrow Class

Abstract In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.

scholarly journals On the Lefschetz trace formula for Lubin–Tate spaces

2016 ◽  
Vol 16 (1) ◽  
Author(s):  
Xu Shen
Keyword(s):  
Trace Formula ◽  
Unitary Group ◽  
Locally Finite ◽  
Lefschetz Trace Formula ◽  
Cell Decompositions ◽  
Finite Cell

AbstractWe give a new proof of the Lefschetz trace formula for Lubin-Tate spaces. Our proof is based on the locally finite cell decompositions of these spaces and on Mieda’s version of the Lefschetz trace formula for certain open adic spaces. This proof is rather different from the proofs of Strauch and Mieda, and it might be generalized to other Rapoport-Zink spaces as soon as there exist suitable cell decompositions. For example, in another paper we have proved a Lefschetz trace formula for some unitary group Rapoport-Zink spaces by using similar ideas as here.

The Weil Character of the Unitary Group Associated to a Finite Local Ring

2002 ◽  
Vol 54 (6) ◽  
pp. 1229-1253 ◽  
Author(s):  
Roderick Gow ◽  
Fernando Szechtman
Keyword(s):  
Local Ring ◽  
Unitary Group ◽  
Quadratic Extension ◽  
Weil Representation ◽  
Finite Local Ring

AbstractLetR/R be a quadratic extension of finite, commutative, local and principal rings of odd characteristic. Denote byUn(R) the unitary group of ranknassociated toR/R. The Weil representation ofUn(R) is defined and its character is explicitly computed.

Regular Elliptic Classes and the Stable Relative Trace Formula

1992 ◽  
Vol 35 (2) ◽  
pp. 230-236
Author(s):  
K. F. Lai
Keyword(s):  
Number Field ◽  
Trace Formula ◽  
Algebraic Number ◽  
Reductive Group ◽  
Algebraic Number Field ◽  
Double Cosets ◽  
Relative Trace Formula

AbstractWe study the relative trace formula of a reductive group over an algebraic number field. Following Langlands we stabilize the geometric side of the relative trace formula contributed by the elliptic regular double cosets.

Poles of Siegel Eisenstein Series on U(n, n)

1999 ◽  
Vol 51 (1) ◽  
pp. 164-175 ◽  
Author(s):  
Victor Tan
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Complex Plane ◽  
Unitary Group

AbstractLet U(n, n) be the rank n quasi-split unitary group over a number field. We show that the normalized Siegel Eisenstein series of U(n, n) has at most simple poles at the integers or half integers in certain strip of the complex plane.

scholarly journals UNITARY GROUPS OVER LOCAL RINGS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350093 ◽  
Author(s):  
J. CRUICKSHANK ◽  
A. HERMAN ◽  
R. QUINLAN ◽  
F. SZECHTMAN
Keyword(s):  
Local Ring ◽  
Unitary Group ◽  
Quadratic Extension ◽  
Unitary Groups ◽  
Weil Representation ◽  
Local Rings ◽  
Hermitian Forms ◽  
Finite Local Ring

We study hermitian forms and unitary groups defined over a local ring, not necessarily commutative, equipped with an involution. When the ring is finite we obtain formulae for the order of the unitary groups as well as their point stabilizers, and use these to compute the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring.

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