Lecture 4: Examples of automorphic forms on the unitary group U(3)

Transcription

1 Lecture 4: Examples of automorphic forms on the unitary group U(3) Lassina Dembélé Department of Mathematics University of Calgary August 9, 2006

2 Motivation The main goal of this talk is to show how one can compute automorphic forms on the unitary group in three variables.

3 Notations F is the field of rationals or a real quadratic field and E is a totally CM quadratic extension of F. The involution in Gal(E/F ) is denoted by a ā, a E. The rings of integers of F and E by O F and O E, respectively. For any prime p in O F, we denote by F p and O F, p the completions of F and O F at p, respectively.

4 Notations F is the field of rationals or a real quadratic field and E is a totally CM quadratic extension of F. The involution in Gal(E/F ) is denoted by a ā, a E. The rings of integers of F and E by O F and O E, respectively. For any prime p in O F, we denote by F p and O F, p the completions of F and O F at p, respectively.

5 Notations F is the field of rationals or a real quadratic field and E is a totally CM quadratic extension of F. The involution in Gal(E/F ) is denoted by a ā, a E. The rings of integers of F and E by O F and O E, respectively. For any prime p in O F, we denote by F p and O F, p the completions of F and O F at p, respectively.

6 Notations F is the field of rationals or a real quadratic field and E is a totally CM quadratic extension of F. The involution in Gal(E/F ) is denoted by a ā, a E. The rings of integers of F and E by O F and O E, respectively. For any prime p in O F, we denote by F p and O F, p the completions of F and O F at p, respectively.

7 Notations For any prime P of E, by E P and O E, P the completions of E and O E at P. The ring of adèles of F and its finite part are denoted by A and A f respectively.

8 Notations For any prime P of E, by E P and O E, P the completions of E and O E at P. The ring of adèles of F and its finite part are denoted by A and A f respectively.

9 Unitary groups For any F -algebra A, the Gal(E/F)-action induces an involution of the matrix group GL 3 (E F A) we denote as before. The unitary group in three variables U(3) on F attached to E is defined as follows. For any F -algebra A, the set of A-rational points on U(3)/F is given by U(3)(A) = { g GL 3 (A F E) : gḡ t = 1 3 }.

10 Unitary groups The unitary group in three variables U(2, 1) on F attached to E is defined as follows. For any F -algebra A, the set of A-rational points on U(2, 1)/F is given by U(2, 1)(A) = g GL 3(A F E) : g ḡ t =

11 Unitary groups We define an integral structure on U(3)/F which we denote the same way by putting U(3)(A) = { g GL 3 (A OF O E ) : gḡ t = 1 3, A }, for any O F -algebra A.

12 Unitary groups We define an integral structure on U(2, 1)/F which we denote the same way by putting U(2, 1)(O E OF A) = g GL 3(A F E) : g ḡ t = 1 3, for any O F -algebra A.

13 U(2, 1) versus SL 2 SL 2 U(2, 1) Global symmetric space: H, Global symmetric space: the Poincaré upper-half plane B = {(z, u) C 2 : 2Re(z) + u 2 < 0} Congruence subgroups: Γ 0 (N) SL 2 (Z) Congruence subroups: Γ 0 (N) U(2, 1)(Z) Γ 0 (N)\H, Γ 0 (N)\B, compact arithmetic curves compact arithmetic surfaces

14 U(2, 1) versus SL 2 SL 2 U(2, 1) Inner forms: Inner forms: quaternions algebras U(3) and (certain) division algebras, with involution of the second kind. Jaquet-Langlands Jacquet-Langlands.

15 Goal We want to construct automorphic forms on U(2, 1) by using its inner form U(3). Study the Galois representations we obtain from those automorphic forms. Assumption: Assume throughout this paper that F has narrow class number 1 and that the quadratic extension E/F is chosen so that the associated group U(3)/F has class number 1.

16 Unitary groups: local case Let p be a prime in F and choose a prime P of E above p. Then, K p = U(3)(O F, p ) is a maximal compact open subgroup in U(3)(F p ). When p is split in E, we choose an isomorphism U(3)(F p ) = GL 3 (E P ) = GL 3 (F p ) s.t. K p = U(3)(O p ) = GL 3 (O p ). When p is inert in E, then U(3)/F p is the unique unitary group in three variables on F p attached to the quadratic extension E P /F p, and K p = U(3)(O p ) is hyperspecial.

17 Unitary groups: local case Let p be a prime in F and choose a prime P of E above p. Then, K p = U(3)(O F, p ) is a maximal compact open subgroup in U(3)(F p ). When p is split in E, we choose an isomorphism U(3)(F p ) = GL 3 (E P ) = GL 3 (F p ) s.t. K p = U(3)(O p ) = GL 3 (O p ). When p is inert in E, then U(3)/F p is the unique unitary group in three variables on F p attached to the quadratic extension E P /F p, and K p = U(3)(O p ) is hyperspecial.

18 Unitary groups: local case Let p be a prime in F and choose a prime P of E above p. Then, K p = U(3)(O F, p ) is a maximal compact open subgroup in U(3)(F p ). When p is split in E, we choose an isomorphism U(3)(F p ) = GL 3 (E P ) = GL 3 (F p ) s.t. K p = U(3)(O p ) = GL 3 (O p ). When p is inert in E, then U(3)/F p is the unique unitary group in three variables on F p attached to the quadratic extension E P /F p, and K p = U(3)(O p ) is hyperspecial.

19 Unitary groups: local case Let p be a prime in F and choose a prime P of E above p. Then, K p = U(3)(O F, p ) is a maximal compact open subgroup in U(3)(F p ). When p is split in E, we choose an isomorphism U(3)(F p ) = GL 3 (E P ) = GL 3 (F p ) s.t. K p = U(3)(O p ) = GL 3 (O p ). When p is inert in E, then U(3)/F p is the unique unitary group in three variables on F p attached to the quadratic extension E P /F p, and K p = U(3)(O p ) is hyperspecial.

20 Automorphic forms We let K be the product K = p K p, and fix a compact open subgroup U of K. Let V be an irreducible algebraic representation of U(3) defined over F.

21 Automorphic forms We let K be the product K = p K p, and fix a compact open subgroup U of K. Let V be an irreducible algebraic representation of U(3) defined over F.

22 Automorphic forms Definition The space of automorphic forms of level U and weight V on U(3) is given by A V (U) = {f : U(3)(A f )/U V : f γ = f, γ U(3)(F )}, where f γ(x) = f (γx)γ for all x U(3)(A f ) and γ U(3)(F ).

23 Hecke operators For any u U(3)(A f ), write UuU = i u iu. Define the Hecke operator [UuU] : A V (U) A V (U) f f [UuU] by f [UuU](x) = i f (xu i), x U(3)(A f ).

24 Hecke operators In the rest of this section, we fix an integral ideal N of F such that (N, disc(e/f )) = (1), and define the level a 1 b 1 c 1 U 0 (N) = a 2 b 2 c 2 K such that a 3 b 3 0 mod N, a 3 b 3 c 3 and to simplify notations, we let A V (N) = A V (U 0 (N)) and T V (N) = T V (U 0 (N)) be the Hecke algebra.

25 Hecke operators Let p be a split prime in F and choose a prime P in E above p. Then, the local algebra of T V (N) at p is isomorphic to the Hecke algebra of GL 3 (F p ) which is generated by the two operators and T 1 (p) = [ GL 3 (O F, p )diag(1, 1, ϖ p )GL 3 (O F, p ) ] = [ 1 (p)] T 2 (p) = [ GL 3 (O F, p )diag(1, ϖ p, ϖ p )GL 3 (O F, p ) ] = [ 2 (p)], where ϖ p is a uniformizer at p.

26 Main result Define the two sets Θ i (p) = U(3)(O F )\{g M 3 (O E ) : gḡ t = π p 1 3 and g i (p)}, where π p is a totally positive generator of p. The quotient U(3)(Ô F )/U 0 (N) is a flag variety over artinian ring O F /N, which we denote by H 0 (N).

27 Main result Theorem There is a natural isomorphism of Hecke modules A V (N) = {f : H 0 (N) V such that f γ = f, γ Γ}, where Γ = U(3)(O F )/O E and the action of the Hecke operators T 1 (p) and T 2 (p) on the right hand side is given by f T i (p)(x) = f (ux)u, x H 0 (N). u Θ i (p)

28 Old and new spaces Let p be a split prime in F such that p N. π : H 0 (N) H 0 (N/p) the natural surjection. Then, the action of the Hecke operators T 1 (p) and T 2 (p) on A V (N/p) is given by f T i (p)(x) = u Θ i (p) f (ux)u, x H 0 (N), where the summation is now restricted to the elements whose action is non-degenerate.

29 Old and new spaces Let p be a split prime in F such that p N. π : H 0 (N) H 0 (N/p) the natural surjection. Then, the action of the Hecke operators T 1 (p) and T 2 (p) on A V (N/p) is given by f T i (p)(x) = u Θ i (p) f (ux)u, x H 0 (N), where the summation is now restricted to the elements whose action is non-degenerate.

30 Old and new spaces There are three degeneracy maps α i (p) : A V (N/p) A V (N), i = 0, 1, 2, where α 0 (p) = π is the pullback map, and α i (p) = π T i (p), i = 1, 2, which combine to give ι p : A V (N/p) 3 A V (N) 2 (f 0, f 1, f 2 ) α i (p)(f i ). i=0

31 Old and new spaces Similarly, when p is inert in E, there are two degeneracy maps α i (p) : A V (N/p) A V (N), i = 0, 2, where α 0 (p) = π, and α 2 (p) = π T 2 (p), and which combine to give ι p : A V (N/p) 2 A V (N) (f 0, f 1 ) α 0 (p)(f 0 ) + α 2 (p)(f 1 ).

32 Old and new spaces Definition The space of oldforms is obtained as A old V (N) := im(ι p ), p N and the space of newforms A new V (N) as its orthogonal complement with respect to any U(3)-invariant Hermitian inner product (, ) on A V (N).

33 Examples The unitary groups U(3)/Q in three variables attached to the quadratic fields Q( 1) and Q( 3) respectively; and also, on the unitary group U(3)/Q( 5) attached to the cyclotomic field Q(ζ 5 ). For each group, we compute the space A 0 (N) of automorphic forms of trivial weight and level N, where Norm(N) 20 and (N, disc(e/f )) = 1. We provide a table for the dimensions of A 0 (N) and A new 0 (N), and the list of all the automorphic forms whose Hecke eigenvalues are rational or defined over a quadratic field.

34 Examples The unitary groups U(3)/Q in three variables attached to the quadratic fields Q( 1) and Q( 3) respectively; and also, on the unitary group U(3)/Q( 5) attached to the cyclotomic field Q(ζ 5 ). For each group, we compute the space A 0 (N) of automorphic forms of trivial weight and level N, where Norm(N) 20 and (N, disc(e/f )) = 1. We provide a table for the dimensions of A 0 (N) and A new 0 (N), and the list of all the automorphic forms whose Hecke eigenvalues are rational or defined over a quadratic field.

35 The unitary group U(3)/Q attached to Q( 1) N dima 0 (N) dima new 0 (N) N p a 1 (p, f 1 ) a 1 (p, f 1 ) a 1 (p, f 1 ) a 1 (p, f 1 ) a 1 (p, f 2 ) a 1 (p, f 3 ) ω ω ω 5 3 6ω ω ω ω ω ω ω ω ω ω ω

36 The unitary group U(3)/Q attached to Q( 3) N dima 0 (N) dima new 0 (N) N p a 1 (p, f 1 ) a 1 (p, f 1 ) a 1 (p, f 1 ) a 1 (p, f 1 ) a 1 (p, f 2 ) a 1 (p, f 1 ) ω ω ω ω ω ω ω ω ω

37 The unitary group U(3)/Q( 5) attached to Q(ζ 5 ) N (4, 2) (9, 3) (11, 3 + ω 5 ) (16, 4) dima 0 (N) dima new 0 (N) N (4, 2) (9, 3) (11, 3 + ω 5 ) (16, 4) N(p) p a 1 (p, f 1 ) a 1 (p, f 1 ) a 1 (p, f 1 ) a 1 (p, f 1 ) a 1 (p, f 2 ) ω ω ω ω ω ω ω ω ω ω 24 ω

38 Residual Galois representations Let f be a newform of level N with eigenvalues in the number field K f, and let O f be the ring of integers of K f. Let l 2 be a prime and choose a prime λ of K f that lies above l. We denote the completions of K f and O f at λ by K f, λ and O f, λ respectively. Let π f = p π p be the automorphic representation attached to f, and let πf E = P πp E be the base change lift of π f to GL(3)/E. We denote the Hecke matrix of π E P by te P.

39 Residual Galois representations Let f be a newform of level N with eigenvalues in the number field K f, and let O f be the ring of integers of K f. Let l 2 be a prime and choose a prime λ of K f that lies above l. We denote the completions of K f and O f at λ by K f, λ and O f, λ respectively. Let π f = p π p be the automorphic representation attached to f, and let πf E = P πp E be the base change lift of π f to GL(3)/E. We denote the Hecke matrix of π E P by te P.

40 Residual Galois representations Let f be a newform of level N with eigenvalues in the number field K f, and let O f be the ring of integers of K f. Let l 2 be a prime and choose a prime λ of K f that lies above l. We denote the completions of K f and O f at λ by K f, λ and O f, λ respectively. Let π f = p π p be the automorphic representation attached to f, and let πf E = P πp E be the base change lift of π f to GL(3)/E. We denote the Hecke matrix of π E P by te P.

41 Residual Galois representations Let f be a newform of level N with eigenvalues in the number field K f, and let O f be the ring of integers of K f. Let l 2 be a prime and choose a prime λ of K f that lies above l. We denote the completions of K f and O f at λ by K f, λ and O f, λ respectively. Let π f = p π p be the automorphic representation attached to f, and let πf E = P πp E be the base change lift of π f to GL(3)/E. We denote the Hecke matrix of π E P by te P.

42 Residual Galois representations Let f be a newform of level N with eigenvalues in the number field K f, and let O f be the ring of integers of K f. Let l 2 be a prime and choose a prime λ of K f that lies above l. We denote the completions of K f and O f at λ by K f, λ and O f, λ respectively. Let π f = p π p be the automorphic representation attached to f, and let πf E = P πp E be the base change lift of π f to GL(3)/E. We denote the Hecke matrix of π E P by te P.

43 Theorem (Kottwitz) There exists a Galois representation ρ f, λ : Gal(Ē/E) GL 3(K f, λ ), associated to f, such that the characteristic polynomial of ρ f, λ (Frob P ) coincides with the one of tp E. The representation ρ f, λ is unramified outside ldisc(e/f)n.

44 By making an appropriate choice of lattice in Kf 3, λ, one can reduce ρ f, λ to get a mod λ representation ρ f, λ. The data we have seem to support the following conjecture. Problem: Numerically study the image of the Galois representation ρ f, l.

45 L. Clozel, M. Harris and R. Taylor; Automorphy for some l-adic lifts of automorphic mod l representations (preprint). L. Dembélé, Explicit computations of Hilbert modular forms on Q( 5). Experimental Mathematics 4, vol. 14 (2005) pp L. Dembélé, Quaternionic M-symbols, Brandt matrices and Hilbert modular forms. To appear in Math. Comp. Hashimoto, Ki-ichiro; Koseki, Harutaka. Class numbers of positive definite binary and ternary unimodular Hermitian forms. Tohoku Math. J. (2) 41 (1989), no. 1, Hashimoto, Ki-ichiro; Koseki, Harutaka. Class numbers of positive definite binary and ternary unimodular Hermitian forms. Tohoku Math. J. (2) 41 (1989), no. 2,

46 Kottwitz, Robert E. Shimura varieties and λ-adic representations. Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), , Perspect. Math., 10, Academic Press, Boston, MA, Lansky, Joshua; Pollack, David. Hecke algebras and automorphic forms. Compositio Math. 130 (2002), no. 1, Gan, Wee Teck; Hanke, Jonathan P.; Yu, Jiu-Kang. On an exact mass formula of Shimura. Duke Math. J. 107 (2001), no. 1,