Related papers: The Prasad conjectures for $\mathrm{GSp_4}$ and $\…

We study GL_2(F)-invariant periods on representations of GL_2(A), where F is a nonarchimedean local field and A/F a product of field extensions of total degree 3. For irreducible representations, a theorem of Prasad shows that the space of…

This article gives a generalization of the work of Y.Ding in the context of $\mathrm{GSp}_4(\mathbb{Q}_p)$, where $p$ is an odd prime number. Let $\rho$ be a 4-dimensional generic non-critical crystalline representations of the absolute…

Let $G$ be a connected reductive group defined over a finite field $\mathbb{F}_q$ with corresponding Frobenius $F$. Let $\iota_G$ denote the duality involution defined by D. Prasad under the hypothesis $2\mathrm{H}^1(F,Z(G))=0$, where…

For a non-Archimedean locally compact field $F$ of odd residue characteristic and characteristic $0$, we prove a conjecture of D. Prasad predicting that, for an integer $n \geq 1$ and a non-split quaternionic $F$-algebra $D$, a discrete…

Shimura's conjectire (1963) concerns the rationality of the generating series for Hecke operators for the symplectic group of genus g. This conjecture wes proved by Andrianov for arbitrary genus g. For genus g=4, we explicify the rational…

We prove that the number of quartic $S_4$--extensions of the rationals of given discriminant $d$ is $O_\eps(d^{1/2+\eps})$ for all $\eps>0$. For a prime number $p$ we derive that the dimension of the space of octahedral modular forms of…

Let F be a number field, A_F its ring of adeles, and let {\pi}_n and {\pi}_{n+1} be irreducible, cuspidal, automorphic representations of SO_n(A_F) and SO_{n+1}(A_F), respectively. In 1991, Benedict Gross and Dipendra Prasad conjectured the…

In this paper we will explore the solutions to the diophantine equation in the Erd\H{o}s-Straus conjecture. For a prime $p$ we are discussing the relationship between the values $x,y,z \in \mathbb{N}$ so that $$ \frac{4}{p} = \frac{1}{x} +…

Scalar field theory is studied by constructing interacting saddle point expansions in the symmetric and broken phase, respectively. Focusing on analytically tractable saddle expansions, it is found that broken and symmetric phases are…

We use the local theta correspondences between the quaternionic Hermitian groups and the quaternionic skew-Hermitian groups to understand the distinction problem for the symmetric pair SL(2,E)/SL(1,D), where E is a quadratic field extension…

We prove an integral formula computing multiplicities of square-integrable representations relative to Galois pairs over $p$-adic fields and we apply this formula to verify two consequences of a conjecture of Dipendra Prasad. One concerns…

We show that the local Langlands conjecture for $Sp(2n)$ follows from that for $GSp(2n)$. In particular, we prove the local Langlands conjecture for $Sp(4)$, based on our previous work on the local Langlands conjecture for $GSp(4)$. We also…

Let $G$ be a split simply laced group defined over a $p$-adic field $F$. In this paper we study the restriction of the minimal representation of $G$ to various dual pairs in $G$. For example, the restriction of the minimal representation of…

One of the important technical tools in Gaitsgory's proof of the Vanishing Conjecture appearing in the geometric Langlands correspondence ([3]) is the theory of Whittaker functors for GL_n. We define Whittaker functors for GSp_4 and study…

We study representations of $GL_{n}(\mathbb{F}_{q})$ that are distinguished with respect to a symmetric subgroup $H=GL_{n}(\mathbb{F}_{q})^{\sigma}$, where $\sigma$ is an involution. We prove that those representations satisfy $\pi \cong…

We prove the classification of discrete automorphic representations of GSp$_4$ explained in [Art04], as well as a compatibility between the local Langlands correspondences for GSp$_4$ and Sp$_4$ .

Fuglede's conjecture in $\mathbb{Z}_{p}^{d}$, $p$ a prime, says that a subset $E$ tiles $\mathbb{Z}_{p}^{d}$ by translation if and only if $E$ is spectral, meaning any complex-valued function $f$ on $E$ can be written as a linear…

We prove a strengthening of the "reciprocity conjecture" of Khare and Wintenberger. The input to the original conjecture is an odd prime p, a CM number field F containing the pth roots of unity, and a pair of primes of the maximal totally…

We prove that the classical theta function $\theta_4$ may be expressed as $$ \theta_4(v,\tau) = \theta_4(0,\tau) \exp[- \sum_{p\geq 1} \sum_{k\geq 0} \frac {1}{p} \bigg(\frac {\sin \pi v}{(\sin (k+{1/2})\pi \tau)}\bigg)^{2p}].$$ We obtain…

In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual…