Nahm s conjecture about modularity of q-series

Transcription

1 (joint work with S. Zwegers) Oberwolfach July, 0

2 Let r and F A,B,C (q) = q nt An+n T B+C n (Z 0 ) r (q) n... (q) nr, q < where A M r (Q) positive definite, symmetric n B Q n, C Q, (q) n = ( q k ) k=

3 Let r and F A,B,C (q) = q nt An+n T B+C n (Z 0 ) r (q) n... (q) nr, q < where A M r (Q) positive definite, symmetric n B Q n, C Q, (q) n = ( q k ) k= Problem (Werner Nahm): for which triples of parameters (A, B, C) the function F A,B,C is a modular form?

4 the case r = Theorem(D. Zagier, M.Terhoeven, 007) All triples (A, B, C) in Q + Q Q for which F A,B,C is modular are given in the following table. A B C F A,B,C (e πiz ) 0 /60 θ 5, (z)/η(z) /60 θ 5, (z)/η(z) 0 /48 η(z) /η( z )η(z) / /4 η(z)/η(z) / /4 η(z)/η(z) / 0 /40 θ 5, ( z 4 )η(z)/η(z)η(4z) / /40 θ 5, ( z 4 )η(z)/η(z)η(4z) Here η(z) = q /4 n= ( q n ), θ 5,j (z) = ( ) [n/0] q n /40. n j (0)

5 Lemma. Let F (q) be a modular form of weight w for a subgroup of finite index Γ SL(, Z). Then when ε 0 ( ) F (e ε ) b ε w e a ε + o(ε N ) N 0 for appropriate numbers a, b C. Moreover, a π Q here.

6 Lemma. Let F (q) be a modular form of weight w for a subgroup of finite index Γ SL(, Z). Then when ε 0 ( ) F (e ε ) b ε w e a ε + o(ε N ) N 0 for appropriate numbers a, b C. Moreover, a π Q here. Strategy: compute the asymptotics of F A,B,C (e ε ) when ε 0.

7 Lemma. The system of equations r Q i = Q A ij j, i =,..., r j= has a unique solution with Q i (0, ) for all i r.

8 Lemma. The system of equations r Q i = Q A ij j, i =,..., r j= has a unique solution with Q i (0, ) for all i r. F A,B,C (q) = n a n (q) a n (q) = q nt An+n T B+C (q) n... (q) nr

9 Lemma. The system of equations r Q i = Q A ij j, i =,..., r j= has a unique solution with Q i (0, ) for all i r. F A,B,C (q) = n a n (q) Let q and n i so that q n i a n+ei (q) a n (q) = qnt Ae i + et i Ae i +e T i B a n (q) = q nt An+n T B+C (q) n... (q) nr Q i. Then q n QAi... Q A ir r =. i + Q i

10 Lemma. The system of equations r Q i = Q A ij j, i =,..., r j= has a unique solution with Q i (0, ) for all i r. F A,B,C (q) = n a n (q) Let q and n i so that q n i a n+ei (q) a n (q) = qnt Ae i + et i Ae i +e T i B a n (q) = q nt An+n T B+C (q) n... (q) nr Q i. Then q n QAi... Q A ir r =. i + Q i For fixed small ε terms a n (e ε ) are maximal around ( n log(q ),..., log(q r ) ). ε ε

11 Theorem. There is an asymptotic expansion F A,B,C (e ε ) βe α ε γε( + with the coefficients given as follows: α = r ( L() L(Qi ) ) > 0, i= β = det à / i c m ε m), ε 0 m= Q B i i ( Q i ) /, γ = C + + Qi, 4 Q i where L(x) = Li (x) + log(x) log( x) is the Rogers dilog, à = A + diag{ξ,..., ξ r }, ξ i = Q i Q i > 0

12 and c m = det Ã/ (π) r/ C m (B, ξ, t)e tt Ãt dt where C m are certain polynomials in 3r variables.

13 and c m = det Ã/ (π) r/ C m (B, ξ, t)e tt Ãt dt where C m are certain polynomials in 3r variables. Namely, we define polynomials in 3 variables D m Q[B, X, T ] by [ exp (B + Q Q )T ε/ m! B ( T ) m Li m ε / (Q) ε m ] m=3 = + ( Q D m B, Q, T ) ε m/. m= Then r C m (B, ξ, t) = D mi (B i, ξ i, t i ). m + +m r =m i=

14 Corollary. If F A,B,C is modular then its weight w = 0 α π Q r i= L(Q i) π Q e γε( + p= c pε p) = c p = γp p! p

15 Corollary. If F A,B,C is modular then its weight w = 0 α π Q r i= L(Q i) π Q e γε( + p= c pε p) = c p = γp p! p Since c m = det Ã/ (π) r/ C m (B, ξ, t)e tt Ãt dt are polynomials in the entries of B, ξ and Ã, we have infinitely many polynomial equations: ( cm m! cm )( B, ξ, Ã ) = 0, m =, 3,...

16 Back to the case r = : let c m (B, ξ, A) = (A + ξ) 3m[ c m m! cm ] (B, ξ, ), m =, 3,... A + ξ

17 Back to the case r = : let c m (B, ξ, A) = (A + ξ) 3m[ c m m! cm ] (B, ξ, ), m =, 3,... A + ξ Using Magma we have got the following decomposition of the radical of I = c, c 3, c 4, c 5 Q[B, ξ, A] into prime ideals Rad(I ) = P... P 4 where the generators of P i are given below:

18 i generators of P i ξ ξ + 3 B /, ξ +, A 4 B, ξ +, A 5 B, ξ +, A 6 B + /, ξ + 3ξ +, A + 7 B /, ξ + 3ξ +, A + 8 B + /, ξ, A 9 B, ξ, A 0 B /, ξ, A B, ξ ξ, A B, ξ ξ, A 3 B /, ξ + ξ, A / 4 B, ξ + ξ, A /

19 A B C F A,B,C (e πiz ) 0 /60 θ 5, (z)/η(z) /60 θ 5, (z)/η(z) 0 /48 η(z) /η( z )η(z) / /4 η(z)/η(z) / /4 η(z)/η(z) / 0 /40 θ 5, ( z 4 )η(z)/η(z)η(4z) / /40 θ 5, ( z 4 )η(z)/η(z)η(4z)

20 ( ) a a Theorem. Consider A =, a > a a, a. Then all pairs (B, C) such that F A,B,C is modular are: B C F A,B,C (e πiz ) ( ) b b ( ) b a 4 8a 4 ( ) ( a a ) a and a a 8 4 η(z) q an / n Z+ b a η(z) q an / n Z+ a η(z) q an / n Z+

21 Theorem. Modular ( functions F A,B,C (z) with the matrix A being a of the form a ) a a exist if and only if a =, a = 3/4 or a = /. Below is the list of all such modular functions. A B C F A,B,C (e πiz ) ( ) ( ) 0 0 ( ) ( ) 0 and (θ 5, 3 (z) + θ 5, 3 (z))η(z)/η(z)η(z/) θ 5, (z)η(z)/η(z) θ 5, (z)η(z)/η(z) +θ 5, 3 (z)θ 5, (z)η(z) 3 /η(z/) η(z) η(0z) ( 34 ) ( ) ( and ( ) ( ) 0 and 0 ( ) 0 ( ) ( ) ( ) 0 and 0 ( ) ) θ 5, ( z 8 )η(z)/η( z )η(z) θ 5, ( z 8 )η(z)/η( z )η(z) (θ 5, ( z 4 )η(z)/η(z)η(4z)) 0 θ 5, ( z 4 )θ 5,( z 4 )(η(z)/η(z)η(4z)) (θ 5, ( z 4 )η(z)/η(z)η(4z))

22 The Bloch group B(K) of a field K is defined as the quotient of the kernel of the map Z[K] Λ K [x] x ( x), [0], [] 0 by a subgroup generated by the elements of the form [x] + [y] + [ xy] + [ x ] [ y ] +. xy xy

23 The Bloch group B(K) of a field K is defined as the quotient of the kernel of the map Z[K] Λ K [x] x ( x), [0], [] 0 by a subgroup generated by the elements of the form [x] + [y] + [ xy] + [ x ] [ y ] +. xy xy If K is a number field then B(K) Z Q = K 3 (K) Z Q

24 The Bloch group B(K) of a field K is defined as the quotient of the kernel of the map Z[K] Λ K [x] x ( x), [0], [] 0 by a subgroup generated by the elements of the form [x] + [y] + [ xy] + [ x ] [ y ] +. xy xy If K is a number field then B(K) Z Q = K 3 (K) Z Q and the regulator map is given explicitly on B(K) by where B(K) R r [x] (D(σ (x)),..., D(σ r (x))) D(x) = I ( Li (x) + log( x) log x ) is the Bloch-Wigner dilogarithm function.

25 If Q i = r j= then [Q ] + + [Q r ] B(K): Q A ij j, i =,..., r

26 If Q i = r j= then [Q ] + + [Q r ] B(K): Q i ( Q i ) = i i Q A ij j, i =,..., r Q i j Q A ij j = i,j A ij Q i Q j = 0.

27 If Q i = r j= then [Q ] + + [Q r ] B(K): Q i ( Q i ) = i i Q A ij j, i =,..., r Q i j Q A ij j = i,j If F A,B,C is modular for some B, C then we must have L(Q ) + + L(Q r ) π Q for the distinguished solution with all Q i (0, ). A ij Q i Q j = 0.

28 Conjecture.(Werner Nahm) For a positive definite symmetric r r matrix with rational coefficients A the following are equivalent: There exist B Q r and C Q such that F A,B,C is a modular function. The element [Q ] + + [Q r ] is torsion in the corresponding Bloch group for every solution of Nahm s equation.

29 Conjecture.(Werner Nahm) For a positive definite symmetric r r matrix with rational coefficients A the following are equivalent: There exist B Q r and C Q such that F A,B,C is a modular function. The element [Q ] + + [Q r ] is torsion in the corresponding Bloch group for every solution of Nahm s equation. True when r = : one can prove that all solutions of Q = Q A are totally real if and only if A {,, }, and for the element [Q] the condition of being torsion is equivalent to being totally real since D(z) = 0 z R.

30 For A = hence ( ) a a all solutions are zero in the Bloch group: a a { Q = Q a Q a, Q = Q a Q a, ( Q Q ) a Q = =, Q Q Q ( Q )( Q ) = Q Q, Q + Q = [Q ] + [Q ] = 0 in B(C).

31 A counterexample to the conjecture: F 3/4 /4 /4 3/4, /4 /4, 80 but there are non-torsion solutions to { Q = Q 3/4 Q /4, Q = Q /4 Q 3/4. = θ 5,( z 8 )η(z) η( z )η(z)

32 A counterexample to the conjecture: F 3/4 /4 /4 3/4, /4 /4, 80 but there are non-torsion solutions to { Q = Q 3/4 Q /4, Q = Q /4 Q 3/4. Let t = Q /4 Q /4, then Q Q / Q Q / = θ 5,( z 8 )η(z) η( z )η(z) = t 3 Q / = t 3 ( Q ), = t 3 Q / = t 3 ( Q ). We substitute these equalities into Q / = t Q / and get

33 t 3 ( Q ) = t t 3 ( Q ), t 4 ( Q ) = Q = t 4 Q, t 4 =. Therefore all solutions are (Q, Q ) = (x, x) where or x = tx /, t 4 = ( x) 4 = x (x 3x + )(x x + ) = 0. Hence (Q, Q ) = ( + 3 not torsion because, + 3 D ( + 3) = ) [ is a solution, and + 3 ] is

34 Counterexample with integer entries: 3 0 A = , B =, C = 5, 0 0 F A,B,C (q) = η(z) θ 5, (z) η(z) 3.

35 Problem: find a correct formulation of Nahm s conjecture.

36 Problem: find a correct formulation of Nahm s conjecture. For a positive definite symmetric r r matrix with rational coefficients A the following are equivalent: There exist B Q r and C Q such that F A,B,C is a modular function. The element [Q ] + + [Q r ] is torsion in the corresponding Bloch group for EVERY (?) solution of Q i = r j= Q A ij j, i =,..., r.