On periods of the third kind for rank 2 Drinfeld module

2012 ◽  
Vol 273 (3-4) ◽  
pp. 921-933
Author(s):  
Chieh-Yu Chang
Keyword(s):  
1995 ◽  
Vol 38 (1) ◽  
pp. 3-10
Author(s):  
Sunghan Bae ◽  
Jakyung Koo

AbstractThe finiteness of K-rational torsion points of a Drinfeld module of rank 2 over a locally compact complete field K with a discrete valuation is proved.


2008 ◽  
Vol 144 (4) ◽  
pp. 827-848 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Chantal David

AbstractLet ϕ be a Drinfeld module of rank 2 over the field of rational functions $F=\mathbb {F}_q(T)$, with $\mathrm {End}_{\bar {F}}(\phi ) = \mathbb {F}_q[T]$. Let K be a fixed imaginary quadratic field over F and d a positive integer. For each prime $\mathfrak {p}$ of good reduction for ϕ, let $\pi _{\mathfrak {p}}(\phi )$ be a root of the characteristic polynomial of the Frobenius endomorphism of ϕ over the finite field $\mathbb {F}_q[T] / \mathfrak {p}$. Let Πϕ(K;d) be the number of primes $\mathfrak {p}$ of degree d such that the field extension $F(\pi _{\mathfrak {p}}(\phi ))$ is the fixed imaginary quadratic field K. We present upper bounds for Πϕ(K;d) obtained by two different approaches, inspired by similar ones for elliptic curves. The first approach, inspired by the work of Serre, is to consider the image of Frobenius in a mixed Galois representation associated to K and to the Drinfeld module ϕ. The second approach, inspired by the work of Cojocaru, Fouvry and Murty, is based on an application of the square sieve. The bounds obtained with the first method are better, but depend on the fixed quadratic imaginary field K. In our application of the second approach, we improve the results of Cojocaru, Murty and Fouvry by considering projective Galois representations.


2017 ◽  
Vol 234 ◽  
pp. 17-45 ◽  
Author(s):  
IMIN CHEN ◽  
YOONJIN LEE

Let $K=\mathbb{F}_{q}(T)$ and $A=\mathbb{F}_{q}[T]$. Suppose that $\unicode[STIX]{x1D719}$ is a Drinfeld $A$-module of rank $2$ over $K$ which does not have complex multiplication. We obtain an explicit upper bound (dependent on $\unicode[STIX]{x1D719}$) on the degree of primes ${\wp}$ of $K$ such that the image of the Galois representation on the ${\wp}$-torsion points of $\unicode[STIX]{x1D719}$ is not surjective, in the case of $q$ odd. Our results are a Drinfeld module analogue of Serre’s explicit large image results for the Galois representations on $p$-torsion points of elliptic curves (Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331; Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Etudes Sci. Publ. Math. 54 (1981), 323–401.) and are unconditional because the generalized Riemann hypothesis for function fields holds. An explicit isogeny theorem for Drinfeld $A$-modules of rank $2$ over $K$ is also proven.


1967 ◽  
Vol 31 ◽  
pp. 177-179
Author(s):  
W. W. Shane

In the course of several 21-cm observing programmes being carried out by the Leiden Observatory with the 25-meter telescope at Dwingeloo, a fairly complete, though inhomogeneous, survey of the regionl11= 0° to 66° at low galactic latitudes is becoming available. The essential data on this survey are presented in Table 1. Oort (1967) has given a preliminary report on the first and third investigations. The third is discussed briefly by Kerr in his introductory lecture on the galactic centre region (Paper 42). Burton (1966) has published provisional results of the fifth investigation, and I have discussed the sixth in Paper 19. All of the observations listed in the table have been completed, but we plan to extend investigation 3 to a much finer grid of positions.


1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.


1988 ◽  
Vol 102 ◽  
pp. 79-81
Author(s):  
A. Goldberg ◽  
S.D. Bloom

AbstractClosed expressions for the first, second, and (in some cases) the third moment of atomic transition arrays now exist. Recently a method has been developed for getting to very high moments (up to the 12th and beyond) in cases where a “collective” state-vector (i.e. a state-vector containing the entire electric dipole strength) can be created from each eigenstate in the parent configuration. Both of these approaches give exact results. Herein we describe astatistical(or Monte Carlo) approach which requires onlyonerepresentative state-vector |RV> for the entire parent manifold to get estimates of transition moments of high order. The representation is achieved through the random amplitudes associated with each basis vector making up |RV>. This also gives rise to the dispersion characterizing the method, which has been applied to a system (in the M shell) with≈250,000 lines where we have calculated up to the 5th moment. It turns out that the dispersion in the moments decreases with the size of the manifold, making its application to very big systems statistically advantageous. A discussion of the method and these dispersion characteristics will be presented.


Author(s):  
Zhifeng Shao

A small electron probe has many applications in many fields and in the case of the STEM, the probe size essentially determines the ultimate resolution. However, there are many difficulties in obtaining a very small probe.Spherical aberration is one of them and all existing probe forming systems have non-zero spherical aberration. The ultimate probe radius is given byδ = 0.43Csl/4ƛ3/4where ƛ is the electron wave length and it is apparent that δ decreases only slowly with decreasing Cs. Scherzer pointed out that the third order aberration coefficient always has the same sign regardless of the field distribution, provided only that the fields have cylindrical symmetry, are independent of time and no space charge is present. To overcome this problem, he proposed a corrector consisting of octupoles and quadrupoles.


Author(s):  
Oktay Arda ◽  
Ulkü Noyan ◽  
Selgçk Yilmaz ◽  
Mustafa Taşyürekli ◽  
İsmail Seçkin ◽  
...  

Turkish dermatologist, H. Beheet described the disease as recurrent triad of iritis, oral aphthous lesions and genital ulceration. Auto immune disease is the recent focus on the unknown etiology which is still being discussed. Among the other immunosupressive drugs, CyA included in it's treatment newly. One of the important side effects of this drug is gingival hyperplasia which has a direct relation with the presence of teeth and periodontal tissue. We are interested in the ultrastructure of immunocompetent target cells that were affected by CyA in BD.Three groups arranged in each having 5 patients with BD. Control group was the first and didn’t have CyA treatment. Patients who had CyA, but didn’t show gingival hyperplasia assembled the second group. The ones displaying gingival hyperplasia following CyA therapy formed the third group. GMC of control group and their granules are shown in FIG. 1,2,3. GMC of the second group presented initiation of supplementary cellular activity and possible maturing functional changes with the signs of increased number of mitochondria and accumulation of numerous dense cored granules next to few normal ones, FIG. 4,5,6.


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