Let $ K$ be a field admitting a Galois extension $L$ of degree $n$, denoting the Galois group as $G = \gal(L/K)$. Our focus lies on the space $\sym_K(L)$ of symmetric $K$-bilinear forms on $L$. We establish a decomposition of $\sym_K(L)$ into direct sum of $K$-subspaces $A^{σ_i}$, where $σ_i \in G$. Notably, these subspaces $ A^{σ_i}$ exhibit nice constant rank properties. The central contribution of this paper is a decomposition theorem for $\sym_K(L)$, revealing a direct sum of $\frac{(n+1)}{2}$ constant rank $n$-subspaces, each having dimension of $n$. This holds particularly when $G$ is cyclic, represented as $G = \gal(L/K) = \langleσ\rangle$. For cyclic extensions of even degree $n = 2m$, we present slightly less precise but analogous results. In this scenario, we enhance and enrich these constant results and show that, the component $ A^σ$ often decomposes directly into a constant rank subspaces. Remarkably, this decomposition is universally valid when $-1 \notin L^{2}$. Consequently, we derive a decomposition of $\sym_K(L)$ into subspaces of constant rank under several situations. Moreover, leveraging these decompositions, we investigate the maximum dimension of an $n$-subspace inside $M(n,K)$ and $S(n,K)$ for various field $K$ where $M(n,K)$ and $ S(n,K)$ denote the vector spaces $(n \times n)$ matrices and symmetric matrices over $K$, respectively.
We say that a field extension $L/F$ has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over $F$ that become isometric (resp. similar) over $L$ are already isometric (resp. similar) over $F$. The famous Artin-Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadr…
We investigate some Galois groups of linearized polynomials over fields such as $\mathbb{F}_q(t)$. The space of roots of such a polynomial is a module for its Galois group. We present a realization of the symmetric powers of this module, as a subspace of the splitting field of another linearized polynomial.
Given an irreducible bivariate polynomial $f(t,x)\in \mathbb{Q}[t,x]$, what groups $H$ appear as the Galois group of $f(t_0,x)$ for infinitely many $t_0\in \mathbb{Q}$? How often does a group $H$ as above appear as the Galois group of $f(t_0,x)$, $t_0\in \mathbb{Q}$? We give an answer for $f$ of large $x$-degree with alternating or symmetric Galois group over $\mathbb{Q}(t)$. This is done by determining the low genus subcovers of coverings $\tilde{X}\rightarrow \mathbb{P}^1_{\mathbb{C}}$ with alternating or symmetric monodromy groups.
The paper is concerned with the following version of Hilbert's irreducibility theorem: if $π: X \to Y$ is a Galois $G$-covering of varieties over a number field $k$ and $H \subset G$ is a subgroup, then for all sufficiently large and sufficiently divisible $n$ there exist a degree $n$ closed point $y \in |Y|$ and $x \in π^{-1}(y)$ for which $k(x)/k(y)$ is a Galois $H$-extension, and $k(y)/k$ is an $S_n$-extension. The result has interesting corollaries when applied to moduli spaces of various kinds. For instance, for every finite group $G$ there is a constant $N$ such that for all $n>N$ there …
We consider distinction of representations in the context of $p$-adic Galois symmetric spaces. We provide new sufficient conditions for distinction of parabolically induced representations in terms of similar conditions on the inducing data and deduce a characterization for distinction of representations parabolically induced from cuspidal. We explicate the results further for classical groups and give several applications, in particular, concerning the preservation of distinction via Langlands functoriality. We relate our results with a conjecture of Dipendra Prasad.
For an abelian number field of odd degree, we study the structure of its 2-Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow class groups in families where the degree and Galois group are fixed.
Let f(x) = x^n + (a[n-1] t + b[n-1]) x^(n-1) + ... + (a[0] t + b[0]) be of constant degree n in x and degree <= 1 in t, where all a[i],b[i] are randomly and uniformly selected from a finite field GF(q) of q elements. Then the probability that the Galois group of f over the rational function field GF(q)(t) is the symmetric group S(n) on n elements is 1 - O(1/q). Furthermore, the probability that the Galois group of f(x) over GF(q)(t) is not S(n) is >= 1/q for n >= 3 and > 1/q - 1/(2q^2) for n = 2.
In this paper, we shall determine the exact number of Hopf-Galois structures on a Galois $S_n$-extension, where $S_n$ denotes the symmetric group on $n$ letters.
A proof of the main theorem of the Galois theory is presented using the main theorem of symmetric polynomials. The idea originated from studying the "Mémoire sur les conditions de résolubilité des équations par radicaux" of Evariste Galois. The motto "Read the masters" pays off.
Galois representations attached to moments of Kloosterman sums and conjectures of Evans
有限域的Kloosterman和是定义在$\Gm$上的某些局部系统的Frobenius迹函数。Kloosterman和的矩计算张量幂（或对称幂，外幂等）上同调的Frobenius迹。这些地方系统。我们表明，当$p$范围在所有素数，相应的Kloosterman和$\mathbb{F}_{p}$的时刻出现作为Frobenius迹上的连续$\ell$ -adic伽罗瓦表示，来自几何。我们还给出了这些伽罗瓦表示的分歧的界限。所有这一切都是在一般性的Kloosterman层附加到还原集团引入\cite{HNY}。作为应用，我们给出了R. Evans（\cite{Evans0}，\cite{Evans}）用显式模形式的傅里叶系数表示经典Kloosterman和的七次和八次对称幂矩.第八对称幂矩猜想的证明依赖于C. D. C. D.
We consider $p$-adic families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zer…
Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$ regular. Our examples include abelian groups, dihedral groups, symmetric groups, general linear groups over finite fields, etc. We also provide a similar conclusion while specializing any given infinitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$ regular of a certain type, under a conjectural "uniform Faltings" theorem".
Let $π$ be a cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbf{A_Q})$, whose archimedean component is a holomorphic discrete series or limit of discrete series representation. If $π$ is not CAP or endoscopic, then we show that its associated $\ell$-adic Galois representations are irreducible and crystalline for $100\%$ of primes $\ell$. If, moreover, $π$ is neither an automorphic induction nor a symmetric cube lift, then we show that, for $100\%$ of primes $\ell$, the image of its mod $\ell$ Galois representation contains $\mathrm{Sp}_4(\mathbf{F}_{\ell})$.
Galois-Orbit Structure, Ramanujan Sums, and Stable-Range Collapse for Cyclotomic Cosine Formulas
Departamento de Matemáticas, Sede Medellín, Universidad Nacional de Colombia（哥伦比亚国立大学麦德林校区数学系）；Departamento de Matemáticas, Universidad EAFIT（伊法蒂大学数学系）
We construct a compatible family of global cohomology classes (an Euler system) for the symmetric square of a modular form, and apply this to bounding Selmer groups of the symmetric square Galois representation and its twists.
