text: BANACH-COLMEZ ET FAISCEAUX COHÉRENTS SUR LA COURBE DE FARGUES-FONTAINE 5 Les mêmes méthodes s’appliquent au disque unité ouvert en dimension quelconque. Elles devraient permettre plus généralement d’étudier la cohomologie pro-étale des espaces rigides Stein lisses déﬁnis sur une extension ﬁnie de Q (7). p BC Terminons cette introduction en soulignant l’analogie frappante entre la catégorie des espaces de Banach-Colmez et celle des groupes quasi-algébriques unipotents (ou proalgé- briques unipotents) de Serre ([55]). Non seulement les déﬁnitions de ces deux catégories ont un air de famille, mais ces objets font leur apparition dans des contextes similaires. Ils interviennent tous deux en théorie du corps de classes : voir respectivement [55] et [22]. Les groupes de cohomologie plate (fppf) de µ sur une variété propre et lisse sur un corps parfait p de caractéristique p sont les points de groupes algébriques unipotents ([45]), de même que la cohomologie syntomique géométrique donne naissance à des espaces de Banach-Colmez ([46]). Il conviendrait d’expliquer et d’explorer cette analogie. Notations. — On note C(cid:91) le basculé de C. Il s’agit d’un corps valué complet algébrique- O ment clos de caractéristique p, dont l’anneau des entiers est C(cid:91) O O := l←im− /p, C(cid:91) (cid:55)→ C x xp O qui s’identiﬁe∈aOussi (comme monoïde m(cid:54)ultiplicatif) à l←im−x(cid:55)→xp C. On ﬁxe ε , avec ε(0) = 1, ε((cid:88)1) = 1, un système compatible de racines de l’unité et C(cid:91) on note ∞ − − ([ε] 1)n t = ( 1)n+1 n n=1 le « 2iπ de Fontaine ». Mes plus vifs remerciements vont à Laurent Fargues qui m’a fait découvrir les espaces de Banach-Colmez et m’a posé la question qui est à l’origine de ce texte. Sans ses nom- breuses suggestions, ses corrections et ses encouragements, ce travail n’aurait probablement pas abouti. Je remercie Peter Scholze pour deux suggestions décisives et une invitation à l’Université de Bonn lors de la rédaction de ce texte, ainsi que Jean-Marc Fontaine pour son intérêt pour mon travail. Je tiens également à remercier Gabriel Dospinescu pour des conversations utiles sur la cohomologie pro-étale, Matthew Morrow pour ses commentaires sur [7], et tous les deux pour leur patience à l’écoute de mes approximations. Bien que je n’aie ﬁnalement pas eu à faire usage des techniques de [9], je suis reconnaissant à Lawrence Breen pour ses explications à leur sujet. Enﬁn, Quentin Guignard et Joaquín Rodrigues Ja- cinto ont eu la gentillesse de me faire part de leurs commentaires sur une version préliminaire de ce texte ; je les en remercie. 2. La catégorie des espaces de Banach-Colmez Dans cette section, nous déﬁnissons les espaces de Banach-Colmez comme faisceaux pro- étales sur la catégorie des espaces perfectoïdes sur Spa(C) (déﬁnition 2.11). Nous expliquons O ensuite pourquoi les revêtements universels de groupes p-divisibles sur donnent naissance C à des espaces de Banach-Colmez, ce qui permet d’en fabriquer des exemples explicites (co- rollaire 2.23). 7. Ce point de vue est esquissé à la ﬁn de [42]. ##LTLine## ##LTLine##

text: BANACH-COLMEZ

text: ET

text: FAISCEAUX

text: COHÉRENTS

text: SUR

text: LA

text: COURBE

text: DE

text: FARGUES-FONTAINE

text: Les

text: mêmes

text: méthodes

text: s’appliquent

text: au

text: disque

text: unité

text: ouvert

text: en

text: dimension

text: quelconque.

text: Elles

text: devraient

text: permettre

text: plus

text: généralement

text: d’étudier

text: la

text: cohomologie

text: pro-étale

text: des

text: espaces

text: rigides

text: Stein

text: lisses

text: déﬁnis

text: sur

text: une

text: extension

text: ﬁnie

text: de

text: Q

text: p

text: BC

text: Terminons

text: cette

text: introduction

text: en

text: soulignant

text: l’analogie

text: frappante

text: entre

text: la

text: catégorie

text: des

text: espaces

text: de

text: Banach-Colmez

text: et

text: celle

text: des

text: groupes

text: quasi-algébriques

text: unipotents

text: (ou

text: proalgé-

text: briques

text: unipotents)

text: de

text: Serre

text: Non

text: seulement

text: les

text: déﬁnitions

text: de

text: ces

text: deux

text: catégories

text: ont

text: un

text: air

text: de

text: famille,

text: mais

text: ces

text: objets

text: font

text: leur

text: apparition

text: dans

text: des

text: contextes

text: similaires.

text: Ils

text: interviennent

text: tous

text: deux

text: en

text: théorie

text: du

text: corps

text: de

text: classes

text: voir

text: respectivement

text: et

text: Les

text: groupes

text: de

text: cohomologie

text: plate

text: (fppf)

text: de

text: µ

text: sur

text: une

text: variété

text: propre

text: et

text: lisse

text: sur

text: un

text: corps

text: parfait

text: p

text: de

text: caractéristique

text: p

text: sont

text: les

text: points

text: de

text: groupes

text: algébriques

text: unipotents

text: de

text: même

text: que

text: la

text: cohomologie

text: syntomique

text: géométrique

text: donne

text: naissance

text: à

text: des

text: espaces

text: de

text: Banach-Colmez

text: Il

text: conviendrait

text: d’expliquer

text: et

text: d’explorer

text: cette

text: analogie.

text: Notations.

text: On

text: note

text: le

text: basculé

text: de

text: C.

text: Il

text: s’agit

text: d’un

text: corps

text: valué

text: complet

text: algébrique-

text: O

text: ment

text: clos

text: de

text: caractéristique

text: p,

text: dont

text: l’anneau

text: des

text: entiers

text: est

text: O

text: O

text: l←im−

text: /p,

text: C

text: x

text: xp

text: O

text: qui

text: s’identiﬁe∈aOussi

text: (comme

text: monoïde

text: à

text: C.

text: On

text: ﬁxe

text: ε

text: avec

text: un

text: système

text: compatible

text: de

text: racines

text: de

text: l’unité

text: et

text: on

text: note

text: ([ε]

text: t

text: n

text: le

text: de

text: Fontaine

text: Mes

text: plus

text: vifs

text: remerciements

text: vont

text: à

text: Laurent

text: Fargues

text: qui

text: m’a

text: fait

text: découvrir

text: les

text: espaces

text: de

text: Banach-Colmez

text: et

text: m’a

text: posé

text: la

text: question

text: qui

text: est

text: à

text: l’origine

text: de

text: ce

text: texte.

text: Sans

text: ses

text: nom-

text: breuses

text: suggestions,

text: ses

text: corrections

text: et

text: ses

text: encouragements,

text: ce

text: travail

text: n’aurait

text: probablement

text: pas

text: abouti.

text: Je

text: remercie

text: Peter

text: Scholze

text: pour

text: deux

text: suggestions

text: décisives

text: et

text: une

text: invitation

text: à

text: l’Université

text: de

text: Bonn

text: lors

text: de

text: la

text: rédaction

text: de

text: ce

text: texte,

text: ainsi

text: que

text: Jean-Marc

text: Fontaine

text: pour

text: son

text: intérêt

text: pour

text: mon

text: travail.

text: Je

text: tiens

text: également

text: à

text: remercier

text: Gabriel

text: Dospinescu

text: pour

text: des

text: conversations

text: utiles

text: sur

text: la

text: cohomologie

text: pro-étale,

text: Matthew

text: Morrow

text: pour

text: ses

text: commentaires

text: sur

text: et

text: tous

text: les

text: deux

text: pour

text: leur

text: patience

text: à

text: l’écoute

text: de

text: mes

text: approximations.

text: Bien

text: que

text: je

text: n’aie

text: ﬁnalement

text: pas

text: eu

text: à

text: faire

text: usage

text: des

text: techniques

text: de

text: je

text: suis

text: reconnaissant

text: à

text: Lawrence

text: Breen

text: pour

text: ses

text: explications

text: à

text: leur

text: sujet.

text: Enﬁn,

text: Quentin

text: Guignard

text: et

text: Joaquín

text: Rodrigues

text: Ja-

text: cinto

text: ont

text: eu

text: la

text: gentillesse

text: de

text: me

text: faire

text: part

text: de

text: leurs

text: commentaires

text: sur

text: une

text: version

text: préliminaire

text: de

text: ce

text: texte

text: je

text: les

text: en

text: remercie.

text: La

text: catégorie

text: des

text: espaces

text: de

text: Banach-Colmez

text: Dans

text: cette

text: section,

text: nous

text: déﬁnissons

text: les

text: espaces

text: de

text: Banach-Colmez

text: comme

text: faisceaux

text: pro-

text: étales

text: sur

text: la

text: catégorie

text: des

text: espaces

text: perfectoïdes

text: sur

text: Spa(C)

text: (déﬁnition

text: Nous

text: expliquons

text: O

text: ensuite

text: pourquoi

text: les

text: revêtements

text: universels

text: de

text: groupes

text: p-divisibles

text: sur

text: donnent

text: naissance

text: C

text: à

text: des

text: espaces

text: de

text: Banach-Colmez,

text: ce

text: qui

text: permet

text: d’en

text: fabriquer

text: des

text: exemples

text: explicites

text: (co-

text: rollaire

text: Ce

text: point

text: de

text: vue

text: est

text: esquissé

text: à

text: la

text: ﬁn

text: de

text: ##LTLine##

text: ##LTLine##


text: BANACH-COLMEZ ET FAISCEAUX COHÉRENTS SUR LA COURBE DE FARGUES-FONTAINE 47 Démonstration. — En eﬀet, le théorème 7.1 et la proposition 5.6 permettent de reconstruire BC Coh à partir de , puisqu’ils montrent que Xsch (cid:39) BC Coh + Xsch (avec les notations du paragraphe ). Or un théorème de Gabriel ([30, Ch VI, §3]) aﬃrme qu’un schéma noethérien Z peut être reconstruit à partir de la catégorie abélienne Coh Z de ses faisceaux cohérents, à équivalence près (Z s’identiﬁe comme espace topologique à I l’ensemble des sous-catégories de Serre irréductibles de Coh avec pour ouverts les D( ), Z I ensemble des sous-catégories de Serre ne contenant pas une sous-catégorie de Serre ﬁxée ; le I I faisceau structural évalué sur l’ouvert D( ) est le centre de la catégorie abélienne Coh / ). Z Corollaire 7.15. — Les espaces de Banach-Colmez sont des diamants sur Spa(C). Démonstration. — On vient de voir que tout espace de Banach-Colmez était le quotient par un Q -Espace Vectoriel de dimension ﬁnie (donc par une relation d’équivalence pro-étale) p du revêtement universel d’un groupe p-divisible, qui est un espace perfectoïde. BC BC Question 7.16. — Est-il vrai que les seuls objets représentables de sont ceux de rep ? Remarque 7.17. — Dans la question précédente, on a laissé de côté les C-Espaces Vecto- riels de dimension ﬁnie, qui ne sont pas représentables par un espace perfectoïde. Si toutefois on se demande quels espaces de Banach-Colmez sont représentables par des espaces adiques BC sur C, on peut s’attendre à obtenir exactement les sommes directes d’un objet de rep et d’un C-Espace Vectoriel de dimension ﬁnie, c’est-à-dire précisément les revêtements univer- sels de groupes analytiques rigides de type p-divisible au sens de [19]. Notons Cohtors la sous-catégorie de Coh formée des faisceaux de torsion, qui est la sous- X X catégorie de Serre des objets de rang rg = 0. Le foncteur ﬁbre générique identiﬁe le quotient Coh /Cohtors à la catégorie des Frac(B )-espaces vectoriels (B = B+ [1/t] et Frac(B ) X X e e cris e est le corps des fonctions de X). En particulier, cette catégorie est semi-simple avec un seul objet simple. − Si l’on fait la même manipulation avec Coh , i.e. si l’on quotiente par la sous-catégorie − X de Serre formée des objets de rang rg = 0, qui est la sous-catégorie des ﬁbrés semi-stables de pente 0 placés en degré 0, on obtient aussi une catégorie semi-simple avec un seul objet simple S à isomorphisme près : cela se déduit immédiatement des suites exactes du lemme 7.3. Par conséquent, C := End − − − (S) Coh /(Coh )rg =0 X X est une algèbre à division. − − − BC − Le théorème 7.1 identiﬁe Coh /(Coh )rg =0 à /(Q EV de d.f.). Comme on peut X X p choisir pour S la classe [i∞ ∗C] de i∞ ∗C dans le quotient, C est l’algèbre à division étudiée , , par Colmez dans [12, §5 et §9]. On a un morphisme d’algèbres → C = End − (i∞ ∗C) End − − − ([i∞ ∗C]), Coh , Coh /(Coh )rg =0 , X X X qui fait de C une sous-algèbre commutative de C . On pourrait également choisir comme ≥ représentant de S la classe de Bϕh=p, h 1, ou celle de i ∗C , x point fermé de x (C x, x x ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BANACH-COLMEZ

label: paragraph

text: ET

text: FAISCEAUX

text: COHÉRENTS

text: SUR

text: LA

text: COURBE

text: DE

text: FARGUES-FONTAINE

text: Démonstration.

text: En

text: eﬀet,

text: le

text: théorème

text: et

text: la

text: proposition

text: permettent

text: de

text: reconstruire

text: BC

text: Coh

text: à

text: partir

text: de

text: puisqu’ils

text: montrent

text: que

text: Xsch

text: BC

text: Coh

text: Xsch

text: (avec

text: les

text: notations

text: du

text: paragraphe

text: Or

text: un

text: théorème

text: de

text: Gabriel

text: Ch

text: VI,

text: aﬃrme

text: qu’un

text: schéma

text: noethérien

text: Z

text: peut

text: être

text: reconstruit

text: à

text: partir

text: de

text: la

text: catégorie

text: abélienne

text: Coh

text: Z

text: de

text: ses

text: faisceaux

text: cohérents,

text: à

text: équivalence

text: près

text: (Z

text: s’identiﬁe

text: comme

text: espace

text: topologique

text: à

text: I

text: l’ensemble

text: des

text: sous-catégories

text: de

text: Serre

text: irréductibles

text: de

text: Coh

text: avec

text: pour

text: ouverts

text: les

text: D(

text: Z

text: I

text: ensemble

text: des

text: sous-catégories

text: de

text: Serre

text: ne

text: contenant

text: pas

text: une

text: sous-catégorie

text: de

text: Serre

text: ﬁxée

text: le

text: I

text: I

text: faisceau

text: structural

text: évalué

text: sur

text: l’ouvert

text: D(

text: est

text: le

text: centre

text: de

text: la

text: catégorie

text: abélienne

text: Coh

text: Z

text: Corollaire

text: Les

text: espaces

text: de

text: Banach-Colmez

text: sont

text: des

text: diamants

text: sur

text: Spa(C).

text: Démonstration.

text: On

text: vient

text: de

text: voir

text: que

text: tout

text: espace

text: de

text: Banach-Colmez

text: était

text: le

text: quotient

text: par

text: un

text: Q

text: -Espace

text: Vectoriel

text: de

text: dimension

text: ﬁnie

text: (donc

text: par

text: une

text: relation

text: d’équivalence

text: pro-étale)

text: p

text: du

text: revêtement

text: universel

text: d’un

text: groupe

text: p-divisible,

text: qui

text: est

text: un

text: espace

text: perfectoïde.

text: BC

text: BC

text: Question

text: Est-il

text: vrai

text: que

text: les

text: seuls

text: objets

text: représentables

text: de

text: sont

text: ceux

text: de

text: rep

text: Remarque

text: Dans

text: la

text: question

text: précédente,

text: on

text: a

text: laissé

text: de

text: côté

text: les

text: C-Espaces

text: Vecto-

text: riels

text: de

text: dimension

text: ﬁnie,

text: qui

text: ne

text: sont

text: pas

text: représentables

text: par

text: un

text: espace

text: perfectoïde.

text: Si

text: toutefois

text: on

text: se

text: demande

text: quels

text: espaces

text: de

text: Banach-Colmez

text: sont

text: représentables

text: par

text: des

text: espaces

text: adiques

text: BC

text: sur

text: C,

text: on

text: peut

text: s’attendre

text: à

text: obtenir

text: exactement

text: les

text: sommes

text: directes

text: d’un

text: objet

text: de

text: rep

text: et

text: d’un

text: C-Espace

text: Vectoriel

text: de

text: dimension

text: ﬁnie,

text: c’est-à-dire

text: précisément

text: les

text: revêtements

text: univer-

text: sels

text: de

text: groupes

text: analytiques

text: rigides

text: de

text: type

text: p-divisible

text: au

text: sens

text: de

text: Notons

text: Cohtors

text: la

text: sous-catégorie

text: de

text: Coh

text: formée

text: des

text: faisceaux

text: de

text: torsion,

text: qui

text: est

text: la

text: sous-

text: X

text: X

text: catégorie

text: de

text: Serre

text: des

text: objets

text: de

text: rang

text: rg

text: Le

text: foncteur

text: ﬁbre

text: générique

text: identiﬁe

text: le

text: quotient

text: Coh

text: /Cohtors

text: à

text: la

text: catégorie

text: des

text: Frac(B

text: )-espaces

text: vectoriels

text: (B

text: B+

text: et

text: Frac(B

text: X

text: X

text: e

text: e

text: cris

text: e

text: est

text: le

text: corps

text: des

text: fonctions

text: de

text: X).

text: En

text: particulier,

text: cette

text: catégorie

text: est

text: semi-simple

text: avec

text: un

text: seul

text: objet

text: simple.

text: Si

text: l’on

text: fait

text: la

text: même

text: manipulation

text: avec

text: Coh

text: i.e.

text: si

text: l’on

text: quotiente

text: par

text: la

text: sous-catégorie

text: X

text: de

text: Serre

text: formée

text: des

text: objets

text: de

text: rang

text: rg

text: qui

text: est

text: la

text: sous-catégorie

text: des

text: ﬁbrés

text: semi-stables

text: de

text: pente

text: placés

text: en

text: degré

text: on

text: obtient

text: aussi

text: une

text: catégorie

text: semi-simple

text: avec

text: un

text: seul

text: objet

text: simple

text: S

text: à

text: isomorphisme

text: près

text: cela

text: se

text: déduit

text: immédiatement

text: des

text: suites

text: exactes

text: du

text: lemme

text: Par

text: conséquent,

text: C

text: End

text: (S)

text: Coh

text: /(Coh

text: )rg

text: X

text: X

text: est

text: une

text: algèbre

text: à

text: division.

text: BC

text: Le

text: théorème

text: identiﬁe

text: Coh

text: /(Coh

text: )rg

text: à

text: /(Q

text: EV

text: de

text: d.f.).

text: Comme

text: on

text: peut

text: X

text: X

text: p

text: choisir

text: pour

text: S

text: la

text: classe

text: [i∞

text: ∗C]

text: de

text: i∞

text: ∗C

text: dans

text: le

text: quotient,

text: C

text: est

text: l’algèbre

text: à

text: division

text: étudiée

text: par

text: Colmez

text: dans

text: §5

text: et

text: On

text: a

text: un

text: morphisme

text: d’algèbres

text: C

text: End

text: (i∞

text: ∗C)

text: End

text: ([i∞

text: ∗C]),

text: Coh

text: Coh

text: /(Coh

text: )rg

text: X

text: X

text: X

text: qui

text: fait

text: de

text: C

text: une

text: sous-algèbre

text: commutative

text: de

text: C

text: On

text: pourrait

text: également

text: choisir

text: comme

text: représentant

text: de

text: S

text: la

text: classe

text: de

text: Bϕh=p,

text: h

text: ou

text: celle

text: de

text: i

text: ∗C

text: x

text: point

text: fermé

text: de

text: x

text: (C

text: x,

text: x

text: x

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 ARTHUR-CÉSAR LE BRAS Dans tout ce paragraphe, K = C ou K = C(cid:91). Déﬁnition 2.1. — Une K-algèbre perfectoïde est une K-algèbre de Banach R uniforme, ◦ i.e. telle que l’ensemble R des éléments de R à puissances bornées soit borné, et telle que ◦ → ◦ le Frobenius Φ : R /p R /p soit surjectif (en particulier, une C(cid:91)-algèbre perfectoïde est simplement une C(cid:91)-algèbre de Banach uniforme et parfaite). Si R est une C-algèbre perfectoïde, le basculement R(cid:91) est déﬁni par ◦ ◦ ◦ ⊗ R(cid:91), = ←lim− R /p ; R(cid:91) = R(cid:91), O C(cid:91). C(cid:91) Φ ◦ ◦ C’est une C(cid:91)-algèbre perfectoïde, et (R(cid:91)) = R(cid:91), . On dispose d’une application continue et → (cid:55)→ multiplicative R(cid:91) = ←lim− (cid:55)→ R R de projection sur la première coordonnée, notée f f(cid:93). x xp Déﬁnition 2.2. — Une K-algèbre aﬃnoïde perfectoïde est un couple (R, R+), avec R une ⊂ ◦ K-algèbre perfectoïde et R+ R un sous-anneau ouvert intégralement clos. O Théorème 2.3. — Soit (R, R+) une K-algèbre perfectoïde et X = Spa(R, R+). Alors ⊂ O X est un faisceau et pour tout ouvert rationnel U X, (U) est encore une K-algèbre X perfectoïde. ∈ ∈ Si (R(cid:91), R(cid:91)+) est la basculée de (R, R+), l’application qui à x X associe x(cid:91) X(cid:91) = | | | | ∈ Spa(R(cid:91), R(cid:91)+), déﬁni par f(x(cid:91)) = f(cid:93)(x) (8), si f R, induit un homémorphisme entre X et X(cid:91) qui identiﬁe les ouverts rationnels. Déﬁnition 2.4. — Un espace perfectoïde sur K est un espace adique sur K recouvert par des ouverts isomorphes à Spa(R, R+), pour certaines K-algèbres perfectoïdes (R, R+). On notera Perf la catégorie des espaces perfectoïdes sur K. K Introduisons maintenant deux topologies de Grothendieck sur la catégorie Perf , la to- K pologie pro-étale et la v-topologie. → Déﬁnition 2.5. — Soit f : Y X un morphisme entre espaces adiques analytiques (9) sur K. Le morphisme f est dit aﬃnoïde pro-étale si X = Spa(R, R+) et Y = Spa(S, S+) sont → aﬃnoïdes et si Y = l←im− Y X s’écrit comme limite projective coﬁltrante de morphismes → i → étales Y X, avec Y = Spa(S , S+) aﬃnoïde. Le morphisme f : X Y est dit pro-étale i i i i s’il est aﬃnoïde pro-étale localement sur X et sur Y . Déﬁnition 2.6. — i) Le gros site pro-étale de K, noté Perf (10), est la topologie de K,pro´et{ → ∈ } Grothendieck sur Perf pour laquelle une famille de morphismes f : S S, i I est K i i | | 8. Rappelons que f(x) est la notation consacrée pour x(f). 9. Rappelons qu’un point d’un espace adique est dit non analytique si la valuation correspondante a un noyau ouvert. Un espace adique est dit analytique si aucun de ses points n’est non analytique. De façon équivalente, il peut être recouvert par des ouverts Spa(R,R+) avec R un anneau de Huber-Tate. En particulier, un espace perfectoïde est analytique. 10. Dans la suite, on aura à considérer des groupes d’extensions entre faisceaux sur le site Perf . C,pro´et Pour éviter les problèmes de théorie des ensembles, on se restreindra aux espaces perfectoïdes X sur C, | | avec card( X ) < κ, κ étant un cardinal inaccessible (donc non dénombrable). Cela suﬃt à garantir que tout objet du site est localement sympathique (voir plus loin). Même convention pour les groupes de cohomologie pro-étale et les groupes de v-cohomologie d’un espace adique analytique. ##LTLine##

text: ARTHUR-CÉSAR

text: LE

text: BRAS

text: Dans

text: tout

text: ce

text: paragraphe,

text: K

text: C

text: ou

text: K

text: Déﬁnition

text: Une

text: K-algèbre

text: perfectoïde

text: est

text: une

text: K-algèbre

text: de

text: Banach

text: R

text: uniforme,

text: i.e.

text: telle

text: que

text: l’ensemble

text: R

text: des

text: éléments

text: de

text: R

text: à

text: puissances

text: bornées

text: soit

text: borné,

text: et

text: telle

text: que

text: le

text: Frobenius

text: Φ

text: R

text: /p

text: R

text: /p

text: soit

text: surjectif

text: (en

text: particulier,

text: une

text: perfectoïde

text: est

text: simplement

text: une

text: de

text: Banach

text: uniforme

text: et

text: parfaite).

text: Si

text: R

text: est

text: une

text: C-algèbre

text: perfectoïde,

text: le

text: basculement

text: est

text: déﬁni

text: par

text: ←lim−

text: R

text: /p

text: O

text: Φ

text: C’est

text: une

text: perfectoïde,

text: et

text: On

text: dispose

text: d’une

text: application

text: continue

text: et

text: multiplicative

text: ←lim−

text: R

text: R

text: de

text: projection

text: sur

text: la

text: première

text: coordonnée,

text: notée

text: f

text: x

text: xp

text: Déﬁnition

text: Une

text: K-algèbre

text: aﬃnoïde

text: perfectoïde

text: est

text: un

text: couple

text: (R,

text: R+),

text: avec

text: R

text: une

text: K-algèbre

text: perfectoïde

text: et

text: R+

text: R

text: un

text: sous-anneau

text: ouvert

text: intégralement

text: clos.

text: O

text: Théorème

text: Soit

text: (R,

text: R+)

text: une

text: K-algèbre

text: perfectoïde

text: et

text: X

text: Spa(R,

text: R+).

text: Alors

text: O

text: X

text: est

text: un

text: faisceau

text: et

text: pour

text: tout

text: ouvert

text: rationnel

text: U

text: X,

text: (U)

text: est

text: encore

text: une

text: K-algèbre

text: X

text: perfectoïde.

text: Si

text: est

text: la

text: basculée

text: de

text: (R,

text: R+),

text: l’application

text: qui

text: à

text: x

text: X

text: associe

text: déﬁni

text: par

text: si

text: f

text: R,

text: induit

text: un

text: homémorphisme

text: entre

text: X

text: et

text: qui

text: identiﬁe

text: les

text: ouverts

text: rationnels.

text: Déﬁnition

text: Un

text: espace

text: perfectoïde

text: sur

text: K

text: est

text: un

text: espace

text: adique

text: sur

text: K

text: recouvert

text: par

text: des

text: ouverts

text: isomorphes

text: à

text: Spa(R,

text: R+),

text: pour

text: certaines

text: K-algèbres

text: perfectoïdes

text: (R,

text: R+).

text: On

text: notera

text: Perf

text: la

text: catégorie

text: des

text: espaces

text: perfectoïdes

text: sur

text: K.

text: K

text: Introduisons

text: maintenant

text: deux

text: topologies

text: de

text: Grothendieck

text: sur

text: la

text: catégorie

text: Perf

text: la

text: to-

text: K

text: pologie

text: pro-étale

text: et

text: la

text: v-topologie.

text: Déﬁnition

text: Soit

text: f

text: Y

text: X

text: un

text: morphisme

text: entre

text: espaces

text: adiques

text: analytiques

text: sur

text: K.

text: Le

text: morphisme

text: f

text: est

text: dit

text: aﬃnoïde

text: pro-étale

text: si

text: X

text: Spa(R,

text: R+)

text: et

text: Y

text: Spa(S,

text: S+)

text: sont

text: aﬃnoïdes

text: et

text: si

text: Y

text: l←im−

text: Y

text: X

text: s’écrit

text: comme

text: limite

text: projective

text: coﬁltrante

text: de

text: morphismes

text: i

text: étales

text: Y

text: X,

text: avec

text: Y

text: Spa(S

text: S+)

text: aﬃnoïde.

text: Le

text: morphisme

text: f

text: X

text: Y

text: est

text: dit

text: pro-étale

text: i

text: i

text: i

text: i

text: s’il

text: est

text: aﬃnoïde

text: pro-étale

text: localement

text: sur

text: X

text: et

text: sur

text: Y

text: Déﬁnition

text: i)

text: Le

text: gros

text: site

text: pro-étale

text: de

text: K,

text: noté

text: Perf

text: est

text: la

text: topologie

text: de

text: K,pro´et{

text: Grothendieck

text: sur

text: Perf

text: pour

text: laquelle

text: une

text: famille

text: de

text: morphismes

text: f

text: S

text: S,

text: i

text: I

text: est

text: K

text: i

text: i

text: Rappelons

text: que

text: f(x)

text: est

text: la

text: notation

text: consacrée

text: pour

text: x(f).

text: Rappelons

text: qu’un

text: point

text: d’un

text: espace

text: adique

text: est

text: dit

text: non

text: analytique

text: si

text: la

text: valuation

text: correspondante

text: a

text: un

text: noyau

text: ouvert.

text: Un

text: espace

text: adique

text: est

text: dit

text: analytique

text: si

text: aucun

text: de

text: ses

text: points

text: n’est

text: non

text: analytique.

text: De

text: façon

text: équivalente,

text: il

text: peut

text: être

text: recouvert

text: par

text: des

text: ouverts

text: Spa(R,R+)

text: avec

text: R

text: un

text: anneau

text: de

text: Huber-Tate.

text: En

text: particulier,

text: un

text: espace

text: perfectoïde

text: est

text: analytique.

text: Dans

text: la

text: suite,

text: on

text: aura

text: à

text: considérer

text: des

text: groupes

text: d’extensions

text: entre

text: faisceaux

text: sur

text: le

text: site

text: Perf

text: C,pro´et

text: Pour

text: éviter

text: les

text: problèmes

text: de

text: théorie

text: des

text: ensembles,

text: on

text: se

text: restreindra

text: aux

text: espaces

text: perfectoïdes

text: X

text: sur

text: C,

text: avec

text: card(

text: X

text: κ,

text: κ

text: étant

text: un

text: cardinal

text: inaccessible

text: (donc

text: non

text: dénombrable).

text: Cela

text: suﬃt

text: à

text: garantir

text: que

text: tout

text: objet

text: du

text: site

text: est

text: localement

text: sympathique

text: (voir

text: plus

text: loin).

text: Même

text: convention

text: pour

text: les

text: groupes

text: de

text: cohomologie

text: pro-étale

text: et

text: les

text: groupes

text: de

text: v-cohomologie

text: d’un

text: espace

text: adique

text: analytique.

text: ##LTLine##


text: (cid:32) (cid:33) V V 2 1 = ρ n (πβ)1/m . (10) 104 10 1 V 2V A A The integer m is the order of the parametric excitation equal to the number of resonances up to the given value of Vn/2VA. 103 The dependence on β1/m is typical for the threshold value of resonant parametric excitation [29]. The order number m at a 10 5 resonance is (cid:32) (cid:33) 102 m = 2ωA (cid:39) n √q = √n Vn , (11) (100, 33) Ω 2 2 2VA 101 (cid:39) hence we can approximate m 0.28n for q = 0.4. Equa- 10 10 101 102 103 104 tion 10 gives realistic values for low-order resonances [30, 31]. The width of the instability tongues decreases rapidly with in- MB/MA creasing order of the parametric resonance. For the Mathieu equation this width is proportional to qm for a resonance ema- nating from (q, a) = (0, m2). Asymptotic approximations work Ftwigouiroen3s:oTfhmeacsusrvMeAs,sMhoBwwthiteh−mthienismamume scph−rairnggecroantisotaQntBs/,Qi.eA. r[eκqBu/iκrAe]d2rtfo=tra1p; well for narrow resonances in the limit of large n as described these are given by Eq. 18 for 10 10 < β < 10 1. The value of damping becomes in c, whereas numerical calculations require a ﬁne grid for such unimportant at large mass ratios and the minimum charge ratio te√nds to the narrow features and hence long computation times. Equations 8 asymptotic value given in Eq. 19. The black dashed line, QB/QA = MB/MA, and 10 are consistent as m → ∞, since (πβ)1/m (cid:39) 1 for large cfroerqreusepnocnydiss tmooEstq.us4efwulithbetlhoewctohnisdiltiinoen. [κTBh/eκAp]o1irnft =(√1M.B/UMsAin,gQBa/sQeAc)on=d values of m. (100,33) corresponds to the numerical simulation in Section c. For large n the critical curve constrains the maximum value of V before the onset of parametric resonance in species A, as 1 determined by Eq. 10, where there is an implicit dependence quantity Vn/2VA = qA is the q-parameter in the Mathieu equa(cid:39)- tion for species A for which we take the typical value of q othfemn o−n1nto=ngΩune/sΩa.reAwtisdmeaalnldn,Etqh.e1s0taibsilnitoyt rveagliido.nsHboewtweveeern, 0.4 (and similarly for qB = V1/2VB(cid:18)) so(cid:19)that ζ0 = ρqA/2 (cid:39)A0.1 under optimal conditions. Substituting Eq. 13 into Eq. 12 gives it is possible to avoid parametric resonances by careful choice of trap parameters on a case-by-case basis, where more general (cid:39) n 2 rules cannot be established. We note that trapping with two η2rf = (ζ0n)2 (πβ)2/m 10 (πβ)2/m . (14) frequencies is most suitable for combinations of species with widely diﬀerent Q/M, for which n will be large. (cid:29) Hence, we require n 10 for the two-frequency scheme 2.2. Enhancing the weakest spring constant to enhance the trapping of two species, i.e. η2rf > 1. As an example consider typical values of n = 100, m = 28, and β = We now discuss the parameter range for which the use of a 10−5, which give (πβ)2/m = 0.48 and η2rf (cid:39) 48. Although the second frequency increases the spring constant for the heavier lower limit for n could be estimated more accurately (using the species, without signiﬁcantly aﬀecting the lighter ions. Com- relationship between m and n in Eq. 11), it is more instructive parison of the ratio of the spring constants for single-frequency to examine the two-frequency scheme from the viewpoint of operation of the Paul trap (Eq. 4) with that given in Eq. 5 shows balancing the spring constants. that the two-frequency(cid:32)scheme e(cid:33)nhan(cid:32)ces t(cid:33)he ratio κB/κA by a 2.3. Range of applicability: balancing the spring constants factor of In this section, we ﬁnd operating parameters for which the V · Ω 2 nV 2 use of a second frequency ensures both species have similar η2rf = Ω1 Vn = V 1 . (12) spring constants in the radial direction of a linear Paul trap (see n n c). The same considerations apply for a quadrupole guide. For this, the two-frequency scheme can be useful only when the ra- The threshold voltag(cid:32)e for(cid:33)parametric excitation of species A, tio of the spring constants for single-frequency operation of the given by Eq. 10, can be rewritten as Paul trap (given in Eq. 4) is less than unity, i.e. weaker trapping VV1n = ρ2 2VVnA (πβ)1/m ≡ ζ0 (πβ)1/m (13) ospfrtihneg hceoanvsitearntssp,e[cκiBe(cid:32)/sκ.AF]2rr(cid:33)ofm= E1,qr.e5q,uwireesﬁnd that balancing the where m is the order number and β = ΓA/Ω describes the nV1 2 · Q2B/MB = 1. (15) damping f(cid:28)or motion driven at angular→frequency Ω, and we as- Vn Q2A/MA sume πβ 1; the function (πβ)1/m 1 as m increases. The 5 ] Q/ Q[ nim A B ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

font: OXPAMV+txex

text: V

text: V

font: SBXDAT+rtxr

text: ρ

font: WJRVWF+rtxmi

text: n

font: default

text: V

text: A

text: A

text: The

text: integer

text: m

text: is

text: the

text: order

text: of

text: the

text: parametric

text: excitation

text: equal

text: to

text: the

text: number

text: of

text: resonances

text: up

text: to

text: the

text: given

text: value

text: of

text: The

text: dependence

text: on

text: is

text: typical

text: for

text: the

text: threshold

text: value

text: of

text: resonant

text: parametric

text: excitation

text: The

text: order

text: number

text: m

text: at

text: a

text: resonance

text: is

text: m

font: GLQDAC+txsy

text: n

text: √q

text: √n

text: Vn

text: Ω

text: hence

text: we

text: can

text: approximate

text: m

text: for

text: q

text: Equa-

text: tion

text: gives

text: realistic

text: values

text: for

text: low-order

text: resonances

text: The

text: width

text: of

text: the

text: instability

text: tongues

text: decreases

text: rapidly

text: with

text: in-

text: MB/MA

text: creasing

text: order

text: of

text: the

text: parametric

text: resonance.

text: For

text: the

text: Mathieu

text: equation

text: this

text: width

text: is

text: proportional

text: to

text: qm

text: for

text: a

text: resonance

text: ema-

text: nating

text: from

text: (q,

text: a)

text: Asymptotic

text: approximations

text: work

text: scph−rairnggecroantisotaQntBs/,Qi.eA.

text: well

text: for

text: narrow

text: resonances

text: in

text: the

text: limit

text: of

text: large

text: n

text: as

text: described

text: these

text: are

text: given

text: by

text: Eq.

text: for

text: β

text: The

text: value

text: of

text: damping

text: becomes

text: in

text: c,

text: whereas

text: numerical

text: calculations

text: require

text: a

text: ﬁne

text: grid

text: for

text: such

text: unimportant

text: at

text: large

text: mass

text: ratios

text: and

text: the

text: minimum

text: charge

text: ratio

text: te√nds

text: to

text: the

text: narrow

text: features

text: and

text: hence

text: long

text: computation

text: times.

text: Equations

text: asymptotic

text: value

text: given

text: in

text: Eq.

text: The

text: black

text: dashed

text: line,

text: QB/QA

text: MB/MA,

text: and

text: are

text: consistent

text: as

text: m

text: since

text: for

text: large

text: cfroerqreusepnocnydiss

text: values

text: of

text: m.

text: corresponds

text: to

text: the

text: numerical

text: simulation

text: in

text: Section

text: c.

text: For

text: large

text: n

text: the

text: critical

text: curve

text: constrains

text: the

text: maximum

text: value

text: of

text: V

text: before

text: the

text: onset

text: of

text: parametric

text: resonance

text: in

text: species

text: A,

text: as

text: determined

text: by

text: Eq.

text: where

text: there

text: is

text: an

text: implicit

text: dependence

text: quantity

text: qA

text: is

text: the

text: q-parameter

text: in

text: the

text: Mathieu

text: tion

text: for

text: species

text: A

text: for

text: which

text: we

text: take

text: the

text: typical

text: value

text: of

text: q

text: othfemn

text: rveagliido.nsHboewtweveeern,

text: (and

text: similarly

text: for

text: qB

text: ζ0

text: under

text: optimal

text: conditions.

text: Substituting

text: Eq.

text: into

text: Eq.

text: gives

text: it

text: is

text: possible

text: to

text: avoid

text: parametric

text: resonances

text: by

text: careful

text: choice

text: of

text: trap

text: parameters

text: on

text: a

text: case-by-case

text: basis,

text: where

text: more

text: general

text: n

text: rules

text: cannot

text: be

text: established.

text: We

text: note

text: that

text: trapping

text: with

text: two

text: frequencies

text: is

text: most

text: suitable

text: for

text: combinations

text: of

text: species

text: with

text: widely

text: diﬀerent

text: Q/M,

text: for

text: which

text: n

text: will

text: be

text: large.

text: Hence,

text: we

text: require

text: n

text: for

text: the

text: two-frequency

text: scheme

text: Enhancing

text: the

text: weakest

text: spring

text: constant

text: to

text: enhance

text: the

text: trapping

text: of

text: two

text: species,

text: i.e.

text: As

text: an

text: example

text: consider

text: typical

text: values

text: of

text: n

text: m

text: and

text: β

text: We

text: now

text: discuss

text: the

text: parameter

text: range

text: for

text: which

text: the

text: use

text: of

text: a

text: which

text: give

text: and

text: Although

text: the

text: second

text: frequency

text: increases

text: the

text: spring

text: constant

text: for

text: the

text: heavier

text: lower

text: limit

text: for

text: n

text: could

text: be

text: estimated

text: more

text: accurately

text: (using

text: the

text: species,

text: without

text: signiﬁcantly

text: aﬀecting

text: the

text: lighter

text: ions.

text: Com-

text: relationship

text: between

text: m

text: and

text: n

text: in

text: Eq.

text: it

text: is

text: more

text: instructive

text: parison

text: of

text: the

text: ratio

text: of

text: the

text: spring

text: constants

text: for

text: single-frequency

text: to

text: examine

text: the

text: two-frequency

text: scheme

text: from

text: the

text: viewpoint

text: of

text: operation

text: of

text: the

text: Paul

text: trap

text: (Eq.

text: with

text: that

text: given

text: in

text: Eq.

text: shows

text: balancing

text: the

text: spring

text: constants.

text: that

text: the

text: ratio

text: κB/κA

text: by

text: a

text: Range

text: of

text: applicability:

text: balancing

text: the

text: spring

text: constants

text: factor

text: of

text: In

text: this

text: section,

text: we

text: ﬁnd

text: operating

text: parameters

text: for

text: which

text: the

text: V

text: Ω

text: nV

text: use

text: of

text: a

text: second

text: frequency

text: ensures

text: both

text: species

text: have

text: similar

text: Ω1

text: Vn

text: V

text: spring

text: constants

text: in

text: the

text: radial

text: direction

text: of

text: a

text: linear

text: Paul

text: trap

text: (see

text: n

text: n

text: c).

text: The

text: same

text: considerations

text: apply

text: for

text: a

text: quadrupole

text: guide.

text: For

text: this,

text: the

text: two-frequency

text: scheme

text: can

text: be

text: useful

text: only

text: when

text: the

text: ra-

text: The

text: threshold

text: excitation

text: of

text: species

text: A,

text: tio

text: of

text: the

text: spring

text: constants

text: for

text: single-frequency

text: operation

text: of

text: the

text: given

text: by

text: Eq.

text: can

text: be

text: rewritten

text: as

text: Paul

text: trap

text: (given

text: in

text: Eq.

text: is

text: less

text: than

text: unity,

text: i.e.

text: weaker

text: trapping

text: ρ2

text: ζ0

text: ospfrtihneg

text: that

text: balancing

text: the

text: where

text: m

text: is

text: the

text: order

text: number

text: and

text: β

text: ΓA/Ω

text: describes

text: the

text: damping

text: motion

text: driven

text: at

text: angular→frequency

text: Ω,

text: and

text: we

text: as-

text: Vn

text: sume

text: πβ

text: the

text: function

text: as

text: m

text: increases.

text: The

text: Q/

text: Q[

text: nim

text: A

text: B

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 34 32 6 30 28 26 24 5 22 20 18 4 16 14 3 50 75 100 125 150 175 200 Ω/2π (kHz) Figure 5: Electrical circuit for two frequency operation of the ion trap. Two of the rf electrodes are driven at a high frequency whereas the second electrode pair is used for the low frequency drive. The dc and rf drives are decoupled Figure 6: Measured values of the threshold voltage versus the excitation fre- using a series of capacitors. quency. Twelve distinct resonances are labeled by their respective order num- ber m = 14,16,18,20,...,34. The higher order resonances, at lower frequen- cies, fade into the background noise. The dots are the threshold voltage for each parametric resonance and the red line is Eq. 10 with best-ﬁt parameters: − is shown in Fig. 6 for the range Ω/2π = 40 − 200 kHz. There VΩ1/2=π9→.3V0. fTohreβd=am1p0in6ghmowusetvbeer itnhteermpraextiemducmaurteiocuosrldyedsinthcreesohnolyldeivse7n.o7rVdearss are twelve distinct resonances in this range, that correspond to are observed which is not predicted by the theory in the text. even orders m = 14, 16, . . . , 34. The threshold for parametric ecxrecaitsaetsi.onWiendﬁutctehde bpyeatkhseoafptphleierdevsoonltaangceeVs 1toindcerteearsmeisn→aesvΩa∞ludees- (cid:88)  (cid:16) (cid:17) of V1 and β. The theoretical limiting voltage as m is V − 0.27qAVn = 7 V for qA = 0.1 and Vn = 250 V. This is close to E(xj; t) = (cid:16) r2κ cos (Ωκt) y(cid:17)eˆy xeˆx (20) the measured maximum threshold 7.7 V, but less than the best κ=1,n 0 ﬁthteveaﬀlueectofofV1th=e ﬁ9.n3itVe.vWalueehoavfeanaortisminagdefraonmy cthoerredcctivoonlstafgoer + Vz20 xeˆx + yeˆy − 2zeˆz . applied to the end-cap electrodes. Moreover, even-order reso- 0 nances were stronger than odd ones which is not expected from The oscillating voltages V provide conﬁnement in the x and the theory for motion along one direction; ﬁtting to even-order k y directions, while the static voltage V conﬁnes ions in the z resonances only, overestimates the threshold voltage. Similar 0 direction. The trajectories of N interacting ions in the electric odd-even staggering was seen in the measurement of [30], but ﬁeld of a Paul trap obey the equa(cid:88)tions of motion not in [31, 33]. Nevertheless, these measurements show that a voltage suﬃcient to conﬁne the heavy ions can be applied in this Ω range for V1 < 7 V. M x¨ = E(x ; t)Q + N 1 Q−iQj (21) Figure 6 also shows that it possible to avoid parametric j j j j i,j=1 8π(cid:15)0 (xj xi)2 resonances by a suitable choice of Ω, although the presence i(cid:44)j of species B may shift the secular oscillation frequencies of for ion j, where E(x ; t) is the electric ﬁeld of Eq. 20 aris- species A. For higher values of m, the resonances are so weak j ing from the two-frequency potential V(t) = V + V cos Ωt + that their inﬂuence becomes comparable to the non-resonant ef- 0 1 fect of an additional dc voltage. Thus, atomic ions are conﬁned Vn cos Ωnt, and (cid:15)0 is the permittivity of vacuum. We numerically integrated the equations of motion using stably in a Paul trap with another quadrupole ﬁeld oscillating at (py)LIon [34], a collection of software tools we developed to in- a lower frequency. vestigate electrodynamic trapping of multiple species in an ion 4. Molecular dynamics simulations trap. These tools provide a wrapper around LAMMPS [35] 2, exposing only a subset of its capabilities relevant to the simula- tion of ion trap dynamics. The eﬀect of collisions with a buﬀer gas is implemented by coupling the ions to a Langevin bath, which provides both a stochastic and viscous damping force We have carried out numerical simulations to conﬁrm the sta- bility of multiple ions in the two-frequency trap, in all three di- mensions. These simulations use a time-varying electric ﬁeld 2LAMMPS is a classical molecular dynamics code developed by Sandia of the form Labs and distributed as open-source software. 7 )V( V 1 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Ω/2π

text: (kHz)

text: Figure

text: Electrical

label: caption

text: circuit

text: for

text: two

text: frequency

text: operation

text: of

text: the

text: ion

text: trap.

text: Two

text: of

text: the

text: rf

text: electrodes

text: are

text: driven

text: at

text: a

text: high

text: frequency

text: whereas

text: the

text: second

text: electrode

text: pair

text: is

text: used

text: for

text: the

text: low

text: frequency

text: drive.

text: The

text: dc

text: and

text: rf

text: drives

text: are

text: decoupled

text: Figure

text: Measured

text: values

text: of

text: the

text: threshold

text: voltage

text: versus

text: the

text: excitation

text: fre-

text: using

text: a

text: series

text: of

text: capacitors.

text: quency.

text: Twelve

text: distinct

text: resonances

text: are

text: labeled

text: by

text: their

text: respective

text: order

text: num-

text: ber

text: m

text: The

text: higher

text: order

text: resonances,

text: at

text: lower

text: frequen-

text: cies,

text: fade

text: into

text: the

text: background

text: noise.

text: The

text: dots

text: are

text: the

text: threshold

text: voltage

text: for

text: each

text: parametric

text: resonance

text: and

text: the

text: red

text: line

text: is

text: Eq.

text: with

text: best-ﬁt

text: parameters:

text: is

text: shown

text: in

text: Fig.

text: for

text: the

text: range

text: Ω/2π

text: kHz.

text: There

text: are

text: twelve

text: distinct

text: resonances

text: in

text: this

text: range,

text: that

text: correspond

text: to

text: are

text: observed

text: which

text: is

text: not

text: predicted

text: by

text: the

text: theory

text: in

text: the

text: text.

text: even

text: orders

text: m

text: The

text: threshold

text: for

text: parametric

text: ecxrecaitsaetsi.onWiendﬁutctehde

text: of

text: and

text: β.

text: The

text: theoretical

text: limiting

text: voltage

text: as

text: m

text: is

text: V

text: V

text: for

text: qA

text: and

text: Vn

text: V.

text: This

text: is

text: close

text: to

text: E(xj;

text: t)

text: cos

text: (Ωκt)

text: xeˆx

text: the

text: measured

text: maximum

text: threshold

text: V,

text: but

text: less

text: than

text: the

text: best

text: cthoerredcctivoonlstafgoer

text: xeˆx

text: yeˆy

text: applied

text: to

text: the

text: end-cap

text: electrodes.

text: Moreover,

text: even-order

text: reso-

text: nances

text: were

text: stronger

text: than

text: odd

text: ones

text: which

text: is

text: not

text: expected

text: from

text: The

text: oscillating

text: voltages

text: V

text: provide

text: conﬁnement

text: in

text: the

text: x

text: and

text: the

text: theory

text: for

text: motion

text: along

text: one

text: direction;

text: ﬁtting

text: to

text: even-order

text: k

text: y

text: directions,

text: while

text: the

text: static

text: voltage

text: V

text: conﬁnes

text: ions

text: in

text: the

text: z

text: resonances

text: only,

text: overestimates

text: the

text: threshold

text: voltage.

text: Similar

text: direction.

text: The

text: trajectories

text: of

text: N

text: interacting

text: ions

text: in

text: the

text: electric

text: odd-even

text: staggering

text: was

text: seen

text: in

text: the

text: measurement

text: of

text: but

text: ﬁeld

text: of

text: a

text: Paul

text: trap

text: obey

text: the

text: of

text: motion

text: not

text: in

text: Nevertheless,

text: these

text: measurements

text: show

text: that

text: a

text: voltage

text: suﬃcient

text: to

text: conﬁne

text: the

text: heavy

text: ions

text: can

text: be

text: applied

text: in

text: this

text: Ω

text: range

text: for

text: V.

text: M

text: x¨

text: E(x

text: t)Q

text: N

text: Q−iQj

text: Figure

text: also

text: shows

text: that

text: it

text: possible

text: to

text: avoid

text: parametric

text: j

text: j

text: j

text: j

text: (xj

text: resonances

text: by

text: a

text: suitable

text: choice

text: of

text: Ω,

text: although

text: the

text: presence

text: of

text: species

text: B

text: may

text: shift

text: the

text: secular

text: oscillation

text: frequencies

text: of

text: for

text: ion

text: j,

text: where

text: E(x

text: t)

text: is

text: the

text: electric

text: ﬁeld

text: of

text: Eq.

text: aris-

text: species

text: A.

text: For

text: higher

text: values

text: of

text: m,

text: the

text: resonances

text: are

text: so

text: weak

text: j

text: ing

text: from

text: the

text: two-frequency

text: potential

text: V(t)

text: V

text: V

text: cos

text: Ωt

text: that

text: their

text: inﬂuence

text: becomes

text: comparable

text: to

text: the

text: non-resonant

text: ef-

text: fect

text: of

text: an

text: additional

text: dc

text: voltage.

text: Thus,

text: atomic

text: ions

text: are

text: conﬁned

text: Vn

text: cos

text: Ωnt,

text: and

text: is

text: the

text: permittivity

text: of

text: vacuum.

text: We

text: numerically

text: integrated

text: the

text: equations

text: of

text: motion

text: using

text: stably

text: in

text: a

text: Paul

text: trap

text: with

text: another

text: quadrupole

text: ﬁeld

text: oscillating

text: at

text: (py)LIon

text: a

text: collection

text: of

text: software

text: tools

text: we

text: developed

text: to

text: in-

text: a

text: lower

text: frequency.

text: vestigate

text: electrodynamic

text: trapping

text: of

text: multiple

text: species

text: in

text: an

text: ion

text: Molecular

text: dynamics

text: simulations

text: trap.

text: These

text: tools

text: provide

text: a

text: wrapper

text: around

text: LAMMPS

text: exposing

text: only

text: a

text: subset

text: of

text: its

text: capabilities

text: relevant

text: to

text: the

text: simula-

text: tion

text: of

text: ion

text: trap

text: dynamics.

text: The

text: eﬀect

text: of

text: collisions

text: with

text: a

text: buﬀer

text: gas

text: is

text: implemented

text: by

text: coupling

text: the

text: ions

text: to

text: a

text: Langevin

text: bath,

text: which

text: provides

text: both

text: a

text: stochastic

text: and

text: viscous

text: damping

text: force

text: We

text: have

text: carried

text: out

text: numerical

text: simulations

text: to

text: conﬁrm

text: the

text: sta-

text: bility

text: of

text: multiple

text: ions

text: in

text: the

text: two-frequency

text: trap,

text: in

text: all

text: three

text: di-

text: mensions.

text: These

text: simulations

text: use

text: a

text: time-varying

text: electric

text: ﬁeld

text: is

text: a

text: classical

text: molecular

text: dynamics

text: code

text: developed

text: by

text: Sandia

text: of

text: the

text: form

text: Labs

text: and

text: distributed

text: as

text: open-source

text: software.

text: )V(

text: V

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:48) ular ions could be followed by controlled deposition of mass- where F is a constant and the secular oscillation frequency − selected particles on a surface for further analysis by other tech- is modiﬁed to ω˜ 2 = ω2 Γ2/4. This is the usual shift of the A A niques. It has also been suggested that a chain of atomic and resonance frequency that occurs in damped simple harmonic − molecular ions can act as a conveyor belt to carry the dark motion. The integrating factor exp( Γt/2) eliminates y˙ for any molecular ions into the focus of an x-ray laser for destruc- value of the constant Γ and is used in numerical calculations of tive measurements [36], as demonstrated in experiments with the stability regions of the Mathieu equation with strong damp- Mg+ and MgH+ which have similar charge-to-mass ratios. Our ing [5, 16]. method opens the way to working with much heavier biomolec- To relate the behaviour of ions in a quadrupole electric ﬁeld ular ions. oscillating at two frequencies (including a dc term) to famil- Paul traps are used to investigate chemical reactions of ions iar properties of a single-frequency Paul trap we consider Eq. 7 at low temperatures [37], and two-frequency operation allows with V = 0 which reduces to a Mathieu equation with parame- 1 much heavier species to be used without a large spatial separa- ter tion. This application is closely related to the use of two fre- (cid:48)(cid:48) quencies for the conﬁnement of antimatter where the creation q(τn) = Vn = QA · 4Vn , (A.2) of overlapping clouds of positrons and anti-protons leads to the A 2V M (nΩ)2 A A formation of anti-hydrogen atoms; work towards this is techni- (cid:48)(cid:48) cally challenging goal is ongoing [14]3. where Vn = Vn/r02 is the curvature of the oscillating electric Sympathetic cooling of biomolecular ions by 138Ba+ in a Paul potential. We drop the minus sign since the steady-state be- trap was demonstrated [12] and heavier ions can be used by ap- haviour does not depend on the sign of q. We take q(τn) = 0.4 A plying two frequencies. The transfer of energy between ions is as a realistic value for operation of a Paul trap since for higher less eﬃcient if the oscillation frequencies are dissimilar. How- values the ions are susceptible to heating and loss from nonlin- ever, it is possible to arrange a system with coupling between ear mixing and parametric excitation even for single frequency axial modes of species A and radial modes of species B in a operation. linear Paul trap [15]. The extension of the techniques devel- The quadrupole ﬁeld proportional to V cos nΩt leads to a n oped for laser-cooled atomic ions to species with much higher pseudopo√tential with a secular oscillation frequency ωA = M/Q that are sympathetically cooled presents new ground for q(τn)nΩ/ 8 so ions of species A undergo simple harmonic mo- A future investigation. Our method allows for cotrapping atomic tion described by y¨+ω2y = 0 where t is real time. We introduce A ions with much heavier charged particles, such as ﬂakes of a voltage V cos Ωt that produces a driving term proportional to (cid:48) 1 graphite [39] or DNA molecules in aqueous solutions [40]. F = QAV1/r02 in Eq. A.1. Using the rescaling Ωt = 2τ as in Acknowledgements Eq. 6 gives (cid:16) (cid:17) This work was supported by the EU H2020 Collaborative d2y dy − { } project QuProCS (Grant Agreement 641277). DT and EB ac- dτ2 + 2βdτ + aeﬀ 2qAτ cos 2τ y = 0, (A.3) knowledge support from the EPSRC. DT is thankful to the Bo- dossaki foundation and St. Peter’s College. We also gratefully where β = ΓA/Ω(cid:32) and(cid:33) (cid:32) (cid:33) (cid:16) (cid:17) acknowledge support from EPSRC (EP/J003670/1). Appendix A. The mathematical treatment of damping a = 2ωA 2 = 1 nVn 2 = 1 nq(τn) 2 (A.4) eﬀ Ω 2 2V 2 A A Equations of motion with damping proportional to veloc- ity, or any second-order diﬀerential equation in which the ﬁrst is the eﬀective value of a static potential equivalent to the and second derivatives have constant coeﬃcients, can be writ- pseudopotential (with V = 0), and ten in the general form y¨ + Γy˙ + H(t)y = 0. The substitution 0 − y = y˜ exp( Γt/2) eliminates the term with th−e ﬁrst derivative − { } V n2 to give y¨˜ + H˜(t)y˜ = 0, where H˜(t) = H(t) Γ2/4. Second- q τ = 1 (A.5) A 2V order equations with linear damping have an associated equa- A tion without damping which is a Hill equation when H˜(t) is a periodic function and can be written as the sum of its Fourier is completely independent from q(Aτn). For q(Aτn) = 0.4 and n = 100 we ﬁnd that a = 800. This is far greater than is components. An illustrative example is the damped simple har- eﬀ usually considered in the theory of Paul traps but the asymptotic monic oscillator with a parametric driving term properties of the Mathieu equation are known in literature [41, (cid:29) d2y F(cid:48) 28]. For a, q 1 the critical line between predominantly stable + ω˜ 2 y = cos (Ωt) y, (A.1) and unstable regions is a = 2q, as shown in Fig. 1a). This passes dt2 A M A through the point (q, a) = (400, 800) in this example. Using | { }| a = 2 q τ , and Equations A.4, A.5, we recover Eq. 8 with the eﬀ A same value ρ = 0.5 as the simpliﬁed argument given previously. 3 Sympathetic cooling of positrons by Be+ ions has been demonstrated ex- Damping modiﬁes aeﬀ however this is not a signiﬁcant eﬀect for perimentally in a Penning trap [38]. Γ/(2ωA) < 0.1 and is neglected here. 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ular

text: ions

text: could

text: be

text: followed

text: by

text: controlled

text: deposition

text: of

text: mass-

text: where

text: F

text: is

text: a

text: constant

text: and

text: the

text: secular

text: oscillation

text: frequency

text: selected

text: particles

text: on

text: a

text: surface

text: for

text: further

text: analysis

text: by

text: other

text: tech-

text: is

text: modiﬁed

text: to

text: ω˜

text: ω2

text: This

text: is

text: the

text: usual

text: shift

text: of

text: the

text: A

text: A

text: niques.

text: It

text: has

text: also

text: been

text: suggested

text: that

text: a

text: chain

text: of

text: atomic

text: and

text: resonance

text: frequency

text: that

text: occurs

text: in

text: damped

text: simple

text: harmonic

text: molecular

text: ions

text: can

text: act

text: as

text: a

text: conveyor

text: belt

text: to

text: carry

text: the

text: dark

text: motion.

text: The

text: integrating

text: factor

text: exp(

text: eliminates

text: y˙

text: for

text: any

text: molecular

text: ions

text: into

text: the

text: focus

text: of

text: an

text: x-ray

text: laser

text: for

text: destruc-

text: value

text: of

text: the

text: constant

text: Γ

text: and

text: is

text: used

text: in

text: numerical

text: calculations

text: of

text: tive

text: measurements

text: as

text: demonstrated

text: in

text: experiments

text: with

text: the

text: stability

text: regions

text: of

text: the

text: Mathieu

text: equation

text: with

text: strong

text: damp-

text: Mg+

text: and

text: MgH+

text: which

text: have

text: similar

text: charge-to-mass

text: ratios.

text: Our

text: ing

text: method

text: opens

text: the

text: way

text: to

text: working

text: with

text: much

text: heavier

text: biomolec-

text: To

text: relate

text: the

text: behaviour

text: of

text: ions

text: in

text: a

text: quadrupole

text: electric

text: ﬁeld

text: ular

text: ions.

text: oscillating

text: at

text: two

text: frequencies

text: (including

text: a

text: dc

text: term)

text: to

text: famil-

text: Paul

text: traps

text: are

text: used

text: to

text: investigate

text: chemical

text: reactions

text: of

text: ions

text: iar

text: properties

text: of

text: a

text: single-frequency

text: Paul

text: trap

text: we

text: consider

text: Eq.

text: at

text: low

text: temperatures

text: and

text: two-frequency

text: operation

text: allows

text: with

text: V

text: which

text: reduces

text: to

text: a

text: Mathieu

text: equation

text: with

text: parame-

text: much

text: heavier

text: species

text: to

text: be

text: used

text: without

text: a

text: large

text: spatial

text: separa-

text: ter

text: tion.

text: This

text: application

text: is

text: closely

text: related

text: to

text: the

text: use

text: of

text: two

text: fre-

text: quencies

text: for

text: the

text: conﬁnement

text: of

text: antimatter

text: where

text: the

text: creation

text: q(τn)

text: Vn

text: QA

text: of

text: overlapping

text: clouds

text: of

text: positrons

text: and

text: anti-protons

text: leads

text: to

text: the

text: A

text: M

text: A

text: A

text: formation

text: of

text: anti-hydrogen

text: atoms;

text: work

text: towards

text: this

text: is

text: techni-

text: cally

text: challenging

text: goal

text: is

text: ongoing

text: where

text: Vn

text: is

text: the

text: curvature

text: of

text: the

text: oscillating

text: electric

text: Sympathetic

text: cooling

text: of

text: biomolecular

text: ions

text: by

text: in

text: a

text: Paul

text: potential.

text: We

text: drop

text: the

text: minus

text: sign

text: since

text: the

text: steady-state

text: be-

text: trap

text: was

text: demonstrated

text: and

text: heavier

text: ions

text: can

text: be

text: used

text: by

text: ap-

text: haviour

text: does

text: not

text: depend

text: on

text: the

text: sign

text: of

text: q.

text: We

text: take

text: q(τn)

text: A

text: plying

text: two

text: frequencies.

text: The

text: transfer

text: of

text: energy

text: between

text: ions

text: is

text: as

text: a

text: realistic

text: value

text: for

text: operation

text: of

text: a

text: Paul

text: trap

text: since

text: for

text: higher

text: less

text: eﬃcient

text: if

text: the

text: oscillation

text: frequencies

text: are

text: dissimilar.

text: How-

text: values

text: the

text: ions

text: are

text: susceptible

text: to

text: heating

text: and

text: loss

text: from

text: nonlin-

text: ever,

text: it

text: is

text: possible

text: to

text: arrange

text: a

text: system

text: with

text: coupling

text: between

text: ear

text: mixing

text: and

text: parametric

text: excitation

text: even

text: for

text: single

text: frequency

text: axial

text: modes

text: of

text: species

text: A

text: and

text: radial

text: modes

text: of

text: species

text: B

text: in

text: a

text: operation.

text: linear

text: Paul

text: trap

text: The

text: extension

text: of

text: the

text: techniques

text: devel-

text: The

text: quadrupole

text: ﬁeld

text: proportional

text: to

text: V

text: cos

text: nΩt

text: leads

text: to

text: a

text: n

text: oped

text: for

text: laser-cooled

text: atomic

text: ions

text: to

text: species

text: with

text: much

text: higher

text: pseudopo√tential

text: with

text: a

text: secular

text: oscillation

text: frequency

text: ωA

text: M/Q

text: that

text: are

text: sympathetically

text: cooled

text: presents

text: new

text: ground

text: for

text: q(τn)nΩ/

text: so

text: ions

text: of

text: species

text: A

text: undergo

text: simple

text: harmonic

text: mo-

text: A

text: future

text: investigation.

text: Our

text: method

text: allows

text: for

text: cotrapping

text: atomic

text: tion

text: described

text: by

text: where

text: t

text: is

text: real

text: time.

text: We

text: introduce

text: A

text: ions

text: with

text: much

text: heavier

text: charged

text: particles,

text: such

text: as

text: ﬂakes

text: of

text: a

text: voltage

text: V

text: cos

text: Ωt

text: that

text: produces

text: a

text: driving

text: term

text: proportional

text: to

text: graphite

text: or

text: DNA

text: molecules

text: in

text: aqueous

text: solutions

text: F

text: in

text: Eq.

text: Using

text: the

text: rescaling

text: Ωt

text: 2τ

text: as

text: in

text: Acknowledgements

text: Eq.

text: gives

text: This

text: work

text: was

text: supported

text: by

text: the

text: EU

text: Collaborative

text: dy

text: project

text: QuProCS

text: (Grant

text: Agreement

text: DT

text: and

text: EB

text: ac-

text: 2βdτ

text: aeﬀ

text: cos

text: 2τ

text: y

text: knowledge

text: support

text: from

text: the

text: EPSRC.

text: DT

text: is

text: thankful

text: to

text: the

text: Bo-

text: dossaki

text: foundation

text: and

text: St.

text: Peter’s

text: College.

text: We

text: also

text: gratefully

text: where

text: β

text: acknowledge

text: support

text: from

text: EPSRC

text: Appendix

text: A.

text: The

text: mathematical

text: treatment

text: of

text: damping

text: a

text: nVn

text: nq(τn)

text: eﬀ

text: Ω

text: A

text: A

text: Equations

text: of

text: motion

text: with

text: damping

text: proportional

text: to

text: veloc-

text: ity,

text: or

text: any

text: second-order

text: diﬀerential

text: equation

text: in

text: which

text: the

text: ﬁrst

text: is

text: the

text: eﬀective

text: value

text: of

text: a

text: static

text: potential

text: equivalent

text: to

text: the

text: and

text: second

text: derivatives

text: have

text: constant

text: coeﬃcients,

text: can

text: be

text: writ-

text: pseudopotential

text: (with

text: V

text: and

text: ten

text: in

text: the

text: general

text: form

text: y¨

text: Γy˙

text: H(t)y

text: The

text: substitution

text: y

text: y˜

text: exp(

text: eliminates

text: the

text: term

text: with

text: th−e

text: ﬁrst

text: derivative

text: V

text: to

text: give

text: y¨˜

text: H˜(t)y˜

text: where

text: H˜(t)

text: H(t)

text: Second-

text: q

text: τ

text: A

text: order

text: equations

text: with

text: linear

text: damping

text: have

text: an

text: associated

text: equa-

text: A

text: tion

text: without

text: damping

text: which

text: is

text: a

text: Hill

text: equation

text: when

text: H˜(t)

text: is

text: a

text: periodic

text: function

text: and

text: can

text: be

text: written

text: as

text: the

text: sum

text: of

text: its

text: Fourier

text: is

text: completely

text: independent

text: from

text: q(Aτn).

text: For

text: q(Aτn)

text: and

text: n

text: we

text: ﬁnd

text: that

text: a

text: This

text: is

text: far

text: greater

text: than

text: is

text: components.

text: An

text: illustrative

text: example

text: is

text: the

text: damped

text: simple

text: har-

text: eﬀ

text: usually

text: considered

text: in

text: the

text: theory

text: of

text: Paul

text: traps

text: but

text: the

text: asymptotic

text: monic

text: oscillator

text: with

text: a

text: parametric

text: driving

text: term

text: properties

text: of

text: the

text: Mathieu

text: equation

text: are

text: known

text: in

text: literature

text: For

text: a,

text: q

text: the

text: critical

text: line

text: between

text: predominantly

text: stable

text: ω˜

text: y

text: cos

text: (Ωt)

text: y,

text: and

text: unstable

text: regions

text: is

text: a

text: as

text: shown

text: in

text: Fig.

text: This

text: passes

text: A

text: M

text: A

text: through

text: the

text: point

text: (q,

text: a)

text: in

text: this

text: example.

text: Using

text: a

text: q

text: τ

text: and

text: Equations

text: we

text: recover

text: Eq.

text: with

text: the

text: eﬀ

text: A

text: same

text: value

text: ρ

text: as

text: the

text: simpliﬁed

text: argument

text: given

text: previously.

text: Sympathetic

text: cooling

text: of

text: positrons

text: by

text: Be+

text: ions

text: has

text: been

text: demonstrated

text: ex-

text: Damping

text: modiﬁes

text: aeﬀ

text: however

text: this

text: is

text: not

text: a

text: signiﬁcant

text: eﬀect

text: for

text: perimentally

text: in

text: a

text: Penning

text: trap

text: and

text: is

text: neglected

text: here.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 ARTHUR-CÉSAR LE BRAS The ring of an(cid:40)alytic functions on Y is called B ; it (cid:41)is the Fréchet algebra obtained by completing the algeb(cid:88)ra ∈ O Bb := [x ]pn, x C(cid:91), (x ) bounded = W( )[1/p, 1/[p(cid:91)]] n n p n n C(cid:91) (cid:29)−∞ p n of “meromorphic functions along t(cid:88)he divisors p = 0 and [p(cid:91)] = 0” with respect to the norms (cid:107) · (cid:107) , for all 0 < ρ < 1, deﬁned by ρ (cid:107) (cid:107) | | [x ]pn = sup x ρn. n ρ n n (cid:88)n (cid:88) There is a Frobenius operator ϕ on Y , whose action on functions is given by the formula (cid:55)→ [x ]pn [xp]pn. n n n n The action of ϕ on Y is properly discontinuous and one can then deﬁne the Fargues-Fontaine curve X (for the local ﬁeld Q ) as the quotient : p X := Y/ϕZ. The set of classical points of X is in bijection with the set of untilts of C(cid:91) in characteristic ∈ | | p 0 (modulo the action of Frobenius) : if x X , the residue ﬁeld C of X at x is an (cid:39) x algebraically closed complete valued extension of Q , with C(cid:91) C(cid:91). In particular, there is ∞ p x p a point called on X, such that C∞ = C . p In fact, the exact same construction works if one replaces C by any algebraically closed p complete valued extension C of Q . In this case, one gets adic spaces which will be called p Y and X and which have exactly the same properties. C(cid:91) C(cid:91) (cid:101) It turns out that the Fargues-Fontaine curve X looks like the non-archimedean analog of the twistor projective line P1 , which is the quotient of the complex projective line by the R (cid:101) antipodal involution : ∼ − − P1 := P1 /z z¯ 1. R C This is a real variety without any real point (exactly like the Fargues-Fontaine curve, whose (cid:101) residue ﬁelds at classical points are all algebraically closed). The analogy between the curve and P1 has so far no prec(cid:101)ise mathematical meaning, but there are some interesting R similarities between these two objects. Here are a few examples : (a) The set of G-torsors over P1 (resp. X) is in bijection with Kottwitz’s set B(G) (as R deﬁned in [27]), for G reductive over R (resp. Q ). In the non-archimedean setting, this p is the main result of [21], see also [1] ; in the archimedean setting, this can be proved for (cid:101) example using the strategy of [1], as explained to us by Anschütz. (b) The function(cid:101) ﬁeld of P1 is conjectured to be (C1) (Lang [28]) ; similarly, Fargues R ([21]) conjectures the function ﬁeld Frac(B[1/t]ϕ=1) of X to be (C1). (c) The variety P1 is the Brauer-Severi variety of Hamilton’s quaternion algebra over R R ; similarly, it was conjectured by Fargues that X should be in some sense the Brauer-Severi C variety of a division algebra introduced by Colmez in [12]. In this paper, we study another instance of this analogy. It is a nowadays classical result (cid:101) of Simpson ([40, §2], see also [31, §3.1-3.4]) that the datum of a real pure Hodge structure can be encoded as a U(1)-equivariant semi-stable vector bundle on P1 . Let Z be a smooth R projective variety over C. The U(1)-equivariant vector bundle attached to the cohomology of degree i of Z is obtained by modifying the U(1)-equivariant vector bundle (cid:101) ⊗ O H∞i (Z) := Hi (Z, R) Betti P1 R

text: ARTHUR-CÉSAR

text: LE

text: BRAS

text: The

text: ring

text: of

text: functions

text: on

text: Y

text: is

text: called

text: B

text: it

text: the

text: Fréchet

text: algebra

text: obtained

text: by

text: completing

text: the

text: O

text: Bb

text: [x

text: ]pn,

text: x

text: (x

text: bounded

text: W(

text: n

text: n

text: p

text: n

text: n

text: p

text: n

text: of

text: “meromorphic

text: functions

text: along

text: divisors

text: p

text: and

text: 0”

text: with

text: respect

text: to

text: the

text: norms

text: for

text: all

text: ρ

text: deﬁned

text: by

text: ρ

text: [x

text: ]pn

text: sup

text: x

text: ρn.

text: n

text: ρ

text: n

text: n

text: There

text: is

text: a

text: Frobenius

text: operator

text: ϕ

text: on

text: Y

text: whose

text: action

text: on

text: functions

text: is

text: given

text: by

text: the

text: formula

text: [x

text: ]pn

text: [xp]pn.

text: n

text: n

text: n

text: n

text: The

text: action

text: of

text: ϕ

text: on

text: Y

text: is

text: properly

text: discontinuous

text: and

text: one

text: can

text: then

text: deﬁne

text: the

text: Fargues-Fontaine

text: curve

text: X

text: (for

text: the

text: local

text: ﬁeld

text: Q

text: as

text: the

text: quotient

text: p

text: X

text: Y/ϕZ.

text: The

text: set

text: of

text: classical

text: points

text: of

text: X

text: is

text: in

text: bijection

text: with

text: the

text: set

text: of

text: untilts

text: of

text: in

text: characteristic

text: p

text: (modulo

text: the

text: action

text: of

text: Frobenius)

text: if

text: x

text: X

text: the

text: residue

text: ﬁeld

text: C

text: of

text: X

text: at

text: x

text: is

text: an

text: x

text: algebraically

text: closed

text: complete

text: valued

text: extension

text: of

text: Q

text: with

text: In

text: particular,

text: there

text: is

text: p

text: x

text: p

text: a

text: point

text: called

text: on

text: X,

text: such

text: that

text: C∞

text: C

text: p

text: In

text: fact,

text: the

text: exact

text: same

text: construction

text: works

text: if

text: one

text: replaces

text: C

text: by

text: any

text: algebraically

text: closed

text: p

text: complete

text: valued

text: extension

text: C

text: of

text: Q

text: In

text: this

text: case,

text: one

text: gets

text: adic

text: spaces

text: which

text: will

text: be

text: called

text: p

text: Y

text: and

text: X

text: and

text: which

text: have

text: exactly

text: the

text: same

text: properties.

text: It

text: turns

text: out

text: that

text: the

text: Fargues-Fontaine

text: curve

text: X

text: looks

text: like

text: the

text: non-archimedean

text: analog

text: of

text: the

text: twistor

text: projective

text: line

text: which

text: is

text: the

text: quotient

text: of

text: the

text: complex

text: projective

text: line

text: by

text: the

text: R

text: antipodal

text: involution

text: /z

text: z¯

text: R

text: C

text: This

text: is

text: a

text: real

text: variety

text: without

text: any

text: real

text: point

text: (exactly

text: like

text: the

text: Fargues-Fontaine

text: curve,

text: whose

text: residue

text: ﬁelds

text: at

text: classical

text: points

text: are

text: all

text: algebraically

text: closed).

text: The

text: analogy

text: between

text: the

text: curve

text: and

text: has

text: so

text: far

text: no

text: mathematical

text: meaning,

text: but

text: there

text: are

text: some

text: interesting

text: R

text: similarities

text: between

text: these

text: two

text: objects.

text: Here

text: are

text: a

text: few

text: examples

text: (a)

text: The

text: set

text: of

text: G-torsors

text: over

text: (resp.

text: X)

text: is

text: in

text: bijection

text: with

text: Kottwitz’s

text: set

text: B(G)

text: (as

text: R

text: deﬁned

text: in

text: for

text: G

text: reductive

text: over

text: R

text: (resp.

text: Q

text: In

text: the

text: non-archimedean

text: setting,

text: this

text: p

text: is

text: the

text: main

text: result

text: of

text: see

text: also

text: in

text: the

text: archimedean

text: setting,

text: this

text: can

text: be

text: proved

text: for

text: example

text: using

text: the

text: strategy

text: of

text: as

text: explained

text: to

text: us

text: by

text: Anschütz.

text: (b)

text: The

text: ﬁeld

text: of

text: is

text: conjectured

text: to

text: be

text: (Lang

text: similarly,

text: Fargues

text: R

text: conjectures

text: the

text: function

text: ﬁeld

text: of

text: X

text: to

text: be

text: (c)

text: The

text: variety

text: is

text: the

text: Brauer-Severi

text: variety

text: of

text: Hamilton’s

text: quaternion

text: algebra

text: over

text: R

text: R

text: similarly,

text: it

text: was

text: conjectured

text: by

text: Fargues

text: that

text: X

text: should

text: be

text: in

text: some

text: sense

text: the

text: Brauer-Severi

text: C

text: variety

text: of

text: a

text: division

text: algebra

text: introduced

text: by

text: Colmez

text: in

text: In

text: this

text: paper,

text: we

text: study

text: another

text: instance

text: of

text: this

text: analogy.

text: It

text: is

text: a

text: nowadays

text: classical

text: result

text: of

text: Simpson

text: see

text: also

text: that

text: the

text: datum

text: of

text: a

text: real

text: pure

text: Hodge

text: structure

text: can

text: be

text: encoded

text: as

text: a

text: semi-stable

text: vector

text: bundle

text: on

text: Let

text: Z

text: be

text: a

text: smooth

text: R

text: projective

text: variety

text: over

text: C.

text: The

text: vector

text: bundle

text: attached

text: to

text: the

text: cohomology

text: of

text: degree

text: i

text: of

text: Z

text: is

text: obtained

text: by

text: modifying

text: the

text: vector

text: bundle

text: O

text: H∞i

text: (Z)

text: Hi

text: (Z,

text: R)

text: Betti

text: R


text: 14 ARTHUR-CÉSAR LE BRAS (cid:16) (cid:17) which identiﬁes to the subcomplex · (cid:104) (cid:105) −→ (cid:104) (cid:105) · ⊗ • X Q X d Q X dX ˆ Lη KM p p Qp t (Z∞) • ⊗ • of Ω (D)ˆ Lη KM . Qp t (Z∞) All in all, we get a quasi-isomorphism : × (cid:48) M (cid:39) • ⊗ • RΓ (Z D , Lη Rν∗ ) Lη KM ˆ Ω (D), ´et C t t (Z∞) Qp as wanted. Now we prove Proposition 4.1, using Lemma 4.2. Let M = B or M = B+ /tr, r ≥ 1. (cid:16) ≥ (cid:17) (cid:16) • (cid:17) ⊗(cid:16) I • dR (cid:17) Choose n 1. The i-th term of the complex Lη KM ˆ Ω (D ) is : t (Z∞) Qp n • ⊗ • i • ⊗ O ⊕ • − ⊗ Lη KM ˆ Ω (D ) = (Lη KM )i ˆ (D ) (Lη KM )i 1 ˆ Ω1(D ) . t (Z∞) Qp n (cid:88) t (Z∞) (cid:88)Qp n t (Z∞) Qp n Any element x in it can thus be written as D (x)Tk + L (x)TkdT, k k ≥ ≥ k 0 k 0 ∈ • ∈ • − − − → with Dk→(x) ∞(LηtKM(Z∞))i, Lk(x) (cid:88)(LηtKM(Z∞))i 1 and p k/nDk(x), p k/nLk(x) 0 when k + . In particular, for any such x, the series L (x) k Tk+1 ≥ k + 1 k 0 ∈ • ⊗ • converges to an element L(x) (Lη KM ˆ Ω (D ))i, and this deﬁnes a morphism L • ⊗ • t (Z∞) Qp • n⊗+1 • of the complex Lη KM ˆ Ω (D(cid:16) ) into Lη KM ˆ Ω(cid:17) (D ). Taking the inductive t (Z∞) Qp n t (Z∞) Qp n+1 limit over n we get a morphism of the complex • ⊗ • l−i→m Lη KM ˆ Ω (D ) t (Z∞) Qp n n into itself, again denoted L. One easily checks the identity : ◦ ◦ − d L + L d = Id D , (cid:16) 0 (cid:17) which proves that the map D is a homotopy inverse to the natural map 0 • → • ⊗ • Lη KM l−i→m Lη KM ˆ Ω (D ) . t (Z∞) t (Z∞) Qp n n This concludes the proof of Proposition 4.1 when M = B or B+ /tr. Let’s now treat the case M = B. I dR Lemma 4.3. — The natural map (cid:48) B → (cid:48) B Lη Rν∗ R l←im− Lη Rν∗ t t I I is an isomorphism. Proof. — We use Lemma 3.10. This reduces us to prove that the map (cid:48) B → (cid:48) B Rν∗ R l←im− Rν∗ I I is an isomorphism, which follows from Lemma 3.5. As derived projective limits commute with RΓ and with derived completed tensor prod- ucts, we have l−i→m RΓ (Z × D , Lη Rν∗(cid:48)(cid:16)B) = −li→m Rl←im− RΓ (Z ×(cid:17)D , Lη Rν∗(cid:48) B ) ´et n,C t ´et n,C t I n n I • ⊗ • = −li→m Rl←im− Lη KB ˆ Ω (D ) , t I(Z∞) Qp n n I ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ARTHUR-CÉSAR

text: LE

text: BRAS

text: which

text: identiﬁes

text: to

text: the

text: subcomplex

text: X

text: Q

text: X

text: d

text: Q

text: X

text: dX

text: ˆ

text: Lη

text: KM

text: p

text: p

text: Qp

text: t

text: (Z∞)

text: of

text: Ω

text: (D)ˆ

text: Lη

text: KM

text: Qp

text: t

text: (Z∞)

text: All

text: in

text: all,

text: we

text: get

text: a

text: quasi-isomorphism

text: M

text: RΓ

text: (Z

text: D

text: Lη

text: Rν∗

text: Lη

text: KM

text: ˆ

text: Ω

text: (D),

text: ´et

text: C

text: t

text: t

text: (Z∞)

text: Qp

text: as

text: wanted.

text: Now

text: we

text: prove

text: Proposition

text: using

text: Lemma

text: Let

text: M

text: B

text: or

text: M

text: B+

text: /tr,

text: r

text: I

text: dR

text: Choose

text: n

text: The

text: i-th

text: term

text: of

text: the

text: complex

text: Lη

text: KM

text: ˆ

text: Ω

text: (D

text: is

text: t

text: (Z∞)

text: Qp

text: n

text: i

text: O

text: Lη

text: KM

text: ˆ

text: Ω

text: (D

text: (Lη

text: KM

text: )i

text: ˆ

text: (D

text: (Lη

text: KM

text: )i

text: ˆ

text: t

text: (Z∞)

text: Qp

text: n

text: t

text: (Z∞)

text: n

text: t

text: (Z∞)

text: Qp

text: n

text: Any

text: element

text: x

text: in

text: it

text: can

text: thus

text: be

text: written

text: as

text: D

text: (x)Tk

text: L

text: (x)TkdT,

text: k

text: k

text: k

text: k

text: with

text: Dk→(x)

text: ∞(LηtKM(Z∞))i,

text: Lk(x)

text: and

text: p

text: k/nDk(x),

text: p

text: k/nLk(x)

text: when

text: k

text: In

text: particular,

text: for

text: any

text: such

text: x,

text: the

text: series

text: L

text: (x)

text: k

text: k

text: k

text: converges

text: to

text: an

text: element

text: L(x)

text: (Lη

text: KM

text: ˆ

text: Ω

text: (D

text: ))i,

text: and

text: this

text: deﬁnes

text: a

text: morphism

text: L

text: t

text: (Z∞)

text: Qp

text: of

text: the

text: complex

text: Lη

text: KM

text: ˆ

text: Ω

text: into

text: Lη

text: KM

text: ˆ

text: (D

text: Taking

text: the

text: inductive

text: t

text: (Z∞)

text: Qp

text: n

text: t

text: (Z∞)

text: Qp

text: limit

text: over

text: n

text: we

text: get

text: a

text: morphism

text: of

text: the

text: complex

text: l−i→m

text: Lη

text: KM

text: ˆ

text: Ω

text: (D

text: t

text: (Z∞)

text: Qp

text: n

text: n

text: into

text: itself,

text: again

text: denoted

text: L.

text: One

text: easily

text: checks

text: the

text: identity

text: d

text: L

text: L

text: d

text: Id

text: D

text: which

text: proves

text: that

text: the

text: map

text: D

text: is

text: a

text: homotopy

text: inverse

text: to

text: the

text: natural

text: map

text: Lη

text: KM

text: l−i→m

text: Lη

text: KM

text: ˆ

text: Ω

text: (D

text: t

text: (Z∞)

text: t

text: (Z∞)

text: Qp

text: n

text: n

text: This

text: concludes

text: the

text: proof

text: of

text: Proposition

text: when

text: M

text: B

text: or

text: B+

text: /tr.

text: Let’s

text: now

text: treat

text: the

text: case

text: M

text: B.

text: I

text: dR

text: Lemma

text: The

text: natural

text: map

text: B

text: B

text: Lη

text: Rν∗

text: R

text: l←im−

text: Lη

text: Rν∗

text: t

text: t

text: I

text: I

text: is

text: an

text: isomorphism.

text: Proof.

text: We

text: use

text: Lemma

text: This

text: reduces

text: us

text: to

text: prove

text: that

text: the

text: map

text: B

text: B

text: Rν∗

text: R

text: l←im−

text: Rν∗

text: I

text: I

text: is

text: an

text: isomorphism,

text: which

text: follows

text: from

text: Lemma

text: As

text: derived

text: projective

text: limits

text: commute

text: with

text: RΓ

text: and

text: with

text: derived

text: completed

text: tensor

text: prod-

text: ucts,

text: we

text: have

text: l−i→m

text: RΓ

text: (Z

text: D

text: Lη

text: −li→m

text: Rl←im−

text: RΓ

text: (Z

text: Lη

text: B

text: ´et

text: n,C

text: t

text: ´et

text: n,C

text: t

text: I

text: n

text: n

text: I

text: −li→m

text: Rl←im−

text: Lη

text: KB

text: ˆ

text: Ω

text: (D

text: t

text: I(Z∞)

text: Qp

text: n

text: n

text: I

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##