Galois groups over function fields of positive characteristic

Conway, John H.; McKay, John; Trojan, Allan

doi:10.1090/s0002-9939-09-10130-2

Cited by 6 publications

(10 citation statements)

References 9 publications

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“…If

p = 2

, then we can apply Serre's linearization method [2]. An almost identical computation, but for the trinomial

x^{24} + x + t

over the field

K (t)

is in [11] We will find an additive polynomial that is divisible by

x^{24} + a x + b

. Consider the following sequence of equalities in

K false(a, b false) false[x false] / false(x^{24} + a x + b false)

\begin{matrix} true \\ right 0 & = & left x^{32} + a x^{9} + b x^{8} = x^{256} + a^{8} x^{72} + b^{8} x^{64} \\ = & left x^{256} + a^{8} {false(a x + b false)}^{3} + b^{8} x^{64} = x^{256} + b^{8} x^{64} + a^{11} x^{3} + a^{10} b x^{2} + a^{9} b^{2} x + a^{8} b^{3} \\ = & left x^{2048} + b^{64} x^{512} + a^{88} x^{24} + a^{80} b^{8} x^{16} + a^{72} b^{16} x^{8} + a^{64} b^{24} \\ = & left x^{2048} + b^{64} x^{512} + a^{80} b^{8} x^{16} + a^{72} b^{16} x^{8} + a^{89} x + a^{88} b + a^{64} b^{24} . \end{matrix}

The last polynomial in this chain of equalities is an additive polynomial up to a constant.…”

Section: Galois Groups Of Trinomialsmentioning

confidence: 99%

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Sectional monodromy groups of projective curves

Kadets

2020

J. London Math. Soc.

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Fix a degree d projective curve X⊂double-struckPr over an algebraically closed field K. Let U⊂(double-struckPr)∗ be a dense open subvariety such that every hyperplane H∈U intersects X in d smooth points. Varying H∈U produces the monodromy action φ:π1étfalse(Ufalse)→Sd. Let GX≔imfalse(φfalse). The permutation group GX is called the sectional monodromy group of X. In characteristic 0, GX is always the full symmetric group, but sectional monodromy groups in characteristic p can be smaller. For a large class of space curves (r⩾3), we classify all possibilities for the sectional monodromy group G as well as the curves with GX=G. We apply similar methods to study a particular family of rational curves in P2, which enables us to answer an old question about Galois groups of generic trinomials.

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“…If

p = 2

, then we can apply Serre's linearization method [2]. An almost identical computation, but for the trinomial

x^{24} + x + t

over the field

K (t)

is in [11] We will find an additive polynomial that is divisible by

x^{24} + a x + b

. Consider the following sequence of equalities in

K false(a, b false) false[x false] / false(x^{24} + a x + b false)

\begin{matrix} true \\ right 0 & = & left x^{32} + a x^{9} + b x^{8} = x^{256} + a^{8} x^{72} + b^{8} x^{64} \\ = & left x^{256} + a^{8} {false(a x + b false)}^{3} + b^{8} x^{64} = x^{256} + b^{8} x^{64} + a^{11} x^{3} + a^{10} b x^{2} + a^{9} b^{2} x + a^{8} b^{3} \\ = & left x^{2048} + b^{64} x^{512} + a^{88} x^{24} + a^{80} b^{8} x^{16} + a^{72} b^{16} x^{8} + a^{64} b^{24} \\ = & left x^{2048} + b^{64} x^{512} + a^{80} b^{8} x^{16} + a^{72} b^{16} x^{8} + a^{89} x + a^{88} b + a^{64} b^{24} . \end{matrix}

The last polynomial in this chain of equalities is an additive polynomial up to a constant.…”

Section: Galois Groups Of Trinomialsmentioning

confidence: 99%

“…Classifying possible Galois groups of generic trinomials is an old problem; see [9, 25–27]. A closely related problem of classifying Galois groups of trinomials over the function field

K (t)

when

a

and

b

are specialized to powers of

t

has also been widely studied; see [1–4, 6 11].…”

Section: Introductionmentioning

confidence: 99%

Sectional monodromy groups of projective curves

Kadets

2020

J. London Math. Soc.

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“…|C(L)| ≥ 2 q m −1 q−1 . For p = 2 one obtains, using [3], the following Theorem 3.4.2. Let p be an odd prime, and q be a power of p. Let m ≥ 2 be an integer.…”

Section: 3mentioning

confidence: 99%

Methods for constructing elliptic and hyperelliptic curves with rational points

Joshi

2017

Preprint

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I provide methods of constructing elliptic and hyperelliptic curves over global fields with interesting rational points over the given fields or over large field extensions. I also provide a elliptic curves defined over any given number field equipped with a rational point, (resp. with two rational points) of infinite order over the given number field, and elliptic curves over the rationals with two rational points over 'simplest cubic fields.' I also provide hyperelliptic curves of genus exceeding any given number over any given number fields with points (over the given number field) which span a subgroup of rank at least g in the group of rational points of the Jacobian of this curve. I also provide a method of constructing hyperelliptic curves over rational function fields with rational points defined over field extensions with large finite simple Galois groups, such as the Mathieu group M 24 .My scheme is far more subtle. Let me outline it for you. Bertie Wooster in [21] CONTENTS * This data has been computed recently using [5], my own computations, for a more modest range of values (d ≤ 20), in 1989-92 were carried out with a patchwork of small search programs, with many stretches of manual and symbolic computations as well, were written mostly in PASCAL for VAX and CYBER "mainframes" which were owned by the Tata Institute at the time. When John Cremona's 'mwrank' became available I was able to extend these computations.

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“…15 ff.]. Recently, Conway, McKay, and Trojan [9] simplified some of Abhyankar's proofs and, additionally, gave particularly nice polynomials with monodromy groups including several Mathieu groups.…”

Section: Introductionmentioning

confidence: 99%