Linearity in Arithmetic Differential Equations Taylor Dupuy UCSD Number Theory Seminar
Part 1: Origins/History Part 2: Arithmetic Differential Equations Part 3: Linear Diff l Eqns
PART 1 Origins of Arithmetic Differential Algebra
Diophantine Geometry A/K abelian variety configuration = finite union of translates of ab subvar = S n i=1 (A i + a i ) A i apple A a i 2 A i (K) A i = A i (K) configurations form a basis of closed sets for a topology on A(K) called the configuration topology
S A(K) Defn. S f.g. f.r. if contained in f.r. f.g. subgroup Mordell-Lang Problem f.r. r S f.g. =) S Conf = S Zar
Mordell-Lang Problem f.r. r S f.g. =) S Conf = S Zar Example: (Mordell Problem) C/Q g(c) 2 Jac(C) =A C,! Jac(C) C(Q) Conf = C(Q) Zar C(K) C(Q) Jac(C)(Q) #C(Q) < 1
example. (Manin-Mumford Problem) C/K K g(c) 2 S = C(K) \ A(K) tors A(K) tors S Zar C(K). A = Jac(C). Mordell-Lang Problem f.r. r S f.g. =) S Conf = S Zar
History of Mordell- Lang Problem Manin - Mordell and Manin-Mumford when K = function field. Raynaud - Manin-Mumford for number fields Faltings, Coleman, Parshin - Lang Conjecture in General Buium - Lang Conjecture over function fields (using differential algebra), char 0 Hrushovski - Relative Lang, char p
Weil Cohomology Theory H : { Smooth Varieties/ k }! { Graded K-algebras} H (X) = 2dim(X) M w=0 H w (X) Finiteness h i (X) < 1 Vanishing h i (X) = 0 for i>2n or i<0 Trace Map Tr : H 2n = K Poincare Duality H i = H 2n i via cup product Kunneth H (X Y )=H (X) H (Y ) Lefschetz Axioms Cycle Map Z i (X)! H 2i has some nice properties Weak Lefschetz Re: pullback of hyperplane sections Hard Lefschetz H i! H i+2, 7! ^! where! =[H], H a hyperplane section.
Setup for Desired Category F1 Z : Sch F1! Sch Z. Spec(Z) F1 in the category Sch F1 to get Spec(Z) F1 Deninger Cohomlogy Theory H D : Sch F1! Graded R-algebras
Deninger Cohomlogy Theory H D : Sch F1! Graded R-algebras Frechetness The spaces H w (X D,j C) Vanishing H w (X) = 0 for w>2n or w<0 Trace Map Tr : H 2n = K Poincare Duality H i = H 2n i via cup product Kunneth H (X Y )=H (X) H (Y ) Lefschetz Axioms???, I m not sure exactly how these should look but there should b Hodge : H w = H 2n w Real Action : R X D! X D, which should be thought of as a replacement of the F
t (induced action on cohomology) =lim t!0 t I t Deninger Says: thm b (s) = guess = Q` s b ( )=0 2 s s 1 acts as zero on H 0. det 1 ( s 2 H1 ) det 1 ( s 2 H0 )det 1 ( s 2 H2 ). acts as identity on H 2
Conditional Proof Pairing: hf,gi := Tr(f 1 ^ f 2 ) How derivatives act: (f ^ g) = (f) ^ g + f ^ (f). Get antisymmetry property: acts trivially on H 2. f 1,f 2 2 H 1 f 1 [ f 2 2 H 2 f 1 ^ f 2 = (f 1 ^ f 2 ) = (f 1 ) ^ f 2 + f 1 ^ ( f 2 ) = (f 1 ) ^ f 2 + f 1 ^ (f 2 )
bhf 1,f 2 i = h (f 1 ),f 2 i + hf 1, (f 2 )i =) 1/2 is antisymmetric on H 1 =) ( 1/2 H 1 ) ir =) ( H 1 ) 1/2+iR
Remarks Cyclic Cohomology (NCG) Topological Hochschild Cohomology De Rham Witt Consistent with random matrix theory statistics (H 1 (Spec(Z) ), ) Algebra with involution F1 Noncommutative Geometric Object? Prototype = BC system?
Borger-Buium Descent
X/K smooth projective K = K char(k) =0 : K! K CLASSICAL Theorem K = {r 2 K : (r) =0} T.F.A.E. 1. KS( )=0 2. J 1 (X) = TX as schemes over X 3. 9X 0 /K such that X 0 K K = X Descent to the constants
Arithmetic Descent R = W (F p ) X/R Theorem (D.) Canonically lifted Frobenius (*) =) Tangent Bundle (FT Xn ) n 0 J 1 (X) n torsor under FT Xn 1. DI n (X/R) =0 2. J 1 (X) n = FT Xn as torsors 3. X n /R n admits a lift of the Frobenius.
Sketch of F 1 Land (???) Monad Geometries Generalized Rings Durov subcat Toen-Vasquies Quillen Set Monoid Lambda Schemes Borger Buium Schemes Algebraic Geometries Connes-Consani Marcolli Soule Z F1 Berkovich Deitmar Z F1 Blueprints Lorsch.
PART 2 Wifferential Algebra
What is a p-derivation? Fermat s Little Theorem 8n 2 Z, 8p prime n n p mod p n n p = p CRAP CRAP = n np p This is a p-derivation p(n) = n np p
p(n) = n np p Zero mod p Zero mod p Product Rule p(ab) = p (a)b p + a p p(b)+p p (a) p (b) Sum Rule p(a + b) = p (a)+ p (b) p 1 X j=1 1 p p j a p j b j Kills Unit non-linear (1) = 0
derivations : A! A ring homomorphisms f : A! A["]/h" 2 i dual numbers infinitesimals p-derivations p : A! A ring homomorphisms f : A! W 1 (A) Witt vectors wittfinitesimals Wittferentiation
Witt Vectors. (x 0,x 1 )(y 0,y 1 )=(x 0 y 0,x 1 y p 0 + y 1x p 0 + px 1y 1 ) (x 0,x 1 )+(y 0,y 1 )=(x 0 + y 0,x 1 + y 1 + C p (x 0,y 0 )) C p (X, Y )= Xp +Y p (X+Y ) p p 2 Z[X, Y ]
: A! B (a) a p mod p p(a) := (a) ap p p-torsion free lifts of the Frobenius p-derivations p : A! B + rules (a) :=a p + p p (a)
Always an A-algebra Defn. (Buium, Joyal 90 s) A p-derivation is a map of sets such that p : A! B 8a, b 2 A p(ab) = p (a)b p + a p p(b)+p p (a) p (b) p 1 X 1 p p(a + b) = p (a)+ p (b) a p p j j=1 j b j
example. R = Z p p(x) = x xp p EXAMPLES example. R = Z p [ ] = root of unity coprime to p p(x) = (x) = = (x) xp p unique lift of the frobenius ( 7! p, identity, on roots of unity else example R = Z ur p = Z p [ : n =1,p- 1] p(x) = (x) xp p
Decreases valuation: p(p) = p p pp =1 p p 1 p(p m )=p m 1 (unitmodp) Decreases valuation: t = d dt t(t n )=n t n 1
Constants Constants of a derivation: (K, ) K = {c 2 K : (c) =0} (ring with der) (subring) Constants of a p-derivation: R = b Z ur p R = {r 2 R : p(r) =0 (ring with p-der) (submonoid)
PART 2 Linear Arithmetic Differential Equations
The poor man s model companion: R = b Z ur p R WILL ALWAYS MEAN THIS RING IN THIS TALK Properties: 1) unramified CDVR, residue field F p 2) bz ur p = Z p [ ; n =1,p- n] b 3) unique lift of the Frobenius: ( ) = p 9! : R! R
Simplest Possible Equation: x =(x ij ) 2 GL n (R) x = x (p) x (p) =(x p ij ) Remarks. x 7! (x)(x (p) ) 1 almost a cocycle
Theorem (existence and uniqueness), 2 gl n (R) u 0 2 GL n (R) ( u = u (p) u u 0 mod p has a unique solution proof. =1+p u = u (p) () Contraction mapping: matrix norms (u) = u (p) f :GL n (R)! GL n (R) f(x) = 1 ( x (p) ) x y p apple 1 =) f(x) f(y) p apple 1 p x y p u = lim n!1 f n (u 0 )
Theorem (coeffs in CDVR) ( u = u (p) u u 0 mod p O = Complete Discrete Valuation Subring u 0 2 GL n (O) and 2 gl n (O) =) u 2 GL n (O) Proof. O = R (characterization) =1+p (u 0 )=u 0, ( ) =, ( ) = (u) = u (p) +1 (u) = ( u (p) ) ( (u)) = ( (u)) (p)
( Theorem (coeffs in valuation delta subring) u = u (p) u u 0 mod p O = valuation delta-subring u 0 2 GL n (O) and 2 gl n (O) =) u 2 GL n (O 0 ) O 0 /O finite extension of delta-subrings Strategy: reformulate as dynamics problem!! Proof. u 0,, =1+p 2 gl n ( b O) complete CDVR regularity Lemma =) 9 0, (u) =u
Theorem (coeffs in valuation delta subring) u 0 2 GL n (O) and 2 gl n (O) =) u 2 GL n (O 0 ) O 0 /O finite extension of delta-subrings Strategy: reformulate as dynamics problem!! u = = (u) 1 ( u (p) ) = = 1 ( )( 2 ( )( ( u (p) ) ) (p) ) (p) w 2 GL n (R) w (m) = pick out mth column properties 1) 2) (w (p) ) (m) =(w (m) ) (p) (vw) (m) =(vw (m) )
Theorem (coeffs in valuation delta subring) u 0 2 GL n (O) and 2 gl n (O) =) u 2 GL n (O 0 ) Strategy: reformulate as dynamics problem!! u = (u) = 1 ( u (p) ) = = 1 ( )( 2 ( )( ( u (p) ) ) (p) ) (p) properties (w (p) ) (m) =(w (m) ) (p) (vw) (m) =(vw (m) ) u (m) = 1 ( 2 ( ( u (p) (m) ) )(p) ) (p) ' : A n F! An F '( ) = 1 ( 2 ( ( (p) ) ) (p) ) (p) u (m) ' : A n (K a )! A n (K a ) fixed point of
Theorem (coeffs in valuation delta subring) u 0 2 GL n (O) and 2 gl n (O) =) u 2 GL n (O 0 ) u (m) = 1 ( 2 ( ( u (p) (m) ) )(p) ) (p) Dynamics! ' : A n F! An F F = Frac(O) '( ) = 1 ( 2 ( ( (p) ) ) (p) ) (p) Lemma. # fixed points of ' : A n (K a )! A n (K a ) < 1
Theorem (coeffs in valuation delta subring) u 0 2 GL n (O) and 2 gl n (O) =) u 2 GL n (O 0 ) Lemma. # fixed points of ' : A n (K a )! A n (K a ) < 1 general principle: '/F =) fixed points of ' : A n (K a )! A n (K a ) A n (F a ) =) columns are in finite algebraic extensions
Lemma. (Fornaes and Sibonay) # fixed points of ' : A n (K a )! A n (K a ) < 1 proof. '(x 1,...,x n )=(x 1,...,x n ) ' built from 7! homogenize j Y : ' j (x 1,...,x n ) x d 0 1 x j =0 7! (p) Claim Y \ {x 0 =0} = ; zero dimensional in =) Y P n fixed points of =) ' : A n (K a )! A n (K a ) ={ (1,u) }
Galois Theory O R, 2 gl n (O) u = u (p) O[u] =O[u ij ] = Picard-Vessiot ring = Ring obtained by adjoining entries of u Galois Group = {c 2 GL n (O) :9 2 Aut O (O[u]), =, (u) =uc} G u/o
Galois Group of = {c 2 GL n (O) :9 2 Aut O (O[u]), =, (u) =uc} G u/o u = u (p) Alternative Descriptions: 1) u/o = { 2 Aut O (O[u]); = 2 GL n (O)} u/o = G u/o 2) 0! I u/o! O[x, 1/ det(x)]! O[u]! 0 Stab GLn (R)(I u/o ) = G u/o
u = u (p) Question: For what c 2 GL n (R) (uc) = (uc) (p) do we get? (uc) =u (p) (c)+ (u)c (p) + p (u) (c)+{u, c} {u, c} = u(p) c (p) p (uc) (p)
(uc) =u (p) (c)+ (u)c (p) + p (u) (c)+{u, c} {u, c} = u(p) c (p) (uc) (p) Claim. p 1) 2) {u, c} =0 c =0 =) (uc) = (uc) (p) Main Example: subgroup of GL n (F a 1) T W N = WT = TW G = maximal torus of diagonals = permutation matrices = matrices with roots of unity entries N = T W = WT = permutation matrices with roots of unity entries
Theorem. O R delta subring O R u 2 O =) G u/o finite u/o = { 2 Aut O (O[u]); = 2 GL n (O)} = G u/o finite to begin with
Theorem A. 9 Q 2 thin set 8 2 Z 2 \, 9u 2 GL n (R) u = u(p) G u/o = finite group containing W Z n2
Theorem B. X = {u 2 GL n (R); u 1 mod p} = ball around identity 9 X of the second category 8u 2 X \, 8O R -closed subring 1) 2) (u)(u (p) ) 1 2 gl n (O) R O =) G u/o = N
Theorem C. 9 GL n (K a ) Zariski closed 8u 2 GL n (R) \ u 0 = u (p) O 3 -closed subring of R dim((z G u/o ) Zar ) apple n
PART 4 Remarks on Picard-Fuchs and Painleve 6 and the Frobenius
Manin Map: E/K K = elliptic curve function field : E(K)! G a (K) (diff alg map) ker( ) =E(K) tors
Setup: E : y 2 = x(x 1)(x t) E universal family A 1 E t (C) = p x(x 1)(x t)
H 1 (E t (C), Z) = Span{ 1, 2}! = dx/y R 1! = 1 R 2! = 2 1 (t), 2 (t) periods
Picard-Fuchs Equations: =4t(1 t) d2 dt 2 + 4(1 2t) d dt 1 kills periods Manin-Map: (X, Y )= R (X,Y ) dx/y
Buium s Picard-Fuchs = 2 + 1 + p 0 Buium s Manin Map (P )= 1 p r log! (P )