We generalize the first part of A. Connes paper (math/9811068) on the zeroes of the Riemann zeta function from a number field $k$ to any simple algebra $M$ over $k$. To a given automorphic representation $π$ of the reductive group $M^\times$ of invertible elements of $M$ we find a Hilbert space $H_π$ and an operator $D_π$ (Polya-Hilbert operator), which is the infinitesimal generator of a canonical flow such that the spectrum of $D_π$ coincides with the purely imaginary zeroes of the function $L(π,\rez{2} +z)$. As a byproduct we get holomorphicity of all automorphic $L$-functions, not only the cuspidal ones.
We prove an automorphic analogue of Deligne's conjecture for symmetric fourth $L$-functions of Hilbert modular forms. We extend the result of Morimoto based on generalization and refinement of the results of Grobner and Lin to cohomological irreducible essentially conjugate self-dual cuspidal automorphic representations of ${\rm GL}_2$ and ${\rm GL}_3$ over CM-fields.
In this paper, we prove Deligne's conjecture for symmetric sixth $L$-functions of Hilbert modular forms. We extend the result of Morimoto based on a different approach. We define automorphic periods associated to globally generic $C$-algebraic cuspidal automorphic representations of ${\rm GSp}_4$ over totally real number fields whose archimedean components are (limits of) discrete series representations. We show that the algebraicity of critical $L$-values for ${\rm GSp}_4 \times {\rm GL}_2$ can be expressed in terms of these periods. In the case of Kim-Ramakrishnan-Shahidi lifts of ${\rm GL}_…
We prove vanishing results for the $0$th cohomology of automorphic line bundles on the Ekedahl--Oort strata of the special fiber of Hilbert--Blumenthal Shimura varieties. This work vastly extends to all Ekedahl--Oort strata previous results of Diamond--Kassaei and Goldring--Koskivirta. Furthermore, our results also cover the case of unitary Shimura varieties of rank $\leq 2$ attached to an arbitrary CM extension. We also prove a conjecture of Goldring--Koskivirta on the set of possible weights of automorphic forms.
In this paper, we decompose the space of nearly holomorphic Hilbert-Siegel automorphic forms as representations of the adele group under certain assumptions. We also give an application for classical holomorphic Hilbert-Siegel modular forms. In particular, we show the surjectivity of the global Siegel operator for certain congruence subgroups with large weights.
We prove a strong form of the trivial zero conjecture at the central point for the $p$-adic $L$-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of $\mathrm{GL}_2$ over a totally real field, which is Iwahori spherical at places above $p$. In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the $p$-adic $L$-function of a nearly finite slope family and on improved $p$-adic $L$-functions that we construct using automorphic symbols and overconvergent cohomology. For higher order zeros we d…
On Hilbert-Polya conjecture: Hermitian operator naturally associated to L-functions
Using as starting point a classical integral representation of a L-function we define a familly of two variables extended functions which are eigenfunctions of a Hermitian operator (having imaginary part of zeros as eigenvalues). This Hermitian operator can take also other forms, more symetric. In the case of particular L-functions, like Zeta function or Dirichlet L-functions, the eigenfunctions defined for this operator have symmetry properties. Moreover, for s zero fo Zeta function (or Dirichlet L-function), the associated eigenfunction has a specific property (a part of eigenfunction is can…
Let $\mathitΠ$ be a cohomological irreducible cuspidal automorphic representation of ${\rm GL}_2(\mathbb{A}_{\mathbb F})$ with central character $ω_{\mathitΠ}$ over a totally real number field ${\mathbb F}$. In this paper, we prove the algebraicity of the near central non-critical value of the symmetric fourth $L$-function of $\mathitΠ$ twisted by $ω_{\mathitΠ}^{-2}$. The algebraicity is expressed in terms of the Petersson norm of the normalized newform of $\mathitΠ$ and the top degree Whittaker period of the Gelbart-Jacquet lift ${\rm Sym}^2\mathitΠ$ of $\mathitΠ$.
Towards a fractal cohomology: Spectra of Polya--Hilbert operators, regularized determinants and Riemann zeros
Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis . This and other conjectures about these zeta functions would come to be called the Weil conjectures, which were proved by Weil for curves and later, by Deligne for varieties over finite fields . Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, which uses the Frobenius operator on a finite field . The zeta function is then expressed…
Siegel modular forms of degree two attached to Hilbert modular forms
设E/Q是实二次域，pi_0是GL（2，A_E）的尖点、不可约、自守表示，具有平凡中心特征标和无穷型（2，2 n +2），其中n是非负整数。我们证明了存在一个非零Siegel仿模新形式F，它的权、水平、Hecke特征值、λ因子和L-函数都由pi_0显式确定.对于每个有理素数p，我们用pi_0的不变量将这些不变量列表。
Towards the Hilbert-Pólya conjecture, in this paper, we present a general construction of Hamiltonian $\hat{H}_{f},$ which leads a general family of Hurwitz zeta functions $(-1)^{z_{n}-1}L(f,z_{n},x+1)$ defined by Mellin transform becomes their eigenstates under a suitable boundary condition, and the eigenvalues $E_{n}$ have the property that $z_{n}=\frac{1}{2}(1-iE_{n})$ are the zeros of a general family of zeta functions $L(f,z)\equiv L(f,z,1)$.
