Fundamental Cone #
Let K be a number field of signature (rโ, rโ). We define an action of the units (๐ K)หฃ on
the mixed space โ^rโ ร โ^rโ via the mixedEmbedding. The fundamental cone is a cone in the
mixed space that is a fundamental domain for the action of (๐ K)หฃ modulo torsion.
Main definitions and results #
NumberField.mixedEmbedding.unitSMul: the action of(๐ K)หฃon the mixed space defined, foru : (๐ K)หฃ, by multiplication component by component withmixedEmbedding K u.NumberField.mixedEmbedding.fundamentalCone: a cone in the mixed space, ie. a subset stable by multiplication by a nonzero real number, seesmul_mem_of_mem, that is also a fundamental domain for the action of(๐ K)หฃmodulo torsion, seeexists_unit_smul_memandtorsion_unit_smul_mem_of_mem.NumberField.mixedEmbedding.fundamentalCone.idealSet: forJan integral ideal, the intersection between the fundamental cone and theidealLatticedefined by the image ofJ.NumberField.mixedEmbedding.fundamentalCone.idealSetEquivNorm: forJan integral ideal andna natural integer, the equivalence between the elements ofidealSet Kof normnand the product of the set of nonzero principal ideals ofKdivisible byJof normnand the torsion ofK.
Tags #
number field, canonical embedding, units, principal ideals
The action of (๐ K)หฃ on the mixed space โ^rโ ร โ^rโ defined, for u : (๐ K)หฃ, by
multiplication component by component with mixedEmbedding K u.
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- One or more equations did not get rendered due to their size.
The map from the mixed space to {w : InfinitePlace K // w โ wโ} โ โ (with wโ the fixed
place from the proof of Dirichlet Unit Theorem) defined in such way that: 1) it factors the map
logEmbedding, see logMap_eq_logEmbedding; 2) it is constant on the sets
{c โข x | c โ โ, c โ 0} if norm x โ 0, see logMap_real_smul.
Equations
- NumberField.mixedEmbedding.logMap x w = โ(โw).mult * (Real.log ((NumberField.mixedEmbedding.normAtPlace โw) x) - Real.log (NumberField.mixedEmbedding.norm x) * (โ(Module.finrank โ K))โปยน)
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The fundamental cone is a cone in the mixed space, ie. a subset fixed by multiplication by
a nonzero real number, see smul_mem_of_mem, that is also a fundamental domain for the action
of (๐ K)หฃ modulo torsion, see exists_unit_smul_mem and torsion_smul_mem_of_mem.
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The intersection between the fundamental cone and the integerLattice.
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If a is in integerSet, then there is a unique algebraic integer in ๐ K such
that mixedEmbedding K x = a.
Alias of NumberField.mixedEmbedding.fundamentalCone.existsUnique_preimage_of_mem_integerSet.
If a is in integerSet, then there is a unique algebraic integer in ๐ K such
that mixedEmbedding K x = a.
For a : integerSet K, the unique nonzero algebraic integer x such that its image by
mixedEmbedding is equal to a. Note that we state the fact that x โ 0 by saying that x is
a nonzero divisors since we will use later on the isomorphism
Ideal.associatesNonZeroDivisorsEquivIsPrincipal, see integerSetEquiv.
Equations
- NumberField.mixedEmbedding.fundamentalCone.preimageOfMemIntegerSet a = โจโฏ.choose, โฏโฉ
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If x : mixedSpace K is nonzero and the image of an algebraic integer, then there exists a
unit such that u โข x โ integerSet K.
The set integerSet K is stable under the action of the torsion.
The action of torsion K on integerSet K.
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- NumberField.mixedEmbedding.fundamentalCone.instMulActionSubtypeUnitsRingOfIntegersMemSubgroupTorsionElemMixedSpaceIntegerSet = MulAction.mk โฏ โฏ
The mixedEmbedding.norm of a : integerSet K as a natural number, see also
intNorm_coe.
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The norm intNorm lifts to a function on integerSet K modulo torsion K.
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The map that sends an element of a : integerSet K to the associates class
of its preimage in (๐ K)โฐ. By quotienting by the kernel of the map, which is equal to the
subgroup of torsion, we get the equivalence integerSetQuotEquivAssociates.
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The equivalence between integerSet K modulo torsion K and Associates (๐ K)โฐ.
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The equivalence between integerSet K and the product of the set of nonzero principal
ideals of K and the torsion of K.
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- One or more equations did not get rendered due to their size.
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For an integer n, The equivalence between the elements of integerSet K of norm n
and the product of the set of nonzero principal ideals of K of norm n and the torsion of K.
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For n positive, the number of principal ideals in ๐ K of norm n multiplied by the order
of the torsion of K is equal to the number of elements in integerSet K of norm n.
The intersection between the fundamental cone and the idealLattice defined by the image of
the integral ideal J.
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The map that sends a : idealSet to an element of integerSet. This map exists because
J is an integral ideal.
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- NumberField.mixedEmbedding.fundamentalCone.idealSetMap K J โจa, haโฉ = โจa, โฏโฉ
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The map idealSetMap is actually an equiv between idealSet K J and the elements of
integerSet K whose preimage lies in J.
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For an integer n, The equivalence between the elements of idealSet K of norm n and
the product of the set of nonzero principal ideals of K divisible by J of norm n and the
torsion of K.
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- One or more equations did not get rendered due to their size.
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For s : โ, the number of principal nonzero ideals in ๐ K divisible par J of norm โค s
multiplied by the order of the torsion of K is equal to the number of elements in idealSet K J
of norm โค s.