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 + C p ( x, y ) , δ ( xy ) = x p δy + y p δx + pδxδy. Here C p ( X, Y ) = X p + Y p − ( X + Y ) p . p View δ as a “ d dp ”

3. Examples Example 1: Fermat quotient δx = x − x p A = Z , . p Example 2. δx = φ ( x ) − x p Z ur A = � p , . p Z ur where φ on A = � is the unique p lift of Frobenius on A/pA and � means p -adic completion. Z ur From now on R = � 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 : Aut δ ( 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 ≃ Aut δ ( B/A ) 2) ρ iso 3) B Γ = A . In particular B Γ = A

7. δ -independence � For u ∈ R { x } ( p ) say u is δ -independent if u, δu, δ 2 u, ... 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 := x φ /x ∈ B . 1) u is δ - independent 2) B/A is a Z × p -extension.

9. Proof Proof. δ n ( x φ /x ) p equals mod x − p n ( x ( n ) ) p − x p n +1 − 2 p n x ( n ) + G n , G n ∈ R [ x, x − 1 , x ′ , ..., x ( n − 1) ] Then one uses (usual) Galois the- ory.

10. δ -rational functions, II Theorem. Let x be one variable and u := ( x φ 3 − x φ )( x φ 2 − x ) ∈ B ( x φ 3 − x φ 2 )( x φ − x ) 1) u is δ - independent 2) B/A is a PGL 2 ( Z p ) -extension. Proof. Same idea but more com- plicated.

11. δ -functions on schemes X ⊂ A N a closed subscheme /R X ( R ) ⊂ R N 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, ..., δ r a ) , a ∈ X ( R ) View f as an arithmetic differen- tial equation

12. Modular curves X 1 ( N ) := modular curve over R of level Γ 1 ( N ) . L:= line bundle on X 1 ( N ) s.t. sections of L ⊗ n are the modular forms of weight n . X ⊂ X 1 ( 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 � n ∈ Z L ⊗ n )/X V := Spec ( M ring of regular functions on V (ring of modular forms on X ) G m 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” := Im ( M ∞ → R (( q )) ∞ ) S ∞ † 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 /p Z ) × -extension of S

18. Result for “ δ -Igusa curve” Theorem. S ∞ is a Z × � S ∞ � p -extension of †

19. Proof . x basis of L on X ; M = S [ x, x − 1 ] . Barcau (Compositio 2003): there exists f ∈ M ∞ f = ϕx φ /x �→ 1 ∈ R (( q )) ∞ . Get surjection Q ∞ := S ∞ { x,x − 1 } ( δ i ( f − 1)) → S ∞ †

20. Proof, continued By computation in the proof of theorem about x φ /x one gets Q ∞ is a Z × S ∞ � p -extension of 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 Z p [[ 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 of various orders.

23. Fermat quotient on R (( q )) Here the Fermat quotient oper- ator on R (( q )) � is defined as � a φ � a n q n ) p n q np − ( � a n q n ) = δ ( 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 X δ + monoid S + subsets ( X s ) , s ∈ S + rings ( O s ) of functions X s → R , such that δ ( O s ) ⊂ O s . Morphisms: naturally defined. Define ring R � X δ � := ∪O s of δ - rational functions

25. Alg. geo ⊂ δ -geo. X/R smooth scheme, L/X line bundle Define δ -sections of L w , 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 X δ := X ( R ) S := { δ -sections �≡ 0 mod p } X s locus where s invertible O s quotients of δ -sections f s v Often R ( X ) ⊂ R � X δ �

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 X δ ← C δ → X δ correspon- dence in δ -geometry.

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

29. Cases with X δ /C δ non-triv Spherical: X = P 1 , C graphs of automorphisms in SL 2 ( Z ) . Flat: X = P 1 , C graph of a post- critically finite dynamical system P 1 → P 1 with negative orbifold Euler char. Hyperbolic: X a modular curve, C a Hecke correspondence.

30. Galois groups A := ( R ( X )“ · ” R � X δ /C δ � ) � B := R � X δ � � Theorem In all 3 cases B/A is a Γ -extension with Γ profinite (often computable)