The Zeta Function of a Pair of Quadratic Forms

2001 ◽  
Vol 44 (2) ◽  
pp. 242-256
Author(s):  
Laura Mann Schueller
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Function Field ◽  
Quadratic Forms ◽  
Arbitrary Characteristic ◽  
Number Of Zeros ◽  
Even Order ◽  
Hyperelliptic Function

AbstractThe zeta function of a nonsingular pair of quadratic forms defined over a finite field, k, of arbitrary characteristic is calculated. A. Weil made this computation when char k ≠ 2. When the pair has even order, a relationship between the number of zeros of the pair and the number of places of degree one in an appropriate hyperelliptic function field is established.

scholarly journals Quadratic form Approach for the Number of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

2017 ◽  
Vol 9 (3) ◽  
pp. 8
Author(s):  
Yasanthi Kottegoda
Keyword(s):  
Quadratic Form ◽  
Finite Field ◽  
Characteristic Polynomial ◽  
Finite Fields ◽  
Quadratic Forms ◽  
Linear Recurring Sequences ◽  
Exact Number ◽  
Number Of Zeros ◽  
Homogeneous Linear ◽  
Form Approach

We consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on an irreducible characteristic polynomial of degree $n$ and order $m$. Let $t=(q^{n}-1)/ m$. We use quadratic forms over finite fields to give the exact number of occurrences of zeros of the sequence within its least period when $t$ has q-adic weight 2. Consequently we prove that the cardinality of the set of zeros for sequences from this category is equal to two.

scholarly journals The -parity conjecture for abelian varieties over function fields of characteristic

2014 ◽  
Vol 150 (4) ◽  
pp. 507-522 ◽  
Author(s):  
Fabien Trihan ◽  
Seidai Yasuda
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Prime Number ◽  
Abelian Variety ◽  
Function Field ◽  
Function Fields ◽  
Selmer Group ◽  
Smooth Variety ◽  
Tate Conjecture ◽  
Parity Conjecture

AbstractLet $A/K$ be an abelian variety over a function field of characteristic $p>0$ and let $\ell $ be a prime number ($\ell =p$ allowed). We prove the following: the parity of the corank $r_\ell $ of the $\ell $-discrete Selmer group of $A/K$ coincides with the parity of the order at $s=1$ of the Hasse–Weil $L$-function of $A/K$. We also prove the analogous parity result for pure $\ell $-adic sheaves endowed with a nice pairing and in particular for the congruence Zeta function of a projective smooth variety over a finite field. Finally, we prove that the full Birch and Swinnerton-Dyer conjecture is equivalent to the Artin–Tate conjecture.

scholarly journals Pencils of Quadratic Forms and Hyperelliptic Function Fields

2000 ◽  
Vol 227 (2) ◽  
pp. 532-548
Author(s):  
David B. Leep ◽  
Laura Mann Schueller
Keyword(s):  
Quadratic Forms ◽  
Function Fields ◽  
Hyperelliptic Function

Higher order Wirtinger inequalities

1980 ◽  
Vol 85 (1-2) ◽  
pp. 87-110 ◽  
Author(s):  
Kurt Kreith ◽  
Charles A. Swanson
Keyword(s):  
Boundary Conditions ◽  
Quadratic Forms ◽  
Differential Operators ◽  
Conjugate Point ◽  
Wirtinger Inequality ◽  
Even Order ◽  
Linear Differential ◽  
Natural Boundary Conditions ◽  
Derivatives Of ◽  
Admissible Functions

SynopsisWirtinger-type inequalities of order n are inequalities between quadratic forms involving derivatives of order k ≦ n of admissible functions in an interval (a, b). Several methods for establishing these inequalities are investigated, leading to improvements of classical results as well as systematic generation of new ones. A Wirtinger inequality for Hamiltonian systems is obtained in which standard regularity hypotheses are weakened and singular intervals are permitted, and this is employed to generalize standard inequalities for linear differential operators of even order. In particular second order inequalities of Beesack's type are developed, in which the admissible functions satisfy only the null boundary conditions at the endpoints of [a, b] and b does not exceed the first systems conjugate point (a) of a. Another approach is presented involving the standard minimization theory of quadratic forms and the theory of “natural boundary conditions”. Finally, inequalities of order n + k are described in terms of (n, n)-disconjugacy of associated 2nth order differential operators.

Generalized Artin'S Conjecture for Primitive Roots and Cyclicity Mod of Elliptic Curves Over Function Fields

1995 ◽  
Vol 38 (2) ◽  
pp. 167-173 ◽  
Author(s):  
David A. Clark ◽  
Masato Kuwata
Keyword(s):  
Finite Field ◽  
Prime Ideal ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Function Fields ◽  
Artin's Conjecture ◽  
Primitive Roots

AbstractLet k = Fq be a finite field of characteristic p with q elements and let K be a function field of one variable over k. Consider an elliptic curve E defined over K. We determine how often the reduction of this elliptic curve to a prime ideal is cyclic. This is done by generalizing a result of Bilharz to a more general form of Artin's primitive roots problem formulated by R. Murty.

scholarly journals Indépendance algébrique de logarithmes en caractéristique P

2006 ◽  
Vol 74 (3) ◽  
pp. 461-470 ◽  
Author(s):  
Laurent Denis
Keyword(s):  
Zeta Function ◽  
Rational Function ◽  
Riemann Zeta Function ◽  
Function Field ◽  
Fundamental Period ◽  
Rational Function Field ◽  
Linearly Independent ◽  
The Riemann Zeta Function ◽  
Carlitz Module ◽  
Riemann Zeta

Let k be the rational function field over the field with q elements with characteristic p. Since the work of Carlitz we know in this situation the function ζ analog of the Riemann zeta function and the function Logφ analog of the usual logarithm. We will show two main results. Firstly, if ξ denotes the fundamental period of Carlitz module, we prove that ξ, ζ(1),…, ζ(p – 2) are algebraically independent over k. Secondly if α1,…, αn are rational elements (of degree less than q/(q − 1) to ensure convergence of the logarithm) such that Logφ α1,…, Logφ αn are linearly independent over k then they are algebraically independent over k. The point is to find suitable functions taking these values and for which Mahler's method can be used.

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