scholarly journals Explicit Calculations of Automorphic Forms for Definite Unitary Groups

2008 ◽  
Vol 11 ◽  
pp. 326-342 ◽  
Author(s):  
David Loeffler
Keyword(s):  
Unitary Group ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Level Structure ◽  
Unitary Groups ◽  
3 Dimensional ◽  
Local Constancy

I give an algorithm for computing the full space of automor-phic forms for definite unitary groups over ℚ, and apply this to calculate the automorphic forms of level G(hat{Z}) and various small weights for an example of a rank 3 unitary group. This leads to some examples of various types of endoscopic lifting from automorphic forms for U1 × U1 × U1 and U1 × U2, and to an example of a non-endoscopic form of weight (3, 3) corresponding to a family of 3-dimensional irreducible ℓ-adic Galois representations. I also compute the 2-adic slopes of some automorphic forms with level structure at 2, giving evidence for the local constancy of the slopes.

scholarly journals On mod local-global compatibility for in the ordinary case

2017 ◽  
Vol 153 (11) ◽  
pp. 2215-2286 ◽  
Author(s):  
Florian Herzig ◽  
Daniel Le ◽  
Stefano Morra
Keyword(s):  
Unitary Group ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Galois Representation ◽  
Number Fields ◽  
Serre's Conjecture ◽  
Ordinary Case ◽  
Serre’S Conjecture ◽  
The Way

Suppose that $F/F^{+}$ is a CM extension of number fields in which the prime $p$ splits completely and every other prime is unramified. Fix a place $w|p$ of $F$. Suppose that $\overline{r}:\operatorname{Gal}(\overline{F}/F)\rightarrow \text{GL}_{3}(\overline{\mathbb{F}}_{p})$ is a continuous irreducible Galois representation such that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ is upper-triangular, maximally non-split, and generic. If $\overline{r}$ is automorphic, and some suitable technical conditions hold, we show that $\overline{r}|_{\operatorname{Gal}(\overline{F}_{w}/F_{w})}$ can be recovered from the $\text{GL}_{3}(F_{w})$-action on a space of mod $p$ automorphic forms on a compact unitary group. On the way we prove results about weights in Serre’s conjecture for $\overline{r}$, show the existence of an ordinary lifting of $\overline{r}$, and prove the freeness of certain Taylor–Wiles patched modules in this context. We also show the existence of many Galois representations $\overline{r}$ to which our main theorem applies.

scholarly journals ON A CERTAIN LOCAL IDENTITY FOR LAPID–MAO'S CONJECTURE AND FORMAL DEGREE CONJECTURE : EVEN UNITARY GROUP CASE

2021 ◽  
pp. 1-55
Author(s):  
Kazuki Morimoto
Keyword(s):  
Explicit Formula ◽  
Unitary Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Unitary Groups ◽  
Reductive Groups ◽  
Local Identity ◽  
Metaplectic Groups ◽  
Group Case ◽  
Formal Degree

Abstract Lapid and Mao formulated a conjecture on an explicit formula of Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups and metaplectic groups as an analogue of the Ichino–Ikeda conjecture. They also showed that this conjecture is reduced to a certain local identity in the case of unitary groups. In this article, we study the even unitary-group case. Indeed, we prove this local identity over p-adic fields. Further, we prove an equivalence between this local identity and a refined formal degree conjecture over any local field of characteristic zero. As a consequence, we prove a refined formal degree conjecture over p-adic fields and get an explicit formula of Whittaker–Fourier coefficients under certain assumptions.

scholarly journals Commuting Involution Graphs for $3$-Dimensional Unitary Groups

10.37236/590 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Alistaire Everett
Keyword(s):  
Conjugacy Class ◽  
Unitary Group ◽  
Unitary Groups ◽  
Special Unitary Group ◽  
3 Dimensional ◽  
Commuting Graph ◽  
Vertex Set

For a group $G$ and $X$ a subset of $G$ the commuting graph of $G$ on $X$, denoted by $\cal{C}(G,X)$, is the graph whose vertex set is $X$ with $x,y\in X$ joined by an edge if $x\neq y$ and $x$ and $y$ commute. If the elements in $X$ are involutions, then $\cal{C}(G,X)$ is called a commuting involution graph. This paper studies $\cal{C}(G,X)$ when $G$ is a 3-dimensional projective special unitary group and $X$ a $G$-conjugacy class of involutions, determining the diameters and structure of the discs of these graphs.

scholarly journals Arithmetic gauge theory: A brief introduction

2018 ◽  
Vol 33 (29) ◽  
pp. 1830012 ◽  
Author(s):  
Minhyong Kim
Keyword(s):  
Moduli Spaces ◽  
Automorphic Forms ◽  
Galois Representations ◽  
Arithmetic Geometry ◽  
Quantum Field ◽  
Principal Bundles ◽  
Langlands Program ◽  
Effective Version ◽  
Diophantine Geometry ◽  
Higher Genus

Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles holds the key to an effective version of Faltings’ theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands program. In this paper, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory.

scholarly journals On some dimension formula for automorphic forms of weight one I

1982 ◽  
Vol 85 ◽  
pp. 213-221 ◽  
Author(s):  
Toyokazu Hiramatsu
Keyword(s):  
Fuchsian Group ◽  
Automorphic Forms ◽  
Galois Representations ◽  
The Other ◽  
Dimension Formula ◽  
Linearly Independent ◽  
Other Hand ◽  
Algebraic Properties ◽  
Lecture Notes ◽  
Weight One

Let Γ be a fuchsian group of the first kind not containing the element . We shall denote by d0 the number of linearly independent automorphic forms of weight 1 for Γ. It would be interesting to have a certain formula for d0. But, Hejhal said in his Lecture Notes 548, it is impossible to calculate d0 using only the basic algebraic properties of Γ. On the other hand, Serre has given such a formula of d0 recently in a paper delivered at the Durham symposium ([7]). His formula is closely connected with 2-dimensional Galois representations.

scholarly journals Fast Constructive Recognition of Black-Box Unitary Groups

2003 ◽  
Vol 6 ◽  
pp. 162-197 ◽  
Author(s):  
Peter A. Brooksbank
Keyword(s):  
Homomorphic Image ◽  
Polynomial Time ◽  
Unitary Group ◽  
Black Box ◽  
Unitary Groups ◽  
Discrete Logarithms ◽  
Cyclic Groups

AbstractIn this paper, the author presents a new algorithm to recognise, constructively, when a given black-box group is a homomorphic image of the unitary group SU(d, q) for known d and q. The algorithm runs in polynomial time, assuming the existence of oracles for handling SL(2, q) subgroups, and for computing discrete logarithms in cyclic groups of order q ± 1.

scholarly journals Galois representations attached to automorphic forms on over fields

2014 ◽  
Vol 150 (4) ◽  
pp. 523-567 ◽  
Author(s):  
Chung Pang Mok
Keyword(s):  
Automorphic Forms ◽  
Galois Representations ◽  
Suitable Condition ◽  
Two Dimensional ◽  
Automorphic Representations ◽  
Cuspidal Automorphic Representations ◽  
Central Characters

AbstractIn this paper we generalize the work of Harris–Soudry–Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on ${\rm GL}_2$ over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semi-simplification.

scholarly journals Commuting elements in central products of special unitary groups

2012 ◽  
Vol 56 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alejandro Adem ◽  
F. R. Cohen ◽  
José Manuel Gómez
Keyword(s):  
Prime Number ◽  
Moduli Space ◽  
Unitary Group ◽  
Unitary Groups ◽  
Connected Components ◽  
Flat Connections ◽  
Isomorphism Classes ◽  
Central Product ◽  
Central Products ◽  
Path Connected

AbstractWe study the space of commuting elements in the central product Gm,p of m copies of the special unitary group SU(p), where p is a prime number. In particular, a computation for the number of path-connected components of these spaces is given and the geometry of the moduli space Rep(ℤn, Gm,p) of isomorphism classes of flat connections on principal Gm,p-bundles over the n-torus is completely described for all values of n, m and p.

scholarly journals LOCAL BORCHERDS PRODUCTS FOR UNITARY GROUPS

2017 ◽  
Vol 234 ◽  
pp. 139-169
Author(s):  
ERIC HOFMANN
Keyword(s):  
Boundary Component ◽  
Unitary Group ◽  
Picard Group ◽  
Unitary Groups ◽  
Weil Representation ◽  
Cusp Forms ◽  
Modular Variety ◽  
Arithmetic Subgroup ◽  
Torsion Element ◽  
Vector Valued

For the modular variety attached to an arithmetic subgroup of an indefinite unitary group of signature $(1,n+1)$, with $n\geqslant 1$, we study Heegner divisors in the local Picard group over a boundary component of a compactification. For this purpose, we introduce local Borcherds products. We obtain a precise criterion for local Heegner divisors to be torsion elements in the Picard group, and further, as an application, we show that the obstructions to a local Heegner divisor being a torsion element can be described by certain spaces of vector-valued elliptic cusp forms, transforming under a Weil representation.

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