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Register a new userOn the Galois group over Q of a truncated binomial expansion
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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 $O(log N)$.
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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 semistable 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 on 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 there 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.
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