On wild ramification in quaternion extensions

Elder, G. Griffith; Hooper, Jeffrey J.

doi:10.5802/jtnb.576

Cited by 3 publications

(3 citation statements)

References 13 publications

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“…Notably, the definition of refined ramification break numbers remains tied to a choice of element and so the refined breaks (beyond the first two) cannot, as yet, be said to be canonical. In addition, it has been observed in the context of quaternion extensions [EH07] that the refined ramification filtration has some influence on breaks in the usual ramification filtration. And so there is much remaining work to determine if/how these two filtrations fit together.…”

Section: Resultsmentioning

confidence: 97%

On the necessity of new ramification breaks

Byott

Elder

2009

Journal of Number Theory

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Ramification invariants are necessary, but not in general sufficient, to determine the Galois module structure of ideals in local number field extensions. This insufficiency is associated with elementary abelian extensions, where one can define a refined ramification filtration-one with more ramification breaks [Nigel P. Byott, G. Griffith Elder, New ramification breaks and additive Galois structure, J. Théor. Nombres Bordeaux 17 (1) (2005) 87-107]. The first refined break number comes from the usual ramification filtration and is therefore necessary. Here we study the second refined break number.

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Section: Resultsmentioning

confidence: 97%

On the necessity of new ramification breaks

Byott

Elder

2009

Journal of Number Theory

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“…When the Galois group is elementary abelian, the Galois module structure of certain ideals is related to the ramification group filtration, see [Byott and Elder 2002;. Such a relation is investigated when the Galois group is quaternion [Elder and Hooper 2007], and hence nonabelian.…”

Section: Introductionmentioning

confidence: 99%

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2015

Pacific J. Math.

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Denote by J the operator of coefficient stripping. We show that for any free convolution semigroup f t W t 0g with finite variance, applying a single stripping produces semicircular evolution with nonzero initial condition, J OE t D ť ; , where ˇ; is the semicircular distribution with meanˇand variance . For more general freely infinitely divisible distributions , expressions of the form Q t arise from stripping Q t , where f. Q t ; t / W t 0g forms a semigroup under the operation of two-state free convolution. The converse to this statement holds in the algebraic setting. Numerous examples illustrating these constructions are computed. Additional results include the formula for generators of such semigroups.

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“…When the Galois group is elementary abelian, the Galois module structure of certain ideals is related to the ramification group filtration, see [3,4,5]. Such a relation is investigated when the Galois group is quaternion [7], hence non-abelian.…”

Section: This Relation Is Close If An Intersection Propertymentioning

confidence: 99%

The ramification group filtrations of certain function field extensions

Castañeda

2015

Pacific J. Math.

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We investigate the ramification group filtration of a Galois extension of function fields, if the Galois group satisfies a certain intersection property. For finite groups, this property is implied by having only elementary abelian Sylow p-subgroups. Note that such groups could be non-abelian. We show how the problem can be reduced to the totally wild ramified case on a p-extension. Our methodology is based on an intimate relationship between the ramification groups of the field extension and those of all degree p sub-extensions. Not only do we confirm that the Hasse-Arf property holds in this setting, but we also prove that the Hasse-Arf divisibility result is the best possible by explicit calculations of the quotients, which are expressed in terms of the different exponents of all those degree p sub-extensions. v ACKNOWLEDGMENTS

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On wild ramification in quaternion extensions

Cited by 3 publications

References 13 publications

On the necessity of new ramification breaks

On the necessity of new ramification breaks

Untitled

The ramification group filtrations of certain function field extensions

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