Free actions on simultaneous moduli schemes #
Let π« and π¬ be moduli problems over Ell/B, and let a group G act on π¬. If an object
V = (E_V/π) of Ell/B represents the simultaneous problem (π«, π¬), then G acts on V
through its action on the second factor (ModuliProblem.simultaneousAction): by the Yoneda lemma
(CategoryTheory.Functor.RepresentableBy.autMulEquiv), g acts as the automorphism Ο_g of V
such that pulling back the universal pair (Ξ±, Ξ²) along Ο_g gives (Ξ±, g β’ Ξ²).
In Katz and Mazur's proof that a relatively representable, affine, rigid moduli problem π« is
representable (KM 4.7.0), π¬ is a rigidifying problem, for instance the naive full level-three
problem [Ξ(3)] with its action of GLβ(π½β), and π« is recovered as the quotient of the moduli
scheme π(π«, π¬) = V.base by G. The key input is that this action is free: if π« is rigid
and G acts freely on π¬ over nonempty bases, then for g β 1 the automorphism of π(π«, π¬)
induced by Ο_g fixes no point with values in a nonempty scheme
(ModuliProblem.IsRigid.eq_one_of_comp_simultaneousAction_base_eq). Indeed, a point
x : T βΆ π(π«, π¬) fixed by g gives an automorphism Ο of the pulled-back curve E_T over T
which carries the pulled-back structure (Ξ±_T, Ξ²_T) to (Ξ±_T, g β’ Ξ²_T). Rigidity of π« forces
Ο = 1, so g β’ Ξ²_T = Ξ²_T, and freeness of the action on π¬ gives g = 1.
Rigidity of π« cannot be dropped. For the trivial problem π« = [Ξ(1)] and π¬ = [Ξ(3)], the
automorphism [-1] of every elliptic curve carries a level-three basis (P, Q) to (-P, -Q),
so the matrix -1 of GLβ(π½β) acts trivially on π([Ξ(3)]), although it acts freely on
level-three bases.
When π(π«, π¬) is affine, freeness says equivalently that for g β 1 the elements Ο_g^*(a) - a
generate the unit ideal of its ring of functions
(ModuliProblem.IsRigid.span_range_appTop_simultaneousAction_sub_eq_top, through
AlgebraicGeometry.Scheme.Hom.span_range_appTop_sub_eq_top_iff). Pullback reverses composition,
so a group G acts on this ring by g β’ a = Ο_{gβ»ΒΉ}^*(a)
(ModuliProblem.simultaneousSectionsAction), and for g β 1 the elements g β’ a - a generate
the unit ideal (ModuliProblem.IsRigid.span_range_smul_sub_eq_top). For a finite group this is
the hypothesis under which the projection to the invariant quotient is a finite Γ©tale torsor
(TauCeti.AffineInvariantQuotient.etale_projection,
TauCeti.AffineInvariantQuotient.isPullback_of_free).
For the naive full level-N problem [Ξ(N)], on which GLβ(β€/Nβ€) acts freely over nonempty
bases (ModuliProblem.eq_one_of_gammaNaiveAction_hom_app_eq_self), this gives the freeness of the
action of GLβ(β€/Nβ€) on π(π«, [Ξ(N)]) for every rigid π«
(ModuliProblem.IsRigid.eq_one_of_comp_simultaneousAction_gammaNaiveAction_base_eq), and its
form for functions on an affine π(π«, [Ξ(N)])
(ModuliProblem.IsRigid.span_range_gammaNaiveAction_smul_sub_eq_top).
Main definitions #
TauCeti.AlgebraicGeometry.ModuliProblem.simultaneousAction: the action ofGon an object representing(π«, π¬), induced by an action ofGonπ¬.TauCeti.AlgebraicGeometry.ModuliProblem.simultaneousSectionsAction: for a groupG, the induced action on the ring of functionsΞ(π(π«, π¬), β€), by pullback along inverses.
Main results #
TauCeti.AlgebraicGeometry.ModuliProblem.homEquiv_comp_simultaneousAction_hom:Ο_gpulls the universal pair(Ξ±, Ξ²)back to(Ξ±, g β’ Ξ²).TauCeti.AlgebraicGeometry.ModuliProblem.IsRigid.eq_one_of_comp_simultaneousAction_base_eq: for rigidπ«and a free action onπ¬, the action onπ(π«, π¬)has no fixed points with values in nonempty schemes.TauCeti.AlgebraicGeometry.ModuliProblem.IsRigid.span_range_appTop_simultaneousAction_sub_eq_top: whenπ(π«, π¬)is affine, the same freeness in terms of its ring of functions.TauCeti.AlgebraicGeometry.ModuliProblem.IsRigid.span_range_smul_sub_eq_top: for a group, the same freeness for the actionsimultaneousSectionsAction, in the form taken byTauCeti.AffineInvariantQuotient.etale_projection.ModuliProblem.IsRigid.eq_one_of_comp_simultaneousAction_gammaNaiveAction_base_eq,ModuliProblem.IsRigid.span_range_gammaNaiveAction_smul_sub_eq_top: the case ofGLβ(β€/Nβ€)acting on naive full level-Nstructures.
References #
- N. M. Katz, B. Mazur, Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies 108, Princeton University Press, 1985, 2.7.2 and Chapter 4, proof of Theorem 4.7.0.
The action on the object representing a simultaneous problem. If G acts on π¬ and V
represents the simultaneous problem (π«, π¬), then G acts on V through its action on the
second factor: g acts as the automorphism Ο_g of V along which the universal pair (Ξ±, Ξ²)
pulls back to (Ξ±, g β’ Ξ²) (homEquiv_comp_simultaneousAction_hom).
Instances For
The automorphism Ο_g of the object V representing (π«, π¬) acts on the pair of level
structures classified by an arrow f : X βΆ V through the action of g on the second factor.
The inverse of the automorphism Ο_g of the object V representing (π«, π¬) acts on the pair
of level structures classified by an arrow f : X βΆ V through the inverse of the action of g on
the second factor.
A rigid problem makes a free action on a rigidifier free on the simultaneous moduli scheme
(the freeness step in Katz and Mazur's proof of Theorem 4.7.0). Let π« be rigid, let G act on
π¬ freely over nonempty bases, and let V represent (π«, π¬). If a point x : T βΆ V.base with
values in a nonempty scheme T is fixed by the automorphism of π(π«, π¬) = V.base induced by
g, then g = 1.
Freeness on an affine simultaneous moduli scheme, in terms of functions. Let π« be rigid,
let G act on π¬ freely over nonempty bases, and let V represent (π«, π¬) with π(π«, π¬) = V.base affine. For g β 1, the elements Ο_g^*(a) - a of Ξ(π(π«, π¬), β€) generate the unit
ideal. For a group, IsRigid.span_range_smul_sub_eq_top restates this for the action
g β’ a = Ο_{gβ»ΒΉ}^*(a), in the form taken by the invariant quotient by a finite group
(TauCeti.AffineInvariantQuotient.etale_projection).
GLβ(β€/Nβ€) acts freely on π(π«, [Ξ(N)]) for every rigid π«. Let V represent the
simultaneous problem (π«, [Ξ(N)]) with the naive full level-N problem, on which GLβ(β€/Nβ€) acts
by change of basis. If a point of V.base with values in a nonempty scheme is fixed by a matrix
g, then g = 1. For N = 3 this is the freeness of the action of GLβ(π½β) on the moduli scheme
of π« rigidified by naive level-three structures.
The action on the functions of the simultaneous moduli scheme. If a group G acts on π¬
and V represents (π«, π¬), then G acts on the ring of functions Ξ(π(π«, π¬), β€) by pullback
along inverses, g β’ a = Ο_{gβ»ΒΉ}^*(a) (simultaneousSectionsAction_smul_def). For finite G
and affine π(π«, π¬), TauCeti.AffineInvariantQuotient forms the quotient by this action.
Instances For
Under simultaneousSectionsAction R Ο, g acts on a function on π(π«, π¬) by pullback along
the automorphism of π(π«, π¬) induced by Ο_{gβ»ΒΉ}.
Freeness on an affine simultaneous moduli scheme, for the action on functions. Let π« be
rigid, let a group G act on π¬ freely over nonempty bases, and let V represent (π«, π¬) with
π(π«, π¬) = V.base affine. For g β 1, the elements g β’ a - a of Ξ(π(π«, π¬), β€) generate the
unit ideal, for the action simultaneousSectionsAction. For finite G this is the freeness
hypothesis of TauCeti.AffineInvariantQuotient.etale_projection and
TauCeti.AffineInvariantQuotient.isPullback_of_free.
GLβ(β€/Nβ€) acts freely on the functions of an affine π(π«, [Ξ(N)]) for every rigid π«.
Let V represent the simultaneous problem (π«, [Ξ(N)]) with V.base affine. For a matrix
g β 1, the elements g β’ a - a of Ξ(V.base, β€) generate the unit ideal, for the action
simultaneousSectionsAction. For N = 3 this is the freeness hypothesis of the invariant quotient
of the moduli scheme of π« rigidified by naive level-three structures by GLβ(π½β).