Gauss points of the discs in the closed unit disc and the points next to them #
Let K be a complete nontrivially normed nonarchimedean field and K⟨T⟩ the coordinate ring of
the closed unit disc closedPolydisc 1 K. For a centre a with ‖a‖ ≤ 1 and a radius
0 < r ≤ 1, the point η_{a,r} of the disc is the Gauss norm of radius r expanded about a:
η_{a,r} : f = ∑ bₙ (T - a)ⁿ ↦ sup_n ‖bₙ‖ rⁿ.
These are the Gauss points of the K-rational discs in the closed unit disc. When K is
algebraically closed they are exactly the points of types (2) (r ∈ |K^×|) and (3)
(r ∉ |K^×|) in Wedhorn's description of the closed unit disc (Example 7.57); over a general K,
Gauss points of discs whose centres need a field extension are not of this form. The point
η_{a,r} is the pullback of the Gauss point η_r = gaussPoint about the origin along the
translation T ↦ T + a of K⟨T⟩.
The main result is that η_{a,r} depends only on the disc D(a, r) = {y : ‖y - a‖ ≤ r}, while
distinct radii give distinct points: η_{a,r} = η_{b,s} exactly when r = s and
‖a - b‖ ≤ r. The key input is that the Gauss norm of radius r is invariant under translation
by any c with ‖c‖ ≤ r, which holds over any complete normed commutative ring with
multiplicative ultrametric norm. In particular, at r = 1 every centre gives the Gauss point of the
disc. Like the Gauss point about the origin, η_{a,r} has trivial support and so is not a
classical point.
The same translation recentres the refined points η_{r⁺} and η_{r⁻} next to η_r, giving
points η_{a,r⁺} (for r < 1) and η_{a,r⁻} of the disc. When K is algebraically closed
and r ∈ |K^×|, these are the points of type (5) in Wedhorn's Example 7.57, written there
a_{>r} and a_{<r}. Both specialize from η_{a,r}, and for r ∈ |K^×| the three points are
pairwise distinct.
The Gauss valuation just above r is invariant under translation by every c with ‖c‖ ≤ r:
translation preserves the last degree in which a series attains its Gauss norm. So η_{a,r⁺}
depends only on the closed disc D(a, r), and there is one such point for each point η_{a,r}.
The Gauss valuation just below r is invariant only under translation by c with ‖c‖ < r,
since translating T by an element of norm r moves the first degree attaining its Gauss norm
from 1 to 0. So η_{a,r⁻} depends exactly on the open disc {y : ‖y - a‖ < r}, and the open
discs of radius r inside D(a, r) give pairwise distinct points next to η_{a,r}. All three
invariance statements come from TauCeti.ValuationSpectrum.valuation_taylorHom_eq: a continuous
valuation that bounds every monomial of a series by the series, and the shifted variable T + c
by T, is invariant under translation by c.
Main definitions #
TauCeti.ValuationSpectrum.discPoint: the pointη_{a,r}of the closed unit disc.TauCeti.ValuationSpectrum.discPointAboveandTauCeti.ValuationSpectrum.discPointBelow: the pointsη_{a,r⁺}andη_{a,r⁻}of the closed unit disc.
Main results #
TauCeti.ValuationSpectrum.closedDiscGaussValuation_taylorHom: the Gauss norm of radiusris invariant under translation bycwith‖c‖ ≤ r.TauCeti.ValuationSpectrum.discPoint_vle_iff:η_{a,r}comparesfandgby the Gauss norms of radiusroff(T + a)andg(T + a).TauCeti.ValuationSpectrum.discPoint_zero: about the origin,η_{0,r}is the Gauss pointη_r.TauCeti.ValuationSpectrum.comap_taylorHom_discPoint: pullingη_{c,r}back along the translationT ↦ T + bgivesη_{c+b,r}.TauCeti.ValuationSpectrum.discPoint_eq_gaussPoint_iff:η_{a,r} = η_sif and only ifr = sand‖a‖ ≤ r.TauCeti.ValuationSpectrum.discPoint_eq_discPoint_iff:η_{a,r} = η_{b,s}if and only ifr = sand‖a - b‖ ≤ r.TauCeti.ValuationSpectrum.discPoint_radius_one: at radius one every centre gives the Gauss point.TauCeti.ValuationSpectrum.supp_discPointandTauCeti.ValuationSpectrum.discPoint_ne_classicalPoint:η_{a,r}has trivial support and is not a classical point.TauCeti.ValuationSpectrum.closedDiscGaussValuationAbove_taylorHomandTauCeti.ValuationSpectrum.closedDiscGaussValuationBelow_taylorHom: the Gauss valuations just above and just belowrare invariant under translation bycwith‖c‖ ≤ r, respectively‖c‖ < r.TauCeti.ValuationSpectrum.discPointAbove_vle_iffandTauCeti.ValuationSpectrum.discPointBelow_vle_iff: howη_{a,r⁺}andη_{a,r⁻}compare series.TauCeti.ValuationSpectrum.discPointAbove_vle_weightedC_iff,TauCeti.ValuationSpectrum.discPointAbove_weightedC_vle_iff,TauCeti.ValuationSpectrum.discPointBelow_vle_weightedC_iffandTauCeti.ValuationSpectrum.discPointBelow_weightedC_vle_iff:η_{a,r⁺}placesT - ajust above the constants of normr, andη_{a,r⁻}just below them.TauCeti.ValuationSpectrum.valuation_weightedX_sub_weightedC_of_lt: a point at whichT - ais never equal in value to a constant takes onT - bthe value ofT - aor ofb - a.TauCeti.ValuationSpectrum.valuation_weightedX_sub_weightedC_discPointAboveandTauCeti.ValuationSpectrum.valuation_weightedX_sub_weightedC_discPointBelow: the values ofη_{a,r±}on every linear polynomialT - b.TauCeti.ValuationSpectrum.discPointAbove_eq_gaussPointAbove_iffandTauCeti.ValuationSpectrum.discPointBelow_eq_gaussPointBelow_iff:η_{a,r⁺} = η_{s⁺}exactly whenr = sand‖a‖ ≤ r, andη_{a,r⁻} = η_{s⁻}exactly whenr = sand‖a‖ < r.TauCeti.ValuationSpectrum.discPointAbove_eq_discPointAbove_iffandTauCeti.ValuationSpectrum.discPointBelow_eq_discPointBelow_iff:η_{a,r⁺} = η_{b,s⁺}exactly whenr = sand‖a - b‖ ≤ r, andη_{a,r⁻} = η_{b,s⁻}exactly whenr = sand‖a - b‖ < r.TauCeti.ValuationSpectrum.discPoint_specializes_discPointAboveandTauCeti.ValuationSpectrum.discPoint_specializes_discPointBelow: both points lie in the closure ofη_{a,r}.TauCeti.ValuationSpectrum.discPointAbove_ne_discPoint,TauCeti.ValuationSpectrum.discPointBelow_ne_discPointandTauCeti.ValuationSpectrum.discPointBelow_ne_discPointAbove: forrthe norm of an element ofK, the three points at centreaand radiusrare pairwise distinct.TauCeti.ValuationSpectrum.supp_discPointAbove,TauCeti.ValuationSpectrum.supp_discPointBelowand the corresponding_ne_classicalPointlemmas: both points have trivial support and are not classical points.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Example 7.57.
- S. Bosch, U. Güntzer, R. Remmert, Non-Archimedean Analysis, §5.1, for the Gauss norm on restricted power series.
- P. Scholze, Perfectoid spaces, Publ. Math. IHÉS 116 (2012), Example 2.20, for the points of type (5).
Translation invariance of the Gauss valuations #
The Gauss norm of radius r is invariant under translation by c with ‖c‖ ≤ r: the
Gauss norms of f(T) and f(T + c) agree. Here c is the sole entry a 0 of the tuple a.
The Gauss valuation just above r is invariant under translation by c with
‖c‖ ≤ r: f(T) and f(T + c) have the same Gauss norm of radius r and the same last degree
attaining it. Here c is the sole entry a 0 of the tuple a.
The Gauss valuation just below r is invariant under translation by c with
‖c‖ < r: f(T) and f(T + c) have the same Gauss norm of radius r and the same first degree
attaining it. Here c is the sole entry a 0 of the tuple a. Translation by c with
‖c‖ = r moves the first degree: T attains its Gauss norm first in degree 1, and T + c in
degree 0.
The points η_{a,r} #
The point η_{a,r} of the closed unit disc, for a centre ‖a‖ ≤ 1 and a radius
0 < r ≤ 1: the Gauss norm of radius r about a, f ↦ sup_n ‖bₙ‖ rⁿ where
f = ∑ bₙ (T - a)ⁿ. It is the pullback of the Gauss point η_r along the translation
T ↦ T + a.
Instances For
The underlying point in Spv K⟨T⟩ is the pullback of the Gauss point η_r along the
translation T ↦ T + a.
The point η_{a,r} compares f and g by the Gauss norms of radius r of f(T + a) and
g(T + a).
The Gauss norm of radius r about a takes the value max r ‖b - a‖ on T - b: this is the
value of η_{a,r} on T - b.
About the origin, η_{0,r} is the Gauss point η_r.
Translating recentres η_{c,r}: pulling η_{c,r} back along the translation
T ↦ T + b gives η_{c+b,r}, since translations compose additively.
Recentring within the disc: η_{a,r} = η_r whenever ‖a‖ ≤ r, by translation invariance
of the Gauss norm of radius r.
η_{a,r} is the Gauss point η_s exactly when r = s and ‖a‖ ≤ r. If ‖a‖ > r, then
η_{a,r} finds T - a strictly smaller than the constant a, while every Gauss point about the
origin finds it at least as large.
η_{a,r} depends only on the disc D(a, r): η_{a,r} = η_{b,s} if and only if r = s
and ‖a - b‖ ≤ r, that is, the radii agree and each centre lies in the other's disc.
At radius one every centre gives the Gauss point of the disc: D(a, 1) is the whole
closed unit disc.
A point η_{a,r} kills only the zero series.
The support of η_{a,r} is trivial.
The points η_{a,r} are not classical points.
The points η_{a,r⁺} and η_{a,r⁻} #
The point η_{a,r⁺} of the closed unit disc, for a centre ‖a‖ ≤ 1 and a radius
0 < r < 1: the Gauss valuation just above r about a, f ↦ (|f|_r, s) with s the last degree
in which f(T + a) attains its Gauss norm |f|_r. It is the pullback of η_{r⁺} along the
translation T ↦ T + a.
Instances For
The point η_{a,r⁻} of the closed unit disc, for a centre ‖a‖ ≤ 1 and a radius
0 < r ≤ 1: the Gauss valuation just below r about a, f ↦ (|f|_r, -s) with s the first
degree in which f(T + a) attains its Gauss norm |f|_r. It is the pullback of η_{r⁻} along the
translation T ↦ T + a.
Instances For
The underlying point of η_{a,r⁺} in Spv K⟨T⟩ is the pullback of η_{r⁺} along the
translation T ↦ T + a.
The underlying point of η_{a,r⁻} in Spv K⟨T⟩ is the pullback of η_{r⁻} along the
translation T ↦ T + a.
The point η_{a,r⁺} compares f and g by the Gauss valuations just above r of
f(T + a) and g(T + a).
The point η_{a,r⁻} compares f and g by the Gauss valuations just below r of
f(T + a) and g(T + a).
About the origin, η_{0,r⁺} is η_{r⁺}.
About the origin, η_{0,r⁻} is η_{r⁻}.
η_{a,r⁺} finds T - a at most a constant c exactly when r < ‖c‖: the value of
T - a lies just above the constants of norm r.
η_{a,r⁺} finds a constant c at most T - a exactly when ‖c‖ ≤ r.
η_{a,r⁻} finds T - a at most a constant c exactly when r ≤ ‖c‖.
η_{a,r⁻} finds a constant c at most T - a exactly when ‖c‖ < r: the value of
T - a lies just below the constants of norm r.
If x(T - a) is strictly larger than x(c) for the constants with Q c and strictly smaller
for the others, then x(T - b) is x(T - a) when Q (b - a) and x(b - a) otherwise: write
T - b = (T - a) - (b - a) and apply the ultrametric equality.
The value of η_{a,r⁺} on T - b is its value on T - a when ‖b - a‖ ≤ r, and the
value of the constant b - a otherwise: T - b = (T - a) - (b - a), and T - a is strictly
larger than the constants of norm at most r and strictly smaller than the others.
The value of η_{a,r⁻} on T - b is its value on T - a when ‖b - a‖ < r, and the
value of the constant b - a otherwise: T - b = (T - a) - (b - a), and T - a is strictly
larger than the constants of norm less than r and strictly smaller than the others.
Translating recentres η_{c,r⁺}: pulling η_{c,r⁺} back along the translation
T ↦ T + b gives η_{c+b,r⁺}.
Translating recentres η_{c,r⁻}: pulling η_{c,r⁻} back along the translation
T ↦ T + b gives η_{c+b,r⁻}.
η_{a,r⁺} is η_{s⁺} exactly when r = s and ‖a‖ ≤ r. If ‖a‖ > r, then η_{a,r⁺}
finds T - a strictly smaller than the constant a, while η_{s⁺} finds a, a monomial of
T - a, at most T - a.
η_{a,r⁻} is η_{s⁻} exactly when r = s and ‖a‖ < r. If ‖a‖ ≥ r, then η_{a,r⁻}
finds T - a strictly smaller than the constant a, while η_{s⁻} finds a, a monomial of
T - a, at most T - a.
η_{a,r⁺} depends only on the closed disc D(a, r): for ‖a‖, ‖b‖ ≤ 1,
η_{a,r⁺} = η_{b,s⁺} if and only if r = s and ‖a - b‖ ≤ r, that is, if and only if
η_{a,r} = η_{b,s} (TauCeti.ValuationSpectrum.discPoint_eq_discPoint_iff).
η_{a,r⁻} depends only on the open disc D°(a, r): for ‖a‖, ‖b‖ ≤ 1,
η_{a,r⁻} = η_{b,s⁻} if and only if r = s and ‖a - b‖ < r. So the open discs of radius r
inside D(a, r) give pairwise distinct points η_{b,r⁻}, all specializations of η_{a,r}.
η_{a,r⁺} is a specialization of η_{a,r}, as η_{r⁺} is of η_r: pullback along the
translation is continuous.
η_{a,r⁻} is a specialization of η_{a,r}, as η_{r⁻} is of η_r: pullback along the
translation is continuous.
η_{a,r⁺} is not η_{a,r} when r is the norm of an element of K.
η_{a,r⁻} is not η_{a,r} when r is the norm of an element of K.
η_{a,r⁻} and η_{a,r⁺} are distinct when r is the norm of an element of K.
The support of η_{a,r⁺} is trivial.
The support of η_{a,r⁻} is trivial.
The points η_{a,r⁺} are not classical points.
The points η_{a,r⁻} are not classical points.