scholarly journals Explicit Weil-pairing for Drinfeld modules

2021 ◽  
pp. 1-22
Author(s):  
Jeffrey Katen
Keyword(s):  
Elementary Proof ◽  
Drinfeld Modules ◽  
Weil Pairing ◽  
Explicit Formulas ◽  
Rank 2

The goal of this paper is to define an analogue of the Weil-pairing for Drinfeld modules using explicit formulas and to deduce its main properties from these formulas. Our result generalizes the formula given for rank 2 Drinfeld modules by van der Heiden and works as a more explicit, elementary proof of the Weil-pairing’s existence given by van der Heiden.

Torsion Points of Drinfeld Modules

1995 ◽  
Vol 38 (1) ◽  
pp. 3-10
Author(s):  
Sunghan Bae ◽  
Jakyung Koo
Keyword(s):  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Locally Compact ◽  
Torsion Points ◽  
Complete Field ◽  
Rank 2

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.

scholarly journals SPHERICAL FUNCTIONS ON THE SPACE OF p-ADIC UNITARY HERMITIAN MATRICES

2014 ◽  
Vol 10 (02) ◽  
pp. 513-558
Author(s):  
YUMIKO HIRONAKA ◽  
YASUSHI KOMORI
Keyword(s):  
Spherical Functions ◽  
Plancherel Formula ◽  
Symmetric Polynomials ◽  
Hermitian Matrices ◽  
Cartan Decomposition ◽  
Explicit Formulas ◽  
Spherical Fourier Transform ◽  
Rank 2 ◽  
Type C ◽  
Residual Characteristic

We investigate the space X of unitary hermitian matrices over 𝔭-adic fields through spherical functions. First we consider Cartan decomposition of X, and give precise representatives for fields with odd residual characteristic, i.e. 2 ∉ 𝔭. From Sec. 2.2 till the end of Sec. 4, we assume odd residual characteristic, and give explicit formulas of typical spherical functions on X, where Hall–Littlewood symmetric polynomials of type Cn appear as a main term, parametrization of all the spherical functions. By spherical Fourier transform, we show that the Schwartz space [Formula: see text] is a free Hecke algebra [Formula: see text]-module of rank 2n, where 2n is the size of matrices in X, and give the explicit Plancherel formula on [Formula: see text].

scholarly journals Algebraic relations among periods and logarithms of rank 2 Drinfeld modules

2011 ◽  
Vol 133 (2) ◽  
pp. 359-391 ◽  
Author(s):  
Chieh-Yu Chang ◽  
Matthew A. Papanikolas
Keyword(s):  
Drinfeld Modules ◽  
Rank 2

scholarly journals Legendre Drinfeld modules and universal supersingular polynomials

2014 ◽  
Vol 10 (05) ◽  
pp. 1277-1289 ◽  
Author(s):  
Ahmad El-Guindy
Keyword(s):  
Normal Form ◽  
Elliptic Curves ◽  
Drinfeld Modules ◽  
Closed Formula ◽  
Explicit Formulas

We introduce a certain family of Drinfeld modules that we propose as analogues of the Legendre normal form elliptic curves. We exhibit explicit formulas for a certain period of such Drinfeld modules as well as formulas for the supersingular locus in that family, establishing a connection between these two kinds of formulas. Lastly, we also provide a closed formula for the supersingular polynomial in the j-invariant for generic Drinfeld modules.

scholarly journals An elementary proof of the group law for elliptic curves

2017 ◽  
Vol 0 (0) ◽  
Author(s):  
Stefan Friedl
Keyword(s):  
Elliptic Curves ◽  
Elementary Proof ◽  
Explicit Formulas ◽  
Group Law

AbstractWe give an elementary proof of the group law for elliptic curves using explicit formulas.

GAUSSIAN LAWS ON DRINFELD MODULES

2009 ◽  
Vol 05 (07) ◽  
pp. 1179-1203 ◽  
Author(s):  
WENTANG KUO ◽  
YU-RU LIU
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Valuation Ring ◽  
Finite Extension ◽  
Drinfeld Modules ◽  
Good Reduction ◽  
Prime Divisors ◽  
Frobenius Automorphism ◽  
Tate Module ◽  
Rank 2

Let A = 𝔽q[T] be the polynomial ring over the finite field 𝔽q, k = 𝔽q(T) the rational function field, and K a finite extension of k. Let ϕ be a Drinfeld A-module over K of rank r. For a place 𝔓 of K of good reduction, write [Formula: see text], where [Formula: see text] is the valuation ring of 𝔓 and [Formula: see text] its maximal ideal. Let P𝔓, ϕ(X) be the characteristic polynomial of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ. Let χϕ(𝔓) = P𝔓, ϕ(1), and let ν(χϕ(𝔓)) be the number of distinct primes dividing χϕ(𝔓). If ϕ is of rank 2 with [Formula: see text], we prove that there exists a normal distribution for the quantity [Formula: see text] For r ≥ 3, we show that the same result holds under the open image conjecture for Drinfeld modules. We also study the number of distinct prime divisors of the trace of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ and obtain similar results.

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