Congruence sheaves and congruence differential equations Beyond hypergeometric functions

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1 Congruence sheaves and congruence differential equations Beyond hypergeometric functions Vasily Golyshev Lille, March 6, 204 / 45

2 Plan of talk Report on joint work in progress with Anton Mellit and Duco van Straten monodromy Calabi-Yau Correspondence Idea of Congruence sheaves Example of Outlook 2 / 45

3 Fano: complex projective variety F with positive K. Classification: dim : P dim 2: del Pezzo surfaces dim 3: H 2 (F,Z) = Z: Iskovskikh, 7 families; higher rank: Mori Mukai, 05 families 3 / 45

4 New approach to Classify Fanos by classifying mirror dual objects: Landau Ginzburg models; respective Picard Fuchs differential equations Golyshev, Classification problems and mirror duality Corti, Coates, Galkin, Golyshev, Kasprzyk, Mirror symmetry and Fano manifolds Pioneered for CY by van Straten van Enckevort. 4 / 45

5 A famous hypergeometric series φ(t) := n=0 5n! n! 5tn Satisfies hypergeometric differential equation Pφ(t) = 0 P := D t(d+ 5 )(D+ 2 5 )(D+ 3 5 )(D+ 4 5 ) D := t t Three singular points Σ := {0, /5 5, } P 5 / 45

6 Candelas, de la Ossa, Green, Parkes (99) Picard-Fuchs equation of family of Calabi-Yau threefolds. π : Y P rankh 3 (Y t ) = 4, h 3,0 = h 2, = h,2 = h 0,3 = Defines symplectic local system R 3 π (Z) on P \Σ Contains enumerative information of general quintic X P 4 : numbers of lines, conics, etc on X. 6 / 45

7 Logarithmic solution: q-coordinate: φ (t) := log(t) φ(t)+f (t), f (t) tq[[t]] q := exp(φ (t)/φ(t) = t+770t t Express P in q-coordinate P D 2 5 K(q) D2 K(q) = 5+ d n d d 3 q d q d n = 2875, n 2 = , n 3 = ,... 7 / 45

8 D 4 Nt(D +α )(D+α 2 )(D+α 3 )(D+α 4 ) α +α 4 = = α 2 +α 3 Mirror to complete intersections in weighted projective spaces. α α 2 α α / 45

9 Monodromy G = h 0,h,h GL n (C) 9 / 45

10 Differential equations D3 Differential operator of type D3 can be written as D 3 +t ( )( ) D + 2 a3 (D 2 +D)+b +t 2 (D+) ( ) a 2 (D +) 2 +b 0 +a t 3 (D +2) ( D + 3 2) (D+) +a 0 t 4 (D +3)(D+2)(D+). Th. [, Przyjalkowski] Regularized quantum differential equations of rank Fano threefolds of index d and degree 2d 2 N are exactly (N,d) modular D3 s: given by the expansions of level N weight 2 modular forms with respect to the d th root of the Hauptmodul on X 0 (N) +N. 0 / 45

11 Calabi-Yau Operators Differential equations D3 Abstraction of Picard-Fuchs for families Y P of Calabi-Yau 3-folds/motives defined over Z. Definition: P Z[t, D] is called Calabi-Yau, when: Fuchsian of order four. MUM (maximal unipotent monodromy) at 0. Monodromy Sp 4 (Z). Integrality: φ(x) Z[[t]], n d Z. / 45

12 Beyond hypergeometrics Simplest family of with four singular points D 4 +t(ad 4 +2aD 3 +bd 2 +cd+e)+ft 2 (D+α )(D+α 2 )(D+α 3 ))(D+α 4 ) singularities are at the roots of +at+ft 2 = 0. b, c, e: accesory parameters. 2 / 45

13 Find all such with coming from geometry. or: with monodromy Sp 4 (Z) or: with an integral solution φ Z[[x]] Seems very difficult: we not know position of the the four points: j-invariant we not know the value of the accesory parameters...but we know many examples ( 85 cases). List complete? 3 / 45

14 Example: Coming from a Calabi-Yau 3-fold in Grass(3, 6): D 4 t(65d 4 +30D 3 +05D 2 +40D+6)+4 3 t 2 (D+/4)(D+) 2 (D+3/4) Solution holomorphic at 0: φ(x) = a n t n n=0 a n = k,l,m( n k)( n l)( m k)( m l )( k+l )( n 2 k m) = k,l( n k) 2 ( n l ) 2 ( k+l ) 2 n 4 / 45

15 A step back Special Heun equations: D 2 +t(ad 2 +ad +λ)+bt 2 (D +) 2 singularities are at the roots of +at+bt 2 = 0, λ: accesory parameter. Basically four cases; studied by F. Beukers, A. Beauville, D. Zagier and others. New systematic approach: Congruence sheaves. Works very nicely for this case. 5 / 45

16 Correspondence:...very Number field case: K Q Gal(K/K)-representations Automorphic representations ρ : Gal(K/K) G π ρ L 2 ( L G(A)/ L G(Q)) Example: Elliptic curve E/Q: H (E,Q l ) Hecke Eigenform n=0 a nq n Tr(F p ) a p 6 / 45

17 Correspondence:...very Function field case: K = k(c), C curve over k := F p Gal(K/K)-representations Automorphic representations l-adic sheaves on C Hecke-Eigensheaves on Bun. 7 / 45

18 Correspondence:...very Example: Family of elliptic curves π : E C = P /F p L = R π (Q l ) is an l-adic sheaf: x : Spec(k) C point (in the scheme sense), k = q = p r. x (L) is Gal(k/k)-representation. Frobenius F x Gal(k/k) (inverse to) a a q. F x Aut(x (L)). Frobenius F x Hecke-operator H x 8 / 45

19 The glueing problem Correspondence:...very Families over π : X P := P Z For p prime get X mod p P mod p and thus l-adic sheaves L p. Relation between L p? 9 / 45

20 Provisional definition: Condition on P x (t) = det(f x t) characteristic polynomial of F x. For L of rank 2 and weight : P x (t) = (t α 0 )(t α ) α 0 = mod N, α = p mod N If this happens for almost all p and x C(F p ), we say: L is a N-congruence sheaf. +p Tr(F x ) 0 mod N 20 / 45

21 For L of rank 4 and weight 3: P(t) = (t α 0 )(t α )(t α 2 )(t α 3 ) α 0 = mod N, α 3 = p 3 mod N α = p mod N 2, α 2 = p 2 mod N 2 If this happens for almost allp and x C(F p ), we say: L is a (N,N 2 )-congruence sheaf. Works as glue between various L p 2 / 45

22 A special Hecke-algebra Kontsevich (2005) Notes on motives in finite characterisic Makes correspondence explicit for SL 2 -local systems on P \{four points} with unipotent local monodromies 22 / 45

23 From data Σ = {roots of t 3 +at 2 +bt+c = 0, } get a polynomial in three variables: P(x,y,z) = (b xy yz xz) 2 4(xyz +c)(x+y +z +a) It is the discriminant of the polynomial (t 3 +at 2 +bt+c) (t x)(t y)(t z) 23 / 45

24 Picture of the surface P(x,y,z) = 0 a = b = c = 0 24 / 45

25 Picture of the surface P(x,y,z) = 0 a > 0,b = c = 0 25 / 45

26 Picture of the surface P(x,y,z) = 0 26 / 45

27 Now take a prime number p and a point x P (F p ). Define a (p+) (p+)-matrix H x by (H x ) yz = 2+#{w w 2 = P(x,y,z)}+correction For x,y P (F p ) one has (miracle!) [H x,h y ] = 0 Hecke-algebra: algebra generated by H x,x P (F). 27 / 45

28 Example p = 7, a = 3,b = 4,c = H = / 45

29 The trace function of L is the Hecke-eigenvector Hecke-algebra acts on functions on P (F p ), written as vectors v = (v x ), x P (F p ) (H x v) y = z (H x ) yz v z Drinfelds theorem tells: SL 2 (F p )-automorphic representations correspond to normalised common eigenvectors H x v = v x.v, v x v y = z H xyz v z and hence to l-adic local systems L on P \Σ. L v = (Tr(F x ) x P ) 29 / 45

30 Let s calculate! The trace function of L is the Hecke-eigenvector Strategy Run over p = 3,5,7,,......and all f = t 3 +at 2 +bt+c F p [t]. Determine rational common eigenvectors for Hecke-algebra. Sort them for various p using congruences. 30 / 45

31 The trace function of L is the Hecke-eigenvector Decomposition of the Hecke-algebra Example: p = 7 a,b,c 0,0, ,0,2 6 0,0,3 6 0,, ,, 7 0,, ,3, ,3, 3 4 0,3,2 7 0,3, / 45

32 The trace function of L is the Hecke-eigenvector The 8 non-trivial rational eigenvectors for p = 7. a, b, c Eigenvector v N j 0,0,2 [ ] 3 0 0,0,3 [ ] 3 0 0,,0 [ ] 4 6 0,3,3 [ ] 5 4 0,3,0 [ ] 6 6 0,3,0 [ ] 6 6 0,3,0 [ ] 8 6 0,0, [ ] 9 0 N: the congruence. j: j-invariant F p of the four points. 32 / 45

33 The trace function of L is the Hecke-eigenvector Observations: Running over p = 3,5,7,,... we see: There are very few Q-Hecke-eigenvectors. All of these have a non-trivial congruence. There are no 7, 0,, 2,... congruences. 3, 4, 5, 6, 8-congruences are persistent. 33 / 45

34 The trace function of L is the Hecke-eigenvector Focus on 5-congruence: p a, b, c j Eigenvector 7 0,3,3 4 [ 2,8, 2,3,3, 2, 2,8] 0,2,0 [2, 3, 3,2,2, 3,2, 3,2,2,2,2] 0,2,0 [2,2,2,2, 3,2, 3,2,2, 3, 3,2] 3 0,2,5 4 [4,4,, 6,,,,4,4,4,4, 6, 6,4] 7 0,,2 [3, 2, 2,3, 2, 7, 2, 2, 2,3, 2, 7,3,3,8,3,8, 9 0,2,4 5 [5,5, 5,0,20,5,0, 20, 20,0,0,0,0, 5,0,0,5, 5, 5 34 / 45

35 The trace function of L is the Hecke-eigenvector j-invariant for 5-congruence trace function p j Reconstruct j Q from reductions mod p (Wang algorithm): j = ! Simplest choice: a =,b =,c = / 45

36 Special Heun equation with Riemann symbol where P = f 2 +f +t λ f = t(t 2 +at+b) Singular points: 0, roots of t 2 +at+b,. λ: the accesory parameter. 36 / 45

37 Automorphic property of Multiplication theorem: Let φ(t) = φ λ (t) = +b (λ)t+b 2 (λ)t be solution of differential equation Then one has: φ(x)φ(y) = 2πi Pφ(t) = 0 K(x,y,z)φ(z) dz z where K(x,y,z) = b P(x,y,b/z) 37 / 45

38 Automorphic property of Complete analogy: for C = P (F p ) v x v y = z H xyz v z Geometric for C = P (C): φ(x)φ(y) = K(x,y,z)φ(z) dz 2πi z 38 / 45

39 Automorphic property of Change from point-basis to power-basis Functions on F p P (F p ) δ x, x F p x n, n = 0,...,p v φ p (x) = p n=0 Inverse Vandermonde-Transformation p.... p... (p ) p a n x n a 0 a. a p = v p.. v v 0 39 / 45

40 Automorphic property of p = 7: p = 3: Eigenvector Transform Eigenvector Transform p = 43: / 45

41 Automorphic property of x = 0 mod 7, x = 4 mod 3, x = 8 mod 43 x = 47 mod We get a series φ(t) = +3t+9t 2 +47t 3 +25t with φ(t) φ p (t) mod p 4 / 45

42 Automorphic property of Apéry numbers Generating function, 3, 9, 47,... a n := n ( ) 2 ( ) n n+k k k k=0 φ(t) := satisfies the differential equation a n t n n=0 (D 2 t(d 2 +D +3)+t 2 (D+) 2 )φ(t) = 0 42 / 45

43 Automorphic property of F. Beukers (98) Modular interpretation Operator arises as Picard-Fuchs operator of modular elliptic surface for Γ (5). π : X P with singular four singular fibres of type I,I,I 5,I 5. Number of points in fibre X t (F p ) divisible by 5. +p Tr(F p ) 0 mod 5 43 / 45

44 Automorphic property of So indeed: R π (Q l ) is a 5-congruence sheaf! What did we? Use explicit SL 2 -Hecke-algebra. Found 5-congruence trace-functions. Found from it Apéry sequence. Found arithmetic D2-equation. It works for all Beukers-Zagier-Beauville. Bonus: Get not only differential equation, but also get the Frobenius traces of all fibres! 44 / 45

45 What one might hope to Find explicit Hecke-algebras and multiplication theorems for general Heun-equations. Same for D3s Find explicit Hecke-algebra for Sp 4 -case. Find bi-congruence trace-functions. We claim: is substitute for modularity of D2 and D3! Find corresponding differential of CY-type. 45 / 45

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