Equivariant Atkin-Lehner Theory Introduction Atkin-Lehner Theory: Atkin-Lehner (1970), Miyake (1971), Li (1975): theory of newforms (+ T-algebra action) a canonical basis for S k (Γ 1 (N)) and hence also for S k (Γ(N)). However: The group G = SL 2 (Z/NZ) acts on the space S k (Γ(N)), but newforms are not compatible with the group action! Problem: (Equivariant A-L) Describe a (canonical) basis of the G-isotypic components of S k (Γ(N)) in terms of oldforms/newforms. Remark: This a variant of Hecke s Problem: construct an explicit basis of the G-isotypic components of S 2 (Γ(p)). (Hecke, 1928)
Applications: 1) Study S k (Γ(N)) as an M-module, where M End C (S k (Γ(N)) is the algebra of all modular operators: M = T, G. How large is M? 2) In particular, for k = 2, how large is M compared to E := End ō Q (J X(N) )? Is M = E? 3) What are the Q-isogeny factors of J X(N)? 4) Calculate the rank rank(ns(z N,1 )) of the Neron Severi group of the modular diagonal quotient surface Z N,1 = \(X(N) X(N)). 5) Study modular forms, particularly T T-eigenforms, on Z N,1. (D. Carlton). 6) Computational: a canonical basis of S k (Γ(N)) can be derived from one of S k (Γ(N)) and S k (Γ 1 (N)) by twisting: f f χ.
1. Fundamental Newforms -joint work with Satya Mohit Fix: k, N and put V = S k (Γ(N)). Recall: Atkin-Lehner Theory (1) V = V new V old such that: V new has a basis of T-eigenforms V old = (V new ) comes from lower level. Caution: The Atkin-Lehner Theory for Γ(N) is transported from that of Γ 1 (N 2 ) via β N = ( N 0 0 1 ) : β 1 N Γ(N)β N = Γ N Γ 1 (N 2 ), where Γ N = { ( a b c d) Γ(1) : a d 1 (N), c 0 (N 2 )}. Thus, the A-L level for Γ(N) is N 2, not N. Example: N = p, k = 2 V old = S 2 (Γ 1 (N)) + S 2 (Γ 1 (N)). Basic Difficulty: G = SL 2 (Z/NZ) acts on V, but (1) is not a decomposition of G-modules, due to the following twisting phenomenon:
Twisting Phenomenon: If f(z) = a n q n N V, where q N = e 2πiz/N, and χ is a Dirichlet character mod N, then its χ-twist f χ = χ(n)a n q n N V, and: 1) f χ is often in V new, even if f V old ; 2) twisting can be done by group elements: f χ = f k R χ, where R χ = g( 1 χ) χ(n)t nn/m ; here M = cond(χ), T = ( 1 1 0 1), g(χ) = Gauss sum. variant of Shimura(1973), Atkin-Li(1978) Definition. A normalized newform f V new is called fundamental if f χ is again a newform, for all characters χ mod N. Notation: a) F = {fundamental newforms}, F CM = {f F : f χ = f, for some χ 1}, V fund = f F Cf V new. b) For any subset S V, let V G (S) = G-module generated by S, and write V G-old = V G (V old ) V old, V G-new = (V G-old ) V new.
Remark: It turns out that a newform f V is fundamental f is p-primitive in the sense of Atkin-Li, for all primes p N. Theorem 1: We have V G-new = V fund, so V G-new and V G-old are M-modules, where M = T, G, and we have the M-module decomposition Corollary. If f F, then V = V G-old V G-new. V G (f) = Cf χ, so V G (f) has a basis consisting of all twists of f, and hence is an M-module. In particular, if f / F CM, then dim V G (f) = φ(n). Theorem 2: If f F \ F CM, then V G (f) is an irreducible, symmetric M-module, and we have: V G (f) V G (f ) V G (f) = V G (f ) f = f χ. Remarks: 1) Since M has an involution, we can define the contragedient W of an M-module W, and W is called symmetric if W W. 2) f F CM V G (f) V G (f).
3) For N = p, Theorem 2 is true for an arbitrary (non-cm) normalized newform f V new, and so we get the following multiplicity 1 decomposition: V = f N V G (f). 4) V G (f) is frequently irreducible as a G-module, but not always. If N = p, then have a classification. (This uses the knowledge of the irreducible representations of G = SL 2 (Z/pZ).) Proof (of Irreducibility). Main Observation: f F R χ acts bijectively on V G (f) V G (f) B = direct sum of irreducible, pairwise non-isomorphic B-modules which are induced from U D. [Here B = Borel subgroup, U =unipotent subgroup, D=diagonal subgroup of G.] This decomposition is incompatible with the T- module decomposition irreducible. Remark. Such induced modules were considered (for SL 2 (F q )) by Gelfand-Graev, who called them fundamental representations. In representation theory, they are also called cuspidal representations.
2. Example: V = S 2 (Γ(p)) Dimension Formulae: dim V = g = 1 24 (p + 2)(p 3)(p 5) dim V new = g 2g 1 = 1 24 (p 5)(p2 3p + 8) dim V G-new = = 1 48 (p 1)(p2 2p 17) + b dim V G-old = p+1 2 g 1 + p 1 2 g 0 = 1 48 (p + 1)(p2 10p + 33) b dim V old = 2g 1 = 1 12 (p 5)(p 7), where g i = g(x i (p)), and b = p 1 p+1 2 a with a = 12 g 0, 0 a 7 6. The G-Generation of V : f N 0 := N (Γ 0 (p)) dim V G (f β p ) = p f N 1 := N (Γ 1 (p)) \ N (Γ 0 (p)) dim V G (f β p ) = p + 1 f N 2 := N (Γ 0 (p 2 )) \ (N N 3 ) dim V G (f β p ) = p 1 f N 3 := CM-forms in N (Γ 0 (p 2 )) dim V G (f β p ) = p 1 2, where N = χ N (Γ 0 (p, χ 2 )) R χ 1. If we let N i = N i / (identifying quadratic twists), then V = V G (f β p ) V G (f β p ). f N 0 N 1 f N 2 N 3
Furthermore, #N 0 = g 0 (p) #N 1 = g 1 (p) g 0 (p) #N 2 = g 0 (p 2 ) g 1 (p) 2g 0 (p) h(p) #N 3 = h(p), where h(p) = { h(q( p)) if p 3 (mod 4) 0 if p 1 (mod 4).
3. Geometric Interpretation (k = 2) Recall: The Shimura Construction: T-eigenform f A f J(N) abelian subvariety Note: dim A f = [K f : Q], where K f = Q({a n (f)}). Put: A f,g = g G g(a f) J(N). Observations: 1) A f,g is defined over Q. 2) T C (A f,g) = σ V G(f σ ) = Γ f \G Q V G (f σ ), where Γ f = {σ G Q : f σ = f χ, for some χ} G f := Gal( Q/K f ). Theorem 3: If f F \ F CM, then dim A f,g = φ(n)[z f : Q] = φ(n)[g f : Γ f ], where Z f = Fix(Γ f ) K f. Furthermore, if M f End 0 Q(A f,g ) denotes the projection of M onto A f,g, then a) Z(M f ) = Z f, b) dim Q M f = φ(n) 2 [Z f : Q]. Remark: Ribet(1980) calls Gal(K f /Z f ) the group of inner twists. Using his results (and Shimura s), one can show:
Theorem 4: If f is a non-cm T-eigenform, then A f,g is a (complete) isogeny factor of J(N) / Q and A f,g B n, for some simple abelian variety B/ Q. Furthermore, if f F, then Z f is the centre of E f := End 0 Q(A f,g ), i.e. Z f = Z(E f ) and dim Q E f = φ(n) 2 [Z f : Q] = dim Q M f. Note: The above assertion is false for f F CM : CM Shimura f F where m = A f E n, E: CM elliptic curve A f,g E m, ) h(p) (if N = p). Thus ( p 1 2 E f = End 0 Q(A f,g ) = M m (K), where K = Q( p), but h M f = M p 1 (K), 2 i=1 since the V G (f σ ) s are M-irreducible and pairwise non-isomorphic.
Application 1: An Isogeny Relation: J(p) J 0 (p) p (J 1 (p)/j 0 (p)) p+1 2 J p 1 f J p 1 2 CM. Here J f J 0 (p 2 ) is the abelian subvariety whose cotangent space is T C (J f) N J f T C (J f) = f N 2 Cf ( dim J f = 1 2 #N 2), and J CM E h, where E is an elliptic curve with End 0 (E) = Q( p). Note: If A J X is an abelian subvariety (here X is any curve), then the polarization induces a surjection N A : J X A and hence an injection NA : T C (A) T C (J X) can H 0 (X, Ω 1 X ). Application 2: Comparison of Algebras: Recall: M = T, G E = End 0 (J(p)). Then: Q dim T = g = p 1 2 (g 0(p 2 ) g 0 (p)) + g 1 (p) dim M = (p 1)g + (p + 1)g 1 (p) g 0 (p) dim E = dim M + 2 1(p 1)2 h(h 1) dim C G (M) = 24 1 (p 1)(p 5) + 1 2 y + h dim C G (E) = dim C G (M) + 2h(h 1), where y = g 0 (p) ( 1) p 1 ( ( )) 2 1 2 1 + 2p.
4. Numerical Examples N = 7: Here g = 3, g 0 = g 1 = 0, dim V G-old = 11+1 2 g 1 + 11 1 2 g 0 = 0, dim V G-new = g dim V G-old = 3; g 0 (7 2 ) = 12 1 (7 1)(7 5) + g 0 = 1, #N 2 = g 0 (7 2 ) g 1 2g 0 h(p) = 0, dim J f = 1 2 #N 2 = 0. Thus, the above isogeny relation becomes J(7) E 3, where E = J CM is the CM-elliptic curve with End 0 (E) = Q( 7). N = 11: In this case we have: g = 26, g 0 = g 1 = 1, dim V G-old = 11+1 2 g 1 + 11 1 2 g 0 = 11, dim V G-new = g dim V G-old = 15; g 0 (11 2 ) = 12 1 (11 1)(11 5) + g 0 = 6, #N 2 = g 0 (11 2 ) g 1 2g 0 h(p) = 2, dim J f = 1 2 #N 2 = 1.
Here the isogeny relation becomes: J(11) E 11 1 E10 2 E5 3 where E 1 = X 0 (11), E 2 = J f and E 3 = J CM. This relation is (essentially) due to Hecke(1928); cf. also Ligozat(1976). N = 13: In this case we have: g = 50, g 0 = 0, g 1 = 24 1 (13 5)(13 7) = 2 dim V G-old = 13+1 2 g 1 + 13 1 2 g 0 = 14, dim V G-new = g dim V G-old = 36; g 0 (13 2 ) = 12 1 (13 1)(13 5) + g 0 = 8, #N 2 = g 0 (11 2 ) g 1 2g 0 h(p) = 6, dim J f = 1 2 #N 2 = 3. Here one has the isogeny relation: J(11) J 1 (13) 7 J 12 f, where dim J f = 3 and dim J 1 (13) = 2.
5. Application to Z N,1 Situation: If G Aut(X) acts on a curve X, G acts diagonally on the surface Y := X X. Then: rk(ns(y )) = 2 + dim End 0 (J X ) rk(ns(g\y )) = 2 + dim C G (End 0 (J X )), where C G (E) = {α E : gα = αg} denotes the centralizer of G in E = End 0 (J X ). Now: if X = X(N), then the quotient Z N,1 = G\(X X) is the modular diagonal quotient surface of determinant 1, and so, by Application 1 above we have Theorem 5: If N = p is a prime, then rk(ns(z N,1 )) = 2 + dim C G (E) = 2 + dim C G (M) + 2h(h 1) = 2 + 1 24 (p 1)(p 5) + 1 2 y + h. In particular, NS(Z N,1 ) Q is generated by modular correspondences either p 1 (4) or p 3 (4) and h(p) = 1.