Relative Fargues–Fontaine Curve

• Relative Fargues-Fontaine curves are Frobenius quotients of period spaces constructed from perfectoid spaces and local fields, used in number theory and arithmetic geometry.
• The curve is defined using Witt vectors and Frobenius maps, and has applications in the study of isocrystals, coherent sheaves, and p-adic cohomology.
• Different usages of relative Fargues–Fontaine curves include pair-relative and family-relative constructions, each with distinct properties and applications.

The relative Fargues–Fontaine curve is a Frobenius quotient of a period space constructed from a perfectoid space of characteristic $p$ and a coefficient local field. In its standard modern form, for a local field $E$ with residue field $\mathbf F_q$ and a perfectoid space $S$ over $\mathbf F_q$, one forms a period space $Y_{E,S}$ from ramified Witt vectors and defines

$X_{E,S}=Y_{E,S}/\phi_S^{\mathbf Z},$

where $\phi_S$ is induced by the $q$-power Frobenius on $S$. The adjective relative has several uses in the literature: it may refer to the pair $E$0, to families $E$1, to an $E$2-based construction, or to cohomological theories valued on an absolute curve. These constructions must be distinguished from one another.

1. Period spaces, Frobenius, and the quotient construction

Let $E$3 be a non-Archimedean local field with residue field $E$4, ring of integers $E$5, and uniformizer $E$6. For an affinoid perfectoid space

$E$7

over $E$8, choose a pseudo-uniformizer $E$9. The ramified Witt-vector ring is

$\mathbf F_q$0

characterized by

$\mathbf F_q$1

In mixed characteristic it is a ramified Witt-vector ring, while in equal characteristic $\mathbf F_q$2 it is of Laurent-series type, concretely $\mathbf F_q$3.

The integral and analytic period spaces are

$\mathbf F_q$4

and

$\mathbf F_q$5

The $\mathbf F_q$6-power Frobenius on $\mathbf F_q$7 induces an automorphism

$\mathbf F_q$8

The relative Fargues–Fontaine curve is

$\mathbf F_q$9

The Frobenius action is free and totally discontinuous, so the quotient exists as an adic space. The associated diamond has the description

$S$0

Consequently, a morphism $S$1 of perfectoid spaces induces

$S$2

Vector bundles, $S$3-bundles, Higgs bundles, and their moduli therefore form objects over the category of perfectoid spaces rather than only over a single field.

The curve also has a schematic incarnation

$S$4

For affinoid perfectoid $S$5, pullback induces a GAGA equivalence between vector bundles on $S$6 and vector bundles on $S$7, with corresponding cohomology groups. The same equivalence holds for $S$8-bundles through their Tannakian description (Fong, 4 Jun 2026).

For a perfectoid field $S$9 of characteristic $\mathbf F_q$0, the notation $\mathbf F_q$1 is often used for the curve associated with the pair $\mathbf F_q$2. Its dependence on $\mathbf F_q$3 is auxiliary in several classification results: the classification of reductive group schemes over the curve is independent of the chosen algebraically closed perfectoid $\mathbf F_q$4 on isomorphism classes, although certain noncanonical global objects can depend on $\mathbf F_q$5 (Anschütz, 2017).

2. Untilts, closed points, and the meaning of relativity

When $\mathbf F_q$6 is a perfectoid field of characteristic $\mathbf F_q$7, characteristic-zero untilts of $\mathbf F_q$8 appear as characteristic-zero points of $\mathbf F_q$9. If $Y_{E,S}$0 is an untilt, its corresponding point has residue field $Y_{E,S}$1. Thus the curve packages different untilts of the same characteristic-$Y_{E,S}$2 field into a single geometric object.

For the usual mixed-characteristic curve associated with $Y_{E,S}$3, the absolute curve is

$Y_{E,S}$4

Its closed points correspond, up to Frobenius equivalence, to untilts of $Y_{E,S}$5. The distinguished point associated with the untilt $Y_{E,S}$6 has residue field $Y_{E,S}$7, but other points can have residue fields that are not topologically isomorphic to $Y_{E,S}$8, even when their tilts are isomorphic to $Y_{E,S}$9. This follows from the existence of non-surjective continuous self-embeddings of completed algebraic closures of characteristic-$X_{E,S}=Y_{E,S}/\phi_S^{\mathbf Z},$0 power-series fields and the failure of tilting to be fully faithful on topological fields (Kedlaya et al., 2015).

The distinction between a characteristic-$X_{E,S}=Y_{E,S}/\phi_S^{\mathbf Z},$1 field and its untilts is therefore essential. The tilt preserves valuation-theoretic and Galois-theoretic information, but it does not determine the untilt as a topological field. A closed point of the curve records an untilt together with its embedding data, not merely an abstract copy of the tilt.

Several usages of “relative Fargues–Fontaine curve” coexist:

1. Pair-relative construction: $X_{E,S}=Y_{E,S}/\phi_S^{\mathbf Z},$2, relative to the coefficient field $X_{E,S}=Y_{E,S}/\phi_S^{\mathbf Z},$3 and perfectoid characteristic-$X_{E,S}=Y_{E,S}/\phi_S^{\mathbf Z},$4 field $X_{E,S}=Y_{E,S}/\phi_S^{\mathbf Z},$5.
2. Family-relative construction: $X_{E,S}=Y_{E,S}/\phi_S^{\mathbf Z},$6, where $X_{E,S}=Y_{E,S}/\phi_S^{\mathbf Z},$7 is an arbitrary perfectoid space.
3. Cohomological relativity: period sheaves and cohomology theories vary over a rigid or perfectoid space while the target curve remains the absolute $X_{E,S}=Y_{E,S}/\phi_S^{\mathbf Z},$8 (Bras, 2018).
4. $X_{E,S}=Y_{E,S}/\phi_S^{\mathbf Z},$9-relative construction: an analytic stack over the analytic geometry of the field with one element, whose base changes recover particular ordinary curves (Bambozzi et al., 2017).
5. Arc-stack relative construction: for a qfd arc-stack $\phi_S$0 in characteristic $\phi_S$1,

$\phi_S$2

in the category of analytic stacks (Anschütz et al., 16 Oct 2025).

These meanings are related through Frobenius, Witt vectors, and untilting, but they are not interchangeable.

3. Period rings and local structure

The period-theoretic construction begins with

$\phi_S$3

in the unramified mixed-characteristic case, or with the corresponding ramified Witt-vector ring $\phi_S$4. The period spaces are obtained by inverting the relevant parameters and removing the loci where the period variables vanish.

For a perfectoid untilt $\phi_S$5 of $\phi_S$6, Fontaine’s map is

$\phi_S$7

Its kernel defines the distinguished divisor associated with the untilt. In the classical $\phi_S$8-case,

$\phi_S$9

The local period rings include

$q$0

where $q$1 is a local generator of $q$2 in the relevant sense. The completed local ring at the distinguished point $q$3 is

$q$4

The complement of $q$5 is governed by the period ring

$q$6

which is a principal ideal domain. The resulting gluing diagram is schematically

$q$7

This decomposes the curve into an affine or punctured part and the formal neighborhood of the distinguished divisor. Topological cyclic homology gives a homotopy-theoretic realization of this decomposition: $q$8 models the formal part, $q$9 models the punctured Laurent-series part, and $S$0, after the crystalline comparison, models the affine complement (Hesselholt, 2022).

For relative bases, period rings become sheaves. If $S$1, one obtains rings $S$2 on annular regions and a global Frobenius-equivariant period object

$S$3

A nuclear Frobenius module

$S$4

on the period space descends to a solid quasi-coherent sheaf on $S$5. Its cohomology is computed by the Frobenius complex

$S$6

This formula is the basic mechanism by which relative period-theoretic cohomology becomes cohomology on the curve (Colmez et al., 2024).

The $S$7-construction gives another period-ring realization. For a bornological monoid $S$8 with a bounded $S$9-action generated by multiplication by $E$00, one defines

$E$01

For

$E$02

base change to a suitable Fréchet-like $E$03-adic Banach ring recovers an ordinary Fargues–Fontaine curve: $E$04 In this setting, “relative” refers to the analytic base $E$05, not to a general perfectoid family (Bambozzi et al., 2017).

4. Vector bundles, slopes, and $E$06-bundles in families

For a vector bundle $E$07 on $E$08, degree, rank, slope, and Harder–Narasimhan data are defined fiberwise and, in suitable settings, relatively. Over an algebraically closed perfectoid field, every rational slope $E$09 determines a stable bundle

$E$10

of rank $E$11 and degree $E$12. Every vector bundle decomposes noncanonically as

$E$13

with uniquely determined rational slopes.

The vector-bundle classification is equivalent to the classification of isocrystals. If $E$14 is the completion of the maximal unramified extension of $E$15, the functor

$E$16

is exact, faithful, tensor-compatible, and induces a bijection on isomorphism classes. An isocrystal of slope $E$17 corresponds to a semistable vector bundle of slope $E$18; the sign convention is essential (Anschütz, 2017).

For coherent sheaves, the HN filtration is split. If

$E$19

then

$E$20

This yields slope criteria for morphisms, quotient bundles, and global generation. In particular, a vector bundle $E$21 is a quotient of $E$22 precisely when its HN truncations satisfy the rank inequalities

$E$23

with equality only when

$E$24

The same theorem implies that $E$25 is generated by $E$26 global sections exactly when all its slopes are nonnegative, $E$27, and equality forces $E$28 (Hong, 2019).

Relative curves enter this theory through moduli of bundle maps. For a perfectoid space $E$29, functors of global sections, maps, surjective maps, and fiberwise-injective maps are represented by locally spatial, partially proper diamonds. Their dimensions are controlled by HN data, for example

$E$30

The absolute quotient classification is proved by stratifying the space of maps according to their image bundle and showing that proper-image strata have smaller dimension. A relative quotient classification for arbitrary families is not established; it would require a relative HN theory, relative semistability, and compatible dimension formulas (Hong, 2019).

For a reductive group $E$31, vector bundles generalize to $E$32-bundles. Reductive group schemes over $E$33 are classified by reductive group schemes over the category of isocrystals. Torsors under a constant reductive group $E$34 are classified by Kottwitz’s set: $E$35 For a reductive group scheme $E$36 over $E$37, the classification becomes

$E$38

where $E$39 is the corresponding group object over isocrystals (Anschütz, 2017).

The relative theory of meromorphic $E$40-bundles refines this classification. A meromorphic vector bundle consists, v-locally, of a vector bundle on the integral period space together with a Frobenius isomorphism on the punctured space that is allowed poles along $E$41. There are maps

$E$42

where $E$43 retains the generic isocrystal and $E$44 retains the special vector bundle. The meromorphic comparison theorem identifies

$E$45

and identifies generic Newton strata with the Fargues–Scholze charts

$E$46

This is a family version of the relation between shtukas, isocrystals, and bundles on the curve (Gleason et al., 2023).

5. Relative cohomology and motivic realizations

The relative curve supports several cohomological constructions, but their degree of relativity differs.

For a rigid analytic space $E$47, relative period sheaves

$E$48

are defined on the pro-étale site of $E$49. Applying the décalage functor $E$50 to the pro-étale period cohomology produces an overconvergent relative de Rham complex

$E$51

Its cohomology gives a Frobenius module and hence a coherent sheaf, or under finiteness hypotheses a vector bundle, on the fixed curve $E$52: $E$53 For smooth proper $E$54, this vector bundle is the Fargues–Fontaine realization of $E$55-adic étale cohomology. In the proper semistable case,

$E$56

and in the good-reduction case,

$E$57

(Bras, 2018).

A genuinely family-valued construction is obtained by starting with an admissible perfectoid space $E$58 of characteristic $E$59. The relative curve

$E$60

supports a motivic functor

$E$61

It is constructed using perfection, Frobenius invariance of motives, homotopy-fixed objects, and descent through the Frobenius quotient. Composing with relative overconvergent de Rham realization gives

$E$62

For a smooth proper $E$63, the resulting complex is split perfect and its cohomology sheaves are vector bundles. For a perfectoid field $E$64, compactness of the motive suffices for split perfectness, without smoothness or properness assumptions (Bras et al., 2021).

Specialization at an untilt $E$65 gives

$E$66

Thus a single object on $E$67 interpolates the de Rham realizations attached to all untilts of $E$68. This is the principal cohomological meaning of the relative curve.

The same framework supports solid quasi-coherent sheaves representing geometric pro-étale cohomology. For a smooth partially proper rigid variety $E$69, one constructs

$E$70

with

$E$71

Syntomic sheaves are defined as mapping fibers of Hyodo–Kato and filtered de Rham sheaves. In the stable range $E$72,

$E$73

The curve-valued formulation carries Poincaré duality: $E$74 and also satisfies Künneth formulas (Colmez et al., 2024).

A further comparison identifies Hyodo–Kato and de Rham Fargues–Fontaine realizations as symmetric monoidal functors on motives: $E$75 After Galois enrichment, the comparison becomes canonical: $E$76 The resulting de Rham Fargues–Fontaine cohomology admits a finite slope-increasing filtration arising from a motivic weight complex. Its graded pieces have slopes determined by motivic weights, and this filtration is not the intrinsic Harder–Narasimhan filtration (Cao, 20 Sep 2025).

6. Moduli, Higgs bundles, and geometric applications

The relative curve provides a setting for moduli problems beyond vector bundles and $E$77-bundles. Let $E$78 be a line bundle on $E$79, and let $E$80 be a connected reductive group over $E$81. An $E$82-twisted $E$83-Higgs bundle is a pair

$E$84

where $E$85 is a $E$86-bundle and

$E$87

For $E$88, this is a rank-$E$89 vector bundle $E$90 equipped with

$E$91

The Hitchin base is

$E$92

For $E$93,

$E$94

The Hitchin map sends a Higgs field to its invariant characteristic polynomial.

Under a regular-semisimple, pairwise-distinct-root condition, a Hitchin parameter defines a spectral curve

$E$95

Locally, if $E$96 is trivial and

$E$97

then

$E$98

The tautological section $E$99 induces an $\mathbf F_q$00-twisted Higgs field on $\mathbf F_q$01. The BNR correspondence gives

$\mathbf F_q$02

The equivalence is proved under the stated distinct-root condition and is compatible with base change in $\mathbf F_q$03 (Fong, 4 Jun 2026).

For a reductive $\mathbf F_q$04, the regular centralizer $\mathbf F_q$05 produces a Picard prestack

$\mathbf F_q$06

The Picard object acts on the Hitchin stack, and on the regular locus the paper proves

$\mathbf F_q$07

At a degree-one divisor, $\mathbf F_q$08-affine Springer fibers describe local modifications of $\mathbf F_q$09-bundles and Higgs fields. For a generically regular-semisimple Hitchin parameter $\mathbf F_q$10, the local affine Springer fiber at a closed point $\mathbf F_q$11 is denoted

$\mathbf F_q$12

Its regular locus is identified with a local Picard object: $\mathbf F_q$13 For all but finitely many $\mathbf F_q$14, the entire affine Springer fiber is regular.

Beauville–Laszlo gluing yields a morphism

$\mathbf F_q$15

This morphism is fully faithful on every affinoid perfectoid test object and becomes an equivalence after evaluation on every geometric point. It is not asserted to be an equivalence of étale stacks globally. The result is a relative Fargues–Fontaine analogue of the product formula for Hitchin fibers (Fong, 4 Jun 2026).

The relative geometry also supports analytic de Rham stacks. For a qfd arc-stack $\mathbf F_q$16 in characteristic $\mathbf F_q$17, define

$\mathbf F_q$18

and then define the Hyodo–Kato stack

$\mathbf F_q$19

The analytic de Rham-stack formalism supplies $\mathbf F_q$20-descent, cohomological properness for proper maps, cohomological smoothness for smooth maps, and cohomological étaleness for étale maps. For $\mathbf F_q$21, a morphism

$\mathbf F_q$22

induces a $\mathbf F_q$23-exact equivalence on perfect complexes and yields a geometric proof of the $\mathbf F_q$24-adic monodromy theorem. A full identification of coherent cohomology on $\mathbf F_q$25 with Hyodo–Kato cohomology is reserved for subsequent work (Anschütz et al., 16 Oct 2025).

The relative Fargues–Fontaine curve is therefore simultaneously a Frobenius quotient, a moduli object for untilts, a geometric realization of isocrystals, a parameter space for families of vector bundles and $\mathbf F_q$26-bundles, and a receptacle for $\mathbf F_q$27-adic cohomology theories. Its central structural feature is the passage

$\mathbf F_q$28

The modern relative theory extends this pattern to perfectoid families, diamonds, arc-stacks, solid sheaves, motivic realizations, meromorphic bundles, and Hitchin-type moduli, while several broader relative classification and cohomological statements remain active areas of research.