The discriminant 10 Shimura curve and its associated Heun functions

2016 ◽  
Vol 48 (6) ◽  
pp. 957-967
Author(s):  
Srinath Baba ◽  
Håkan Granath
Keyword(s):  
Shimura Curve ◽  
Heun Functions

scholarly journals Diagrams in the mod p cohomology of Shimura curves

2021 ◽  
Vol 157 (8) ◽  
pp. 1653-1723
Author(s):  
Andrea Dotto ◽  
Daniel Le
Keyword(s):  
Shimura Curves ◽  
Shimura Curve ◽  
Langlands Program ◽  
Mod P

Abstract We prove a local–global compatibility result in the mod $p$ Langlands program for $\mathrm {GL}_2(\mathbf {Q}_{p^f})$ . Namely, given a global residual representation $\bar {r}$ appearing in the mod $p$ cohomology of a Shimura curve that is sufficiently generic at $p$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $p$ completed cohomology is determined by the restrictions of $\bar {r}$ to decomposition groups at $p$ . If these restrictions are moreover semisimple, we show that the $(\varphi ,\Gamma )$ -modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $\bar {r}$ to decomposition groups at $p$ .

scholarly journals Analytical investigation of wave absorption by a rotating black hole analogue

2020 ◽  
Vol 29 (11) ◽  
pp. 2041018
Author(s):  
Leandro A. Oliveira ◽  
Carolina L. Benone ◽  
Amanda L. Almeida ◽  
Luís C. B. Crispino
Keyword(s):  
Black Hole ◽  
Excellent Agreement ◽  
Analytical Investigation ◽  
Absorption Length ◽  
Wave Absorption ◽  
Rotating Black Hole ◽  
Analogue Models ◽  
Ideal Fluids ◽  
Heun Functions ◽  
Models Of Gravity

Perturbations in a draining vortex can be described analytically in terms of confluent Heun functions. In the context of analogue models of gravity in ideal fluids, we investigate analytically the absorption length of waves in a draining bathtub, a rotating black hole analogue, using confluent Heun functions. We compare our analytical results with the corresponding numerical ones, obtaining excellent agreement.

scholarly journals Klein–Gordon particle near RN black hole, generalized sl(2) algebra and harmonic oscillator energy

2019 ◽  
Vol 34 (31) ◽  
pp. 1950196
Author(s):  
J. Sadeghi ◽  
M. R. Alipour
Keyword(s):  
Differential Equation ◽  
Information Theory ◽  
Black Hole ◽  
Energy Spectrum ◽  
Harmonic Oscillator ◽  
Three Dimensional ◽  
The Other ◽  
Thermodynamical Properties ◽  
Heun Functions ◽  
Klein Gordon

In this paper, we consider Klein–Gordon particle near Reissner–Nordström black hole. The symmetry of such a background led us to compare the corresponding Laplace equation with the generalized Heun functions. Such relations help us achieve the generalized [Formula: see text] algebra and some suitable results for describing the above-mentioned symmetry. On the other hand, in case of [Formula: see text], which is near the proximity black hole, we obtain the energy spectrum. When we compare the equation of RN background with Laguerre differential equation, we show that the obtained energy spectrum is same as the three-dimensional harmonic oscillator. So, finally we take advantage of harmonic oscillator energy and make suitable partition function. Such function help us to obtain all thermodynamical properties of black hole. Also, the structure of obtained entropy lead us to have some bit and information theory in the RN black hole.

scholarly journals Non-linear integral equations for Heun functions

1969 ◽  
Vol 16 (4) ◽  
pp. 281-289 ◽  
Author(s):  
B. D. Sleeman
Keyword(s):  
Integral Equations ◽  
Heun Equation ◽  
Mathieu Functions ◽  
Special Cases ◽  
Mathieu’S Equation ◽  
Heun Functions ◽  
Non Linear ◽  
Linear Integral ◽  
Image Position ◽  
Linear Integral Equations

Some years ago Lambe and Ward (1) and Erdélyi (2) obtained integral equations for Heun polynomials and Heun functions. The integral equations discussed by these authors were of the formFurther, as is well known, the Heun equation includes, among its special cases, Lamé's equation and Mathieu's equation and so (1.1) may be considered a generalisation of the integral equations satisfied by Lamé polynomials and Mathieu functions. However, integral equations of the type (1.1) are not the only ones satisfied by Lamé polynomials; Arscott (3) discussed a class of non- linear integral equations associated with these functions. This paper then is concerned with discussing the existence of non-linear integral equations satisfied by solutions of Heun's equation.

Exact solutions of a quartic potential

2019 ◽  
Vol 34 (26) ◽  
pp. 1950208 ◽  
Author(s):  
Qian Dong ◽  
Guo-Hua Sun ◽  
M. Avila Aoki ◽  
Chang-Yuan Chen ◽  
Shi-Hai Dong
Keyword(s):  
Exact Solutions ◽  
Quantum System ◽  
Analytical Solutions ◽  
Wave Functions ◽  
Negative Potential ◽  
Potential Parameter ◽  
Real Numbers ◽  
Quartic Potential ◽  
Heun Functions ◽  
Potential Parameters

We find that the analytical solutions to quantum system with a quartic potential [Formula: see text] (arbitrary [Formula: see text] and [Formula: see text] are real numbers) are given by the triconfluent Heun functions [Formula: see text]. The properties of the wave functions, which are strongly relevant for the potential parameters [Formula: see text] and [Formula: see text], are illustrated. It is shown that the wave functions are shrunk to the origin for a given [Formula: see text] when the potential parameter [Formula: see text] increases, while the wave peak of wave functions is concaved to the origin when the negative potential parameter [Formula: see text] increases or parameter [Formula: see text] decreases for a given negative potential parameter [Formula: see text]. The minimum value of the double well case ([Formula: see text]) is given by [Formula: see text] at [Formula: see text].

scholarly journals Analytic solutions of the quantum two-state problem in terms of the double, bi- and triconfluent Heun functions

2015 ◽  
Vol 50 (3) ◽  
pp. 211-226 ◽  
Author(s):  
T. A. Shahverdyan ◽  
T. A. Ishkhanyan ◽  
A. E. Grigoryan ◽  
A. M. Ishkhanyan
Keyword(s):  
Analytic Solutions ◽  
State Problem ◽  
Heun Functions
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