On Artin’s conjecture for pairs of diagonal forms

2021 ◽  
pp. 1-16
Author(s):  
João Campos Vargas
Keyword(s):  
Artin's Conjecture ◽  
Diagonal Forms

Let [Formula: see text] be an odd prime and [Formula: see text]. In the spirit of Artin’s conjecture, consider the system of two diagonal forms of degree [Formula: see text] in [Formula: see text] variables given by [Formula: see text] [Formula: see text] with [Formula: see text]. For [Formula: see text], this paper shows that this system has a non-trivial [Formula: see text]-adic solution for every [Formula: see text], and for every [Formula: see text], where [Formula: see text]. Moreover, for [Formula: see text], this system will have a non-trivial [Formula: see text]-adic solution for every [Formula: see text].

scholarly journals Pairs of diagonal forms of degree 3 τ .2 and Artin's conjecture

2017 ◽  
Vol 177 ◽  
pp. 211-247
Author(s):  
Hemar Godinho ◽  
Luciana Ventura
Keyword(s):  
Artin's Conjecture ◽  
Diagonal Forms

GENERALIZING THE TITCHMARSH DIVISOR PROBLEM

2012 ◽  
Vol 08 (03) ◽  
pp. 613-629 ◽  
Author(s):  
ADAM TYLER FELIX
Keyword(s):  
Natural Number ◽  
Asymptotic Relation ◽  
Divisor Problem ◽  
Implied Constant ◽  
Artin's Conjecture ◽  
Primitive Roots

Let a be a natural number different from 0. In 1963, Linnik proved the following unconditional result about the Titchmarsh divisor problem [Formula: see text] where c is a constant dependent on a. Titchmarsh proved the above result assuming GRH for Dirichlet L-functions in 1931. We establish the following asymptotic relation: [Formula: see text] where Ck is a constant dependent on k and a, and the implied constant is dependent on k. We also apply it a question related to Artin's conjecture for primitive roots.

Detecting Large Simple Rational Hecke Modules for Γ0(N) via Congruences

2018 ◽  
Vol 2020 (19) ◽  
pp. 6149-6168
Author(s):  
Michael Lipnowski ◽  
George J Schaeffer
Keyword(s):  
Modular Forms ◽  
Class Number ◽  
Target Space ◽  
Large Set ◽  
Hecke Eigenvalues ◽  
Artin's Conjecture ◽  
Number Problem ◽  
Novel Method ◽  
Hecke Modules ◽  
Primitive Roots

Abstract We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_{2}(\Gamma _{0}(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_{0}(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. We prove conditional bounds, the strongest of which is $d\gg _{\epsilon } N^{1/2-\epsilon }$ over a large set of primes $N$, contingent on Soundararajan’s heuristics for the class number problem and Artin’s conjecture on primitive roots. For prime levels $N\equiv 7\mod 8,$ our method yields an unconditional bound of $d\geq \log _{2}\log _{2}(\frac{N}{8})$, which is larger than the known bound of $d\gg \sqrt{\log \log N}$ due to Murty–Sinha and Royer. A stronger unconditional bound of $d\gg \log N$ can be obtained in more specialized (but infinitely many) cases. We also propose a number of Maeda-style conjectures based on our data, and we outline a possible congruence-based approach toward the conjectural Hecke simplicity of $S_{k}(\textrm{SL}_{2}(\mathbb{Z});\mathbb{Q})$.

On Artin's Conjecture

1972 ◽  
Vol s2-5 (1) ◽  
pp. 91-94 ◽  
Author(s):  
Richard Warlimont
Keyword(s):  
Artin's Conjecture

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.

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