841 research outputs found
The density of primes in orbits of z^d + c
Given a polynomial f(z) = z^d + c over a global field K and a_0 in K, we
study the density of prime ideals of K dividing at least one element of the
orbit of a_0 under f. The density of such sets for linear polynomials has
attracted much study, and the second author has examined several families of
quadratic polynomials, but little is known in the higher-degree case. We show
that for many choices of d and c this density is zero for all a_0, assuming K
contains a primitive dth root of unity. The proof relies on several new
results, including some ensuring the number of irreducible factors of the nth
iterate of f remains bounded as n grows, and others on the ramification above
certain primes in iterated extensions. Together these allow for nearly complete
information when K is a global function field or when K=Q(zeta_d).Comment: 27 page
Analysis of the classical cyclotomic approach to fermat's last theorem
We give again the proof of several classical results concerning the
cyclotomic approach to Fermat's last theorem using exclusively class field
theory (essentially the reflection theorems), without any calculations. The
fact that this is possible suggests a part of the logical inefficiency of the
historical investigations. We analyze the significance of the numerous
computations of the literature, to show how they are probably too local to get
any proof of the theorem. However we use the derivation method of Eichler as a
prerequisite for our purpose, a method which is also local but more effective.
Then we propose some modest ways of study in a more diophantine context using
radicals; this point of view would require further nonalgebraic investigations.Comment: Publications Math\'ematiques UFR Sciences Techniques Besan\c{c}on
2010 (2010) 85-11
An examination of the structure of extension families of irreducible polynomials over finite fields
In this paper we examine the behavior of particular family of polynomial over a
nite eld. The family studied is that obtained by composing an irreducible poly-
nomial with prime power monomials. We examine methods of testing irreducibility
via a new method of discriminant calculation. We also provide new incite into how
the members of the given family factor when not irreducible. Further, we provided
a nite eld generalization to "Roots Appearing in Quanta", an article presented by
Perlis
On l-adic representations for a space of noncongruence cuspforms
This paper is concerned with a compatible family of 4-dimensional \ell-adic
representations \rho_{\ell} of G_\Q:=\Gal(\bar \Q/\Q) attached to the space of
weight 3 cuspforms S_3 (\Gamma) on a noncongruence subgroup \Gamma \subset \SL.
For this representation we prove that: 1.)It is automorphic: the L-function
L(s, \rho_{\ell}^{\vee}) agrees with the L-function for an automorphic form for
\text{GL}_4(\mathbb A_{\Q}), where \rho_{\ell}^{\vee} is the dual of
\rho_{\ell}. 2.) For each prime p \ge 5 there is a basis h_p = \{h_p ^+, h_p ^-
\} of S_3 (\Gamma) whose expansion coefficients satisfy 3-term Atkin and
Swinnerton-Dyer (ASD) relations, relative to the q-expansion coefficients of a
newform f of level 432. The structure of this basis depends on the class of p
modulo 12. The key point is that the representation admits a
quaternion multiplication structure in the sense of a recent work of Atkin, Li,
Liu and Long.Comment: Second revised version. To appear: Proceedings of the American
Mathematical Societ
- …