Three-dimensional imprimitive representations of PSL 2 (Z) and their associated vector-valued modular forms U-M Automorphic forms workshop, March 2015
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Let Γ = PSL 2 (Z) Write ( 0 1 S = 1 0 Γ has a presentation ) ( 1 1, T = 0 1 Γ = R, S R 3, S 2. ) ( 0 1, R = ST = 1 1 In particular, Γ is a quotient of the free nonabelian group on two generators ).
Let ρ: Γ GL n (C) be a complex representation of Γ Let k be an integer. Let H = {τ C Iτ > 0} denote the complex upper half plane. Definition A vector-valued modular function of weight k with respect to ρ is a holomorphic function F : H C n such that ( ) F (γτ) = ρ(γ)(cτ + d) k a b F (τ) for all γ = Γ, c d and such that F satisfies a condition at infinity (explained on next slide)
If F is vector-valued modular for a rep. ρ, = F (τ + 1) = F (T τ) = ρ(t )F (τ) for all τ H. Matrix exponential is surjective, so write ρ(t ) = e 2πiL for some matrix L. Then F (τ) = e 2πiLτ F (τ) satisfies F (τ + 1) = e 2πiLτ e 2πiL ρ(t )F (τ) = F (τ). Meromorphy condition at infinity: insist F has a left finite Fourier expansion for all choices of logarithm L. Can use Deligne s canonical compactification of a vector bundle with a regular connection on a punctured sphere to define holomorphic forms in a natural way.
Example: Let ρ denote the trivial representation Then: vector-valued forms are scalar forms of level 1 Two examples are E 4 = 1 + 240 σ 3 (n)q n, E 6 = 1 504 σ 5 (n)q n. n 1 n 1 The ring generated by the (holomorphic) forms of level 1 in all (integer) weights is C[E 4, E 6 ].
Example: More generally let ρ be a 1-dim rep of Γ = PSL 2 (Z) ρ factors through abelianization of Γ, which is Z/6Z Let χ be the character of Γ such that χ(t ) = e 2πi/6. Then ρ = χ r for some 0 r < 6. The C[E 4, E 6 ]-module generated by vvmfs of all weights for χ r is free of rank 1: H(χ r ) = C[E 4, E 6 ]η 4r, where η is the Dedekind η-function η(q) = q 1/24 n 1(1 q n ).
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The following Free-module theorem is very useful: Theorem (Marks-Mason, Knopp-Mason, Bantay-Gannon) Let ρ denote an n dimensional complex representation of Γ. Let H(ρ) denote the C[E 4, E 6 ]-module generated by all vvmfs of varying weight. Then H(ρ) is free of rank n as a C[E 4, E 6 ]-module. Note: we stated this previously for 1-dim reps!
Example: two-dimensional irreducibles Let ρ be a 2-dim irrep ρ(t ) must have distinct eigenvalues, otherwise ρ factors through abelianization of Γ Assume that ρ(t ) is diagonal and of finite order (to avoid introducing logarithmic terms), and write ( ) e 2πir 1 0 ρ(t ) = 0 e 2πir 2 with r 1, r 2 [0, 1). Let H(ρ) denote the C[E 4, E 6 ]-module of vector-valued modular forms for ρ.
Theorem (F-Mason, 2013) Let notation be as on the previous slide, and let K = 1728/j where j is the usual j-function. Then where: F = η 2k H(ρ) = C[E 4, E 6 ]F C[E 4, E 6 ]DF K 6(r1 r2)+1 12 2F 1 ( 6(r1 r 2 )+1 12, 6(r 1 r 2 )+5 k = 6(r 1 + r 2 ) 1, D = q d dq k 12 E 2. ) 12 ; r 1 r 2 + 1; K ) K 6(r 2 r 1 )+1 12 2F 1 ( 6(r2 r 1 )+1 12, 6(r 2 r 1 )+5 12 ; r 2 r 1 + 1; K,
Example: three-dimensional irreducibles Let ρ be a 3-dim irrep Again, ρ(t ) must have distinct eigenvalues Assume that ρ(t ) is diagonal and of finite order (to avoid introducing logarithmic terms), and write with r 1, r 2, r 3 [0, 1). ρ(t ) = diag(e 2πir 1, e 2πir 2, e 2πir 3 ).
Theorem (F-Mason, 2013) Let notation be as on the previous slide. Then where: F = η 2k H(ρ) = C[E 4, E 6 ]F C[E 4, E 6 ]DF C[E 4, E 6 ]D 2 F K a 1 +1 K a 2 +1 6 3F 2 ( a1 +1 6 3F 2 ( a2 +1 6, a 1+3 6, a 1+5 6, a 2+3 6, a 2+5 K a 3 +1 6 3F 2 ( a3 +1 6, a 3+3 6, a 3+5 k = 4(r 1 + r 2 + r 3 ) 2, 6 ; r 1 r 2 + 1, r 1 r 3 + 1; K ) 6 ; r 2 r 1 + 1, r 2 r 3 + 1; K ) 6 ; r 3 r 2 + 1, r 3 r 1 + 1; K ), and for {i, j, k} = {1, 2, 3} we write a i = 4r i 2r j 2r k.
We used our results on 2-dim vvmfs to verify the unbounded denominator conjecture in those cases Unfortunately, no noncongruence examples arise there! 3-dim case: infinitely many noncongruence examples Results of next section were motivated by the question: can we use our explicit formulae to prove congruences for the Fourier coefficients of noncongruence modular forms? Answer: in some cases we can prove congruences, but not yet using the explicit formulae Instead use a result of Katz on crystalline cohomology
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First we mention some definitions and facts, and explain why we restrict attention to imprimitive representations of dimension 3.
Representations of Γ = PSL 2 (Z): Γ is discrete and its irreps of fixed dimension are parameterized by an algebraic variety (character variety) Most irreps are of infinite image and the corresponding vvmfs are weird (the compoments are modular with respect to a thin subgroup of Γ) We ll focus on reps with finite image Equivalently: we consider irreps ρ with ker ρ a finite index subgroup of Γ Components of vvmfs for ρ are then scalar forms for ker ρ
Representations of Γ of finite image: Finite image reps come in two flavours: primitive and imprimitive Imprimitive means it s induced from a nontrivial subgroup Primitive means it s not There are finitely many primitives of each dimension we d like to classify the (infinitely many) 3-dimensional imprimitive representations of Γ = PSL 2 (Z) with finite image. all but finitely many of these imprimitive ρ have a noncongruence subgroup as kernel.
Three-dimensional imprimitive irreps of Γ of finite image: A 3-dim imprimitive is induced from an index-3 subgroup Lemma Γ contains exactly 4 subgroups of index 3. One is a normal congruence subgroup of level 3, while the others are conjugate with Γ 0 (2). The normal subgroup has finite abelianization and gives rise to a finite number of congruence representations The other index 3 subgroups have infinite abelianization and many characters Since they re conjugate, we can assume we re inducing a character from Γ 0 (2).
Characters of Γ 0 (2) Let U.= ( 1 0 2 1 ) (, V.= TU 1 1 1 = 2 1 Then Γ 0 (2) = T, U = T 2, U V and Γ 0 (2)/Γ 0 (2) = Z (Z/2Z) U generates the copy of Z and V generates Z/2Z ). Thus χ: Γ 0 (2) C with finite image is classified by data χ(u) = λ χ(v ) = ε where λ n = 1 for some n 1 and ε 2 = 1.
The representation ρ = Ind Γ Γ 0 (2) (χ): Let χ: Γ 0 (2) C be a finite order character, with χ(u) = λ and χ(v ) = ε. If ρ = Ind Γ Γ 0 (2)(χ), one checks that ρ(t ) has eigenvalues {ελ, σ, σ} where σ 2 = λ. Further, one can prove the following. Proposition (F-Mason, 2014) Let n be the order of the root of unity λ = χ(u). Then the following hold: 1 ρ is irreducible if and only if n 3; 2 ker ρ is a congruence subgroup if and only if n 24. Thus: previous formulae describe an infinite collection of noncongruence modular forms in terms of η, j and 3 F 2
Next: restrict to certain χ so that we can prove congruences among Fourier coefficients of these noncongruence forms Let n 5 be an odd positive integer and let n/2 < r < n be another integer Define χ n,r : Γ 0 (2) C by χ n,r (U) = e 2πir/n and χ n,r (V ) = 1. If gcd(n, r) = 1 then ker χ n,r = U n Γ 0 (2) (independent of r) let X /C be a projective model of the quotient ker χ n,r \H. Proposition (F-Mason) The curve X is isomorphic (as an algebraic curve over C) with the smooth projective hyperelliptic curve defined by y 2 = x n + 64.
Idea of proof. Let X 0 be the curve corresponding to (ker χ n,r ) V. Then there s a diagram (arrow labels are degrees of maps): X n X (2) 2 2 X 0 n X 0 (2) Hauptmodul for X 0 (2) is K 0 (τ) = ( η(τ) η(2τ)) 24. Bottom arrow only ramified at 0 and, can take n K 0 as hauptmodul for X 0 = P 1 Study ramification of left arrow to deduce X is y 2 = x n K 0 ((1 + i)/2). Use Chowla-Selberg to show K 0 ((1 + i)/2) = 64.
Riemann-Roch + previous results now imply the following. Theorem (F-Mason) Let n 5 be an odd integer and let H = U n Γ 0 (2). Then H is a noncongruence subgroup of PSL 2 (Z) of finite index, and there is a decomposition of the space of cusp forms of weight 2 as follows: S 2 (H) = n 1 r= n+1 2 S 2 (Γ 0 (2), χ n,r ). Further, each subspace S 2 (Γ 0 (2), χ n,r ) is one-dimensional, with a basis given by ( ) r 1728 f n,r = η 4 n 2 ( 3 r 3F 2 j n 2 3, r n 1 3, r n ; 3r 2n, 3r 2n 1 2 ; 1728 ). j
Finally, we can state a congruence result for these forms. Theorem (F-Mason) Let n 5 be an odd integer and let n/2 < r < n denote another integer. Let f n,r be as on the previous slide. Then f n,r has an expansion in terms of q N = e 2πiτ/N, where N = 2n, of the form f n,r (q N ) = m 1 a mq m N, where a m Q for all m. For all primes p 1 (mod n) and for all indices m 1, the coefficient a m is a p-adic integer, and one has a p 2 m + pa m 0 (mod p 2+vp(m) ).
Idea of proof. interpret f n,r as rational differential on X = hyperelliptic curve y 2 = x n + 64 (or one of its finitely many twists mod p) standard results on crystalline cohomology already yield an n term congruence relation for coefficients of f n,r For p 1 (mod n) (supersingular primes), the numerator of the L-function of X mod p is (1 + pt 2 ) (n 1)/2. Factorization of this numerator + decomposition of S 2 (X ) into 1-dimensional pieces, suggest that one can use the CM-automorphisms of X to obtain more refined congruences The χ n,r -equivariant L-function associated to X /F p 2 equals 1 + pt (linear b/c we work over F p 2, field of def. of the CM-automorphisms) Congruence follows by Th. 6.1 of Katz s paper Crystalline cohomology, Dieudonné modules, and Jacobi sums
It would be interesting to obtain a proof of the preceding congruence result that uses the explicit formulae for f n,r in place of Katz s theorem. We would expect such a proof to generalize to some of the higher weight forms obtained from 3-dimensional representations of PSL 2 (Z) (where Katz doesn t apply)
Thanks for listening!