Related papers: Root Number Equidistribution for Self-Dual Automor…

Let $G=\SL(2,\R)\ltimes(\R^2)^{k}$, let $\Gamma$ be a congruence subgroup of $\SL(2,\Z)\ltimes(\Z^2)^{k}$, and let $u_{\R}=(u_x)_{x\in\R}$ be the one-parameter subgroup of $G$ given by $u_x=\left(\matr 1x01,0\right)$. We prove polynomially…

Suppose that R is an ordered ring, G_n(R) is a subsemigroup of $GL_n(R)$, consisting of all matrices with nonnegative elements. A.V. Mikhalev and M.A. Shatalova described all automorphisms of G_n(R), where R is a linearly ordered skewfield…

We prove that the Galois equidistribution of torsion points of the algebraic torus $\mathbb{G}_{m}^d$ extends to the singular test functions of the form $\log{|P|}$, where $P$ is a Laurent polynomial having algebraic coefficients that…

Let $\pi$ be a cuspidal automorphic representation for $\mathrm{GL}(n)$ over a number field. We establish a conditional upper bound on the number of cuspidal isobaric summands in the symmetric $k$-th power lift of $\pi$, assuming that the…

Let $F$ be a local non-archimedian field of odd residue characteristic and let $G=PGL(2)$. In this paper we study an analog of irreducible cuspidal representations of the group $G(F)$ when $F$ is replaced by the field $K=F((t))$. The story…

We study the specializations $\mathcal{L}_{g,n}^\epsilon$ at roots of unity $\epsilon$ of odd order of the graph algebras, associated to a simply-connected complex semi-simple algebraic group $G$ and a compact oriented surface…

We prove that over totally real fields, the $p$-adic Galois representations attached to non-self-dual regular algebraic cuspidal automorphic representations of $\mathrm{GL}(4)$ are irreducible. We then develop the theory of extra-twists in…

Given two elliptic curves $E_1$ and $E_2$ defined over the field of rational numbers, $\mathbb{Q}$, with good reduction at an odd prime $p$ and equivalent mod $p$ Galois representation, we compare the $p$-Selmer rank, global and local root…

We find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as $\mathrm{GSp}_2(\mathbb{A})$, $\mathrm{SO}(4,3)(\mathbb{\mathbb{A}})$ and $\mathrm{SO}(5,4)(\mathbb{A})$, where the…

We prove the existence of GSpin-valued Galois representations corresponding to cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a…

Let $G$ be the group of $F$-points of a split connected reductive $F$-group over a non-Archimedean local field $F$ of characteristic 0. Let $\pi$ be an irreducible smooth self-dual representation of $G$. The space $W$ of $\pi$ carries a…

We give combinatorial models for complex, smooth, non-spherical, generic, irreducible representations of the group G=PGL(2,F), where F is a non-archimedean locally compact field. They use the graphs X_k lying above the tree of G, introduced…

Let pi be a regular algebraic cuspidal automorphic representation of GL(2) over an imaginary quadratic number field K such that the central character of pi is invariant under the non-trivial automorphism of K. We show that pi is associated…

We obtain nonvanishing estimates for central values of certain self-dual Rankin-Selberg $L$-functions on $\operatorname{GL}_2({\bf{A}}_F) \times \operatorname{GL}_2({\bf{A}}_F)$, and more generally $\operatorname{GL}_r({\bf{A}}_F) \times…

We show that certain paramodular cuspidal automorphic irreducible representations of $\mathrm{GSp}(4,\mathbb{A}_\mathbb{Q})$, which are not CAP, are globally generic. This implies a multiplicity one theorem for paramodular cuspidal…

For a cuspidal automorphic representation \Pi of GL(4,A), H. Kim proved that the exterior square transfer \wedge^2\Pi is an isobaric automorphic representation of GL(6,A). In this paper we characterize those representations \Pi for which…

Given a cuspidal automorphic representation $\pi$ for GL(3) over a number field and a positive integer $k$, assume that the symmetric $m$th power lifts of $\pi$ are isobaric automorphic for $m \leq k$, cuspidal for $m \leq k-1$, and that…

Let $F$ be a non-Archimedean locally compact field and let $D$ be a central division algebra over $F$. Let $\pi_1$ and $\pi_2$ be respectively two smooth irreducible representations of ${\rm GL}(n_1,D)$ and ${\rm GL}(n_2,F)$, $n_1, n_2 \geq…

In this paper, we use type theory to construct a family of depth $\frac{1}{N}$ minimax supercuspidal representations of $\text{GL}(2N, F)$ which we call middle supercuspidal representations. These supercuspidals may be viewed as a natural…

Let $\pi$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${\rm U}_{\mathfrak n}$ defined over a number field $F$. Under the assumption that $\pi$ has a generic global Arthur parameter, we establish the…