Remarks on the rightmost critical value of the triple product L-function

2021 ◽  
pp. 1-16
Author(s):  
Kengo Fukunaga ◽  
Kohta Gejima
Keyword(s):  
Critical Value ◽  
Product Formula ◽  
Orthogonal Basis ◽  
Triple Product ◽  
Explicit Formulas ◽  
Arithmetic Expression ◽  
Weighted Averages ◽  
Hecke Eigenforms ◽  
Hecke Eigenform ◽  
The Individual

Let [Formula: see text] be a normalized cuspidal Hecke eigenform. We give explicit formulas for weighted averages of the rightmost critical values of triple product [Formula: see text]-functions [Formula: see text], where [Formula: see text] and [Formula: see text] run over an orthogonal basis of [Formula: see text] consisting of normalized cuspidal Hecke eigenforms. Those explicit formulas provide us an arithmetic expression of the rightmost critical value of the individual triple product [Formula: see text]-functions.

Polynomial identities between Hecke eigenforms

2019 ◽  
Vol 15 (10) ◽  
pp. 2135-2150
Author(s):  
Dianbin Bao
Keyword(s):  
Eisenstein Series ◽  
Product Formula ◽  
Inner Product ◽  
Polynomial Identities ◽  
Number Of Solutions ◽  
Hecke Eigenforms

In this paper, we study solutions to [Formula: see text], where [Formula: see text] are Hecke newforms with respect to [Formula: see text] of weight [Formula: see text] and [Formula: see text]. We show that the number of solutions is finite for all [Formula: see text]. Assuming Maeda’s conjecture, we prove that the Petersson inner product [Formula: see text] is nonzero, where [Formula: see text] and [Formula: see text] are any nonzero cusp eigenforms for [Formula: see text] of weight [Formula: see text] and [Formula: see text], respectively. As a corollary, we obtain that, assuming Maeda’s conjecture, identities between cusp eigenforms for [Formula: see text] of the form [Formula: see text] all are forced by dimension considerations. We also give a proof using polynomial identities between eigenforms that the [Formula: see text]-function is algebraic on zeros of Eisenstein series of weight [Formula: see text].

The tensor product of supersymmetric N = (1,1) spectral data

2018 ◽  
Vol 15 (12) ◽  
pp. 1850207 ◽  
Author(s):  
Satyajit Guin
Keyword(s):  
Quantum Theory ◽  
Tensor Product ◽  
Spectral Data ◽  
Noncommutative Geometry ◽  
Product Formula ◽  
Unitary Connection ◽  
The Individual

We define the notion of tensor product of supersymmetric [Formula: see text] spectral data in the context of supersymmetric quantum theory and noncommutative geometry. We explain in which sense our definition is canonical and also establish its compatibility with the tensor product of [Formula: see text] spectral data defined earlier by Connes. As an application, we show that the unitary connections on the individual [Formula: see text] spectral data give rise to a unitary connection on the product [Formula: see text] spectral data.

2176. On the Vector Triple Product Formula

1950 ◽  
Vol 34 (310) ◽  
pp. 295
Author(s):  
G. J. Whitrow
Keyword(s):  
Product Formula ◽  
Triple Product

scholarly journals Poles of triple product L-functions involving monomial representations

2021 ◽  
pp. 1-8
Author(s):  
Heekyoung Hahn
Keyword(s):  
Tensor Product ◽  
Product Formula ◽  
Triple Product ◽  
Natural Setting ◽  
Automorphic Representations ◽  
Cuspidal Automorphic Representations ◽  
Beyond Endoscopy ◽  
Monomial Representations ◽  
Point Of Reference

In this paper, we study the order of the pole of the triple tensor product [Formula: see text]-functions [Formula: see text] for cuspidal automorphic representations [Formula: see text] of [Formula: see text] in the setting where one of the [Formula: see text] is a monomial representation. In the view of Brauer theory, this is a natural setting to consider. The results provided in this paper give crucial examples that can be used as a point of reference for Langlands’ beyond endoscopy proposal.

scholarly journals On the Central Critical Value of the Triple Product L–Function

Number Theory ◽  
1996 ◽  
pp. 1-46 ◽  
Author(s):  
S. Böcherer ◽  
R. Schulze–Pillot
Keyword(s):  
Critical Value ◽  
Triple Product

Modifications of the Classical Notion of Panum's Fusional Area

Perception ◽  
1980 ◽  
Vol 9 (6) ◽  
pp. 671-682 ◽  
Author(s):  
Peter Burt ◽  
Bela Julesz
Keyword(s):  
Visual Field ◽  
Angular Separation ◽  
Critical Value ◽  
Binocular Fusion ◽  
Classical Notion ◽  
Disparity Gradient ◽  
The Absolute ◽  
The Difference ◽  
The Individual ◽  
Limiting Case

It is generally believed that there is an absolute disparity limit for binocular fusion; objects with disparities within this limit, known as Panum's fusional area, will appear fused and single, while objects with disparities outside the limit appear double. It is demonstrated, however, that the disparity gradient, rather than the disparity magnitude, dictates binocular fusion when several objects occur near one another in the visual field. The disparity gradient is defined as the difference between the disparities of neighboring objects divided by their angular separation. If this ratio exceeds a critical value (∼1) then fusion does not occur, even though the absolute disparities of the individual objects may be well within the classical Panum's area. This discovery leads to the reinterpretation of several enigmatic phenomena in stereopsis, including Panum's limiting case.

The Bombieri–Vinogradov Theorem on Higher Rank Groups and its Applications

2019 ◽  
Vol 72 (4) ◽  
pp. 928-966
Author(s):  
Yujiao Jiang ◽  
Guangshi Lü
Keyword(s):  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Maass Form ◽  
Hecke Eigenforms ◽  
Hecke Eigenform ◽  
Symmetric Square ◽  
Ramanujan Conjecture ◽  
Higher Rank ◽  
Image Position ◽  
Holomorphic Hecke Eigenforms

AbstractWe study the analogue of the Bombieri–Vinogradov theorem for $\operatorname{SL}_{m}(\mathbb{Z})$ Hecke–Maass form $F(z)$. In particular, for $\operatorname{SL}_{2}(\mathbb{Z})$ holomorphic or Maass Hecke eigenforms, symmetric-square lifts of holomorphic Hecke eigenforms on $\operatorname{SL}_{2}(\mathbb{Z})$, and $\operatorname{SL}_{3}(\mathbb{Z})$ Maass Hecke eigenforms under the Ramanujan conjecture, the levels of distribution are all equal to $1/2,$ which is as strong as the Bombieri–Vinogradov theorem. As an application, we study an automorphic version of Titchmarch’s divisor problem; namely for $a\neq 0,$$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D70C}(n)d(n-a)\ll x\log \log x,\end{eqnarray}$$ where $\unicode[STIX]{x1D70C}(n)$ are Fourier coefficients $\unicode[STIX]{x1D706}_{f}(n)$ of a holomorphic Hecke eigenform $f$ for $\operatorname{SL}_{2}(\mathbb{Z})$ or Fourier coefficients $A_{F}(n,1)$ of its symmetric-square lift $F$. Further, as a consequence, we get an asymptotic formula $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D706}_{f}^{2}(n)d(n-a)=E_{1}(a)x\log x+O(x\log \log x),\end{eqnarray}$$ where $E_{1}(a)$ is a constant depending on $a$. Moreover, we also consider the asymptotic orthogonality of the Möbius function against the arithmetic function $\unicode[STIX]{x1D70C}(n)d(n-a)$.

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