اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn the Galois group over Q of a truncated binomial expansion
128
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
For positive integers $n$, the truncated binomial expansions of $(1+x)^n$ which consist of all the terms of degree $le r$ where $1 le r le n-2$ appear always to be irreducible. For fixed $r$ and $n$ sufficiently large, this is known to be the case. We show here that for a fixed positive integer $r e 6$ and $n$ sufficiently large, the Galois group of such a polynomial over the rationals is the symmetric group $S_{r}$. For $r = 6$, we show the number of exceptional $n le N$ for which the Galois group of this polynomial is not $S_r$ is at most $O(log N)$.
قيم البحث
اقرأ أيضاً
In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C) has semista
ble reduction with toric dimension 1 at p. We provide an algorithm to compute a list of primes l (if they exist) such that the Galois representation attached to the l-torsion of J(C) is surjective onto the group GSp(2n, l). In particular we realize GSp(6, l) as a Galois group over Q for all primes l in [11, 500000].
We discuss the $ell$-adic case of Mazurs Program B over $mathbb{Q}$, the problem of classifying the possible images of $ell$-adic Galois representations attached to elliptic curves $E$ over $mathbb{Q}$, equivalently, classifying the rational points o
n the corresponding modular curves. The primes $ell=2$ and $ellge 13$ are addressed by prior work, so we focus on the remaining primes $ell = 3, 5, 7, 11$. For each of these $ell$, we compute the directed graph of arithmetically maximal $ell$-power level modular curves, compute explicit equations for most of them, and classify the rational points on all of them except $X_{{rm ns}}^{+}(N)$, for $N = 27, 25, 49, 121$, and two level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$. Aside from the $ell$-adic images that are known to arise for infinitely many $overline{mathbb{Q}}$-isomorphism classes of elliptic curves $E/mathbb{Q}$, we find only 22 exceptional subgroups that arise for any prime $ell$ and any $E/mathbb{Q}$ without complex multiplication; these exceptional subgroups are realized by 20 non-CM rational $j$-invariants. We conjecture that this list of 22 exceptional subgroups is complete and show that any counterexamples must arise from unexpected rational points on $X_{rm ns}^+(ell)$ with $ellge 17$, or one of the six modular curves noted above. This gives us an efficient algorithm to compute the $ell$-adic images of Galois for any non-CM elliptic curve over $mathbb{Q}$. In an appendix with John Voight we generalize Ribets observation that simple abelian varieties attached to newforms on $Gamma_1(N)$ are of ${rm GL}_2$-type; this extends Kolyvagins theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$.
Let $X$ be a smooth projective connected curve of genus $gge 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Let $G$ be a finite group, $P$ a Sylow $p$-subgroup of $G$ and $N_G(P)$ its normalizer in $G$. We show that if the
re exists an etale Galois cover $Yto X$ with group $N_G(P)$, then $G$ is the Galois group wan etale Galois cover $mathcal{Y}tomathcal{X}$, where the genus of $mathcal{X}$ depends on the order of $G$, the number of Sylow $p$-subgroups of $G$ and $g$. Suppose that $G$ is an extension of a group $H$ of order prime to $p$ by a $p$-group $P$ and $X$ is defined over a finite field $mathbb{F}_q$ large enough to contain the $|H|$-th roots of unity. We show that integral idempotent relations in the group ring $mathbb{C}[H]$ imply similar relations among the corresponding generalized Hasse-Witt invariants.
Let $ksubseteq K$ be a finite Galois extension of fields with Galois group $G$. Let $mathscr{G}$ be the automorphism $k$-group scheme of $K$. We construct a canonical $k$-subgroup scheme $underline{G}subsetmathscr{G}$ with the property that $Spec_k(K
)$ is a $k$-torsor for $underline{G}$. $underline{G}$ is a constant $k$-group if and only if $G$ is abelian, in which case $G=underline{G}$.
In this paper we show an explicit polynomial in Q[x] that has Galois group SL2(F16), filling in a gap in the tables of Juergen Klueners and Gunther Malle. The computation of this polynomial uses modular forms and their Galois representations.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
