Galois groups arising from arithmetic differential equations - - PDF document

Galois groups arising from arithmetic differential equations

ALEXANDRU BUIUM University of New Mexico

• 1. Aim of the talk

1) Briefly introduce arithmetic differential equations (Reference: AB, Inventiones 1995; AB, book, AMS 2005) 2) Show how Galois groups arise in this context (Reference: book above + preprint by AB and A. Saha, 2009.)

• 2. p-derivations

A p-derivation δ : A → A on a ring A is a map such that δ(x + y) = δx + δy + Cp(x, y), δ(xy) = xpδy + ypδx + pδxδy. Here Cp(X, Y ) = Xp + Y p − (X + Y )p p . View δ as a “ d

dp”

• 3. Examples

Example 1: Fermat quotient A = Z, δx = x − xp p . Example 2. A =

Zur

p ,

δx = φ(x) − xp p . where φ on A =

Zur

p

is the unique lift of Frobenius on A/pA and

• means p-adic completion.

From now on R =

Zur

p , R = R/pR

• 4. Examples, continued

Example 3: δ-polynomials A = R{x} := R[x, x′, x′′, ...], δx = x′, δx′ = x′′, ... Here x is a tuple of variables. Example 4: δ-rational functions A =

• R{x}(p), induced

δ

• 5. Galois groups

A ⊂ B δ-rings, pB ∩ A = pA p-ad. compl., p non-zero div. A := A/pA ⊂ B := B/pB ρ : Autδ(B/A) → Aut(B/A) Note: ρ injective

• 6. Γ-extensions

Let Γ be a group. B/A called a Γ-extension if 1) Γ ≃ Autδ(B/A) 2) ρ iso 3) BΓ = A. In particular BΓ = A

• 7. δ-independence

For u ∈

• R{x}(p) say u is

δ-independent if u, δu, δ2u, ... are algebraically in- dependent in R(x, x′, x′′, ...). In this case we have A =:

• R{u}(p) ⊂
• R{x}(p) := B

Write F φ := F p + pδF for F ∈ B

• 8. δ-rational functions
• Theorem. Let x be one variable

and u := xφ/x ∈ B. 1) u is δ- independent 2) B/A is a Z×

p -extension.

• 9. Proof
• Proof. δn(xφ/x)

mod p equals x−pn(x(n))p − xpn+1−2pnx(n) + Gn, Gn ∈ R[x, x−1, x′, ..., x(n−1)] Then one uses (usual) Galois the-

• ry.

• 10. δ-rational functions, II
• Theorem. Let x be one variable

and u := (xφ3 − xφ)(xφ2 − x) (xφ3 − xφ2)(xφ − x) ∈ B 1) u is δ- independent 2) B/A is a PGL2(Zp)-extension.

• Proof. Same idea but more com-

plicated.

• 11. δ-functions on schemes

X ⊂ AN a closed subscheme /R X(R) ⊂ RN set of R-points f : X(R) → R called a δ-function if there exists F ∈ R[x, x′, ..., x(r)], r ≥ 0, x an N-tuple of variables, such that f(a) = F(a, δa, ..., δra), a ∈ X(R) View f as an arithmetic differen- tial equation

• 12. Modular curves

X1(N) := modular curve over R

• f level Γ1(N).

L:= line bundle on X1(N) s.t. sections of L⊗n are the modular forms of weight n. X ⊂ X1(N) affine open set dis- joint from cusps and supersingu- lar locus; restriction of L to X denoted again by X.

• 13. Modular forms

S ring of regular functions on X V :=Spec(

• n∈Z L⊗n)/X

M ring of regular functions on V (ring of modular forms on X)

Gm acts on V/X hence on M/S

• 14. δ-modular forms

A δ-modular function (on X) is a δ-function f : V (R) → R M∞ := δ-ring of δ-modular func- tions. R× acts on V hence on M∞

• 15. δ-Fourier expansion

R((q))[q′, q′′, ..., q(n)]: rings R((q))∞ their union: a δ-ring M∞ → R((q))∞ δ-Fourier expan- sion map: the unique ring homo- morphism extending usual Fourier expansion map M → R((q)) and commuting with δ

• 16. “δ-Igusa curve”

S∞

†

:=Im(M∞ → R((q))∞) Morally viewed as the ring of func- tions on a“δ-Igusa curve” (which we do not define)

• 17. Motivation: Igusa curve

Let S :=S/pS, M:=M/pM S† :=Im(M → R((q))) Spec(S†) ⊂ classical Igusa curve.

• Theorem. (well known)

S† is a (Z/pZ)×-extension of S

• 18. Result for “δ-Igusa curve”

Theorem.

• S∞

†

is a Z×

p -extension of

• S∞

• 19. Proof. x basis of L on X;

M = S[x, x−1]. Barcau (Compositio 2003): there exists f ∈ M∞ f = ϕxφ/x → 1 ∈ R((q))∞. Get surjection Q∞ :=S∞{x,x−1}

(δi(f−1)) → S∞ †

• 20. Proof, continued

By computation in the proof of theorem about xφ/x one gets Q∞ is a Z×

p -extension of

• S∞

left to prove: above surjection an isomorphism enough to show: Q∞ an integral domain because S∞ → Q∞ → S∞

†

is injective and first map is an in- tegral extension

• 21. Proof, continued

The latter follows from an anal- ysis of the Z×

p -equivariant map

M∞ → W

W= Katz’s ring of generalized p-

adic modular forms. argument is geometric; needs aux- iliary construction

22. Application to classical modular forms Corollary. Any divided congru- ence in Zp[[q]] (in the sense of Katz) can be represented as a restricted power series in clas- sical modular forms over R and their (iterated) Fermat quotients

• f various orders.

• 23. Fermat quotient on R((q))

Here the Fermat quotient oper- ator on R((q)) is defined as δ(

anqn) = aφ

nqnp − (

anqn)p

p

• 24. Moral

The above (plus other results,

• cf. AB, book, AMS 2005) sug-

gest that: Some of Number Theory is gov- erned by a “new” geometry (δ- geometry). The latter is obtained from algebraic geometry by re- placing algebraic equations with arithmetic differential equations

• 25. δ-geometry

Objects: δ-sets: sets Xδ + monoid S + subsets (Xs), s ∈ S + rings (Os) of functions Xs → R, such that δ(Os) ⊂ Os. Morphisms: naturally defined. Define ring RXδ:= ∪Os of δ- rational functions

• 25. Alg. geo ⊂ δ-geo.

X/R smooth scheme, L/X line bundle Define δ-sections of Lw, w ∈ Z[φ]; Zariski locally: δ-functions. Define R(X)= ∪O(X′), X′ mod p = ∅, ring of rational functions

• 26. Alg. geo ⊂ δ-geo. II

Xδ:= X(R) S:= {δ-sections ≡ 0 mod p} Xs locus where s invertible Os quotients of δ-sections f

sv

Often R(X) ⊂ RXδ

• 27. Correspondences

X ← C → X correspondence in alg.geo /R; assume etale and as- sume L line bundle on X with iso pull backs on C. Get Xδ ← Cδ → Xδ correspon- dence in δ-geometry.

• 28. Quotients

X/C cat. quot. in {schemes} Xδ/Cδ cat. quot. in {δ-sets} N.B. There are interesting cases when X/C is trivial but Xδ/Cδ is not

• 29. Cases with Xδ/Cδ non-triv

Spherical: X = P1, C graphs of automorphisms in SL2(Z). Flat: X = P1, C graph of a post- critically finite dynamical system

P1 → P1 with negative orbifold

Euler char. Hyperbolic: X a modular curve, C a Hecke correspondence.

• 30. Galois groups

A:= (R(X)“ · ”RXδ/Cδ) B:= RXδ Theorem In all 3 cases B/A is a Γ-extension with Γ profinite (often computable)