Mock modular forms and their shadows
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1 Mock modular forms and their shadows Zachary A. Kent Emory University
2 Classical Eichler-Shimura Theory Modular Forms Basic Definitions
3 Classical Eichler-Shimura Theory Modular Forms Basic Definitions Notation: Throughout, let z = x + iy H with x, y R and y > 0.
4 Classical Eichler-Shimura Theory Modular Forms Basic Definitions Notation: Throughout, let z = x + iy H with x, y R and y > 0. Definition A modular form of weight k is any holomorphic function f on H satisfying:
5 Classical Eichler-Shimura Theory Modular Forms Basic Definitions Notation: Throughout, let z = x + iy H with x, y R and y > 0. Definition A modular form of weight k is any holomorphic function f on H satisfying: 1 For all ( a b c d ) SL2 (Z), we have f ( ) az + b = (cz + d) k f (z). cz + d
6 Classical Eichler-Shimura Theory Modular Forms Basic Definitions Notation: Throughout, let z = x + iy H with x, y R and y > 0. Definition A modular form of weight k is any holomorphic function f on H satisfying: 1 For all ( a b c d ) SL2 (Z), we have f ( ) az + b = (cz + d) k f (z). cz + d 2 We have that f is holomorphic at the cusps.
7 Classical Eichler-Shimura Theory Modular Forms Fourier expansions
8 Classical Eichler-Shimura Theory Modular Forms Fourier expansions Lemma If f is a weight k modular form, then f (z) = n 0 c f (n)q n where q := exp(2πiz).
9 Classical Eichler-Shimura Theory Modular Forms Fourier expansions Lemma If f is a weight k modular form, then f (z) = n 0 c f (n)q n where q := exp(2πiz). Notation M k := weight k modular forms.
10 Classical Eichler-Shimura Theory Modular Forms Fourier expansions Lemma If f is a weight k modular form, then f (z) = n 0 c f (n)q n where q := exp(2πiz). Notation M k := weight k modular forms. S k := weight k cusp forms (subspace of M k whose forms have vanishing constant terms)
11 Classical Eichler-Shimura Theory Eichler-Shimura-Manin Periods
12 Classical Eichler-Shimura Theory Eichler-Shimura-Manin Periods The Setting k 4 is an even integer
13 Classical Eichler-Shimura Theory Eichler-Shimura-Manin Periods The Setting k 4 is an even integer Definition If f S k and 0 n k 2, then the nth period of f is r n (f ) := 0 f (it)t n dt.
14 Classical Eichler-Shimura Theory Eichler-Shimura-Manin Periods and L-values
15 Classical Eichler-Shimura Theory Eichler-Shimura-Manin Periods and L-values Remark For these n, we have the following relation with critical values L(f, n + 1) = (2π)n+1 n! r n (f ).
16 Classical Eichler-Shimura Theory Eichler-Shimura-Manin Period Polynomials
17 Classical Eichler-Shimura Theory Eichler-Shimura-Manin Period Polynomials Lemma If f S k and r(f ; z) := i 0 f (τ)(z τ) k 2 dτ,
18 Classical Eichler-Shimura Theory Eichler-Shimura-Manin Period Polynomials Lemma If f S k and then r(f ; z) := n=0 i 0 f (τ)(z τ) k 2 dτ, k 2 ( ) k 2 r(f ; z) = i n+1 r n (f ) z k 2 n. n
19 Classical Eichler-Shimura Theory Eichler-Shimura-Manin Abstract Framework
20 Classical Eichler-Shimura Theory Eichler-Shimura-Manin Abstract Framework Definition If P V := V k 2 (C) = polynomials of degree k 2, and γ := ( ) a b c d SL2 (Z), then let ( ) az + b P γ := (cz + d) k 2 P. cz + d
21 Classical Eichler-Shimura Theory Eichler-Shimura-Manin Abstract Framework Definition If P V := V k 2 (C) = polynomials of degree k 2, and γ := ( ) a b c d SL2 (Z), then let ( ) az + b P γ := (cz + d) k 2 P. cz + d Lemma Let S := ( ) ( and U := 1 1 ) 1 0, and let W := {P V : P + P S = P + P U + P U 2 = 0}.
22 Classical Eichler-Shimura Theory Eichler-Shimura-Manin Abstract Framework Definition If P V := V k 2 (C) = polynomials of degree k 2, and γ := ( ) a b c d SL2 (Z), then let ( ) az + b P γ := (cz + d) k 2 P. cz + d Lemma Let S := ( ) ( and U := 1 1 ) 1 0, and let W := {P V : P + P S = P + P U + P U 2 = 0}. Then W is the cohomology group H 1 (SL 2 (Z), V).
23 Classical Eichler-Shimura Theory Eichler-Shimura-Manin, Kohnen-Zagier Eichler-Shimura Isomorphism
24 Classical Eichler-Shimura Theory Eichler-Shimura-Manin, Kohnen-Zagier Eichler-Shimura Isomorphism Remark The map r : f r(f ; z) defines a homomorphism r : S k W.
25 Classical Eichler-Shimura Theory Eichler-Shimura-Manin, Kohnen-Zagier Eichler-Shimura Isomorphism Remark The map r : f r(f ; z) defines a homomorphism r : S k W. Theorem (Eichler-Shimura) If W 0 W is the codim. 1 space not containing z k 2 1, then is an isomorphism. r : S k W 0
26 Classical Eichler-Shimura Theory Eichler-Shimura-Manin, Kohnen-Zagier A larger theory?
27 Classical Eichler-Shimura Theory Eichler-Shimura-Manin, Kohnen-Zagier A larger theory? Question Is this all part of a larger theory?
28 Harmonic Maass forms Harmonic Maass Forms
29 Harmonic Maass forms Harmonic Maass Forms The theory of harmonic Maass forms has many applications.
30 Harmonic Maass forms Harmonic Maass Forms The theory of harmonic Maass forms has many applications. Partitions and q-series identities Moonshine for affine Lie superalgebras Borcherds products Donaldson invariants (Moore-Witten Conjecture)
31 Harmonic Maass forms Definition Basic Definitions
32 Harmonic Maass forms Definition Basic Definitions Hyperbolic Laplacian: ( ) ( ) k := y 2 2 x y 2 + iky x + i. y
33 Harmonic Maass forms Definition Harmonic weak Maass forms Definition A harmonic Maass form is any smooth function F on H satisfying:
34 Harmonic Maass forms Definition Harmonic weak Maass forms Definition A harmonic Maass form is any smooth function F on H satisfying: 1 For all ( a b c d ) SL2 (Z), we have F ( ) az + b = (cz + d) k F (z). cz + d
35 Harmonic Maass forms Definition Harmonic weak Maass forms Definition A harmonic Maass form is any smooth function F on H satisfying: 1 For all ( a b c d ) SL2 (Z), we have F ( ) az + b = (cz + d) k F (z). cz + d 2 We have that k F = 0.
36 Harmonic Maass forms Definition Harmonic weak Maass forms Definition A harmonic Maass form is any smooth function F on H satisfying: 1 For all ( a b c d ) SL2 (Z), we have F ( ) az + b = (cz + d) k F (z). cz + d 2 We have that k F = 0. Notation The space of weight k harmonic Maass forms is denoted H k.
37 Harmonic Maass forms Fourier expansions Fourier expansions
38 Harmonic Maass forms Fourier expansions Fourier expansions Lemma (Bruinier, Funke) If F H 2 k and Γ(a, x) is the incomplete Γ-function, then F (z) = c + F (n)qn + c F (n)γ(k 1, 4π n y)qn. n n<0 Holomorphic part F + Nonholomorphic part F
39 Harmonic Maass forms Fourier expansions Fourier expansions Lemma (Bruinier, Funke) If F H 2 k and Γ(a, x) is the incomplete Γ-function, then F (z) = c + F (n)qn + c F (n)γ(k 1, 4π n y)qn. n n<0 Holomorphic part F + Nonholomorphic part F Remark The function F + is called a mock modular form.
40 Harmonic Maass forms Differential Operators Relation to classical modular forms
41 Harmonic Maass forms Differential Operators Relation to classical modular forms M! k := weight k weakly holomorphic modular forms.
42 Harmonic Maass forms Differential Operators Relation to classical modular forms M! k := weight k weakly holomorphic modular forms. S k! := weight k weakly holomorphic cusp forms (subspace of M k! whose forms have vanishing constant terms.)
43 Harmonic Maass forms Differential Operators Relation to classical modular forms M! k := weight k weakly holomorphic modular forms. S k! := weight k weakly holomorphic cusp forms (subspace of M k! whose forms have vanishing constant terms.) Lemma If D := 1 2πi d dz, then D k 1 : M! 2 k S! k (Bol),
44 Harmonic Maass forms Differential Operators Relation to classical modular forms M! k := weight k weakly holomorphic modular forms. S k! := weight k weakly holomorphic cusp forms (subspace of M k! whose forms have vanishing constant terms.) Lemma If D := 1 2πi d dz, then D k 1 : M! 2 k S! k D k 1 : H 2 k S! k (Bol), (Bruinier, Ono, Rhoades).
45 Harmonic Maass forms Differential Operators Relation to classical modular forms
46 Harmonic Maass forms Differential Operators Relation to classical modular forms Lemma (Bruinier, Funke) If ξ w := 2iy w z, then ξ 2 k : H 2 k S k.
47 Harmonic Maass forms Differential Operators Relation to classical modular forms Lemma (Bruinier, Funke) If ξ w := 2iy w z, then ξ 2 k : H 2 k S k. Remark The cusp form ξ 2 k (F ) is called the shadow of F +.
48 Extended Eichler-Shimura Theory Period polynomials Period polynomials revisited Definition For f S k, the period polynomial is given by r(f ; z) := i 0 f (τ)(z τ) k 2 dτ,
49 Extended Eichler-Shimura Theory Period polynomials Period polynomials revisited Definition For f S k, the period polynomial is given by r(f ; z) := i 0 f (τ)(z τ) k 2 dτ, Remark For f S k!, the above integral may be divergent.
50 Extended Eichler-Shimura Theory Period polynomials The Regularized integral
51 Extended Eichler-Shimura Theory Period polynomials The Regularized integral Consider a continuous function f : H C and for some c R + as Im z. f (z) = O(e c Im z )
52 Extended Eichler-Shimura Theory Period polynomials The Regularized integral Consider a continuous function f : H C and for some c R + as Im z. f (z) = O(e c Im z ) Definition (Fricke, Rankin-Selberg) For t 0, if i i e itz f (z) dz has an analytic continuation to t = 0, then we define R. i i f (z) dz := [ i i ] e itz f (z) dz. t=0
53 Extended Eichler-Shimura Theory Generalized Period Polynomials More period polynomials
54 Extended Eichler-Shimura Theory Generalized Period Polynomials More period polynomials If f S k, then r(f ; z) := i 0 f (τ)(z τ) k 2 dτ.
55 Extended Eichler-Shimura Theory Generalized Period Polynomials More period polynomials If f S k, then r(f ; z) := i 0 f (τ)(z τ) k 2 dτ. Definition For f S k!, define r(f ; z) := R. i 0 f (τ)(z τ) k 2 dτ.
56 Extended Eichler-Shimura Theory Generalized Period Polynomials Mock modular forms and Eichler integrals
57 Extended Eichler-Shimura Theory Generalized Period Polynomials Mock modular forms and Eichler integrals Remark Recall the extended Bol-type identity: D k 1 :H 2 k S! k (Bruinier, Ono, Rhoades).
58 Extended Eichler-Shimura Theory Generalized Period Polynomials Mock modular forms and Eichler integrals Remark Recall the extended Bol-type identity: D k 1 :H 2 k S! k (Bruinier, Ono, Rhoades). Remark Mock modular forms are regularized iterated integrals of weakly holomorphic cusp forms.
59 Extended Eichler-Shimura Theory Generalized Period Polynomials Mock modular periods generate critical L-values
60 Extended Eichler-Shimura Theory Generalized Period Polynomials Mock modular periods generate critical L-values Theorem (Bringmann, Guerzhoy, Kent, Ono) If F H 2 k and g = ξ 2 k (F ) S k, then k 2 F + (z) z k 2 F + n L(g, n + 1) ( 1/z) = ( 1) (k 2 n)! (2πiz)k 2 n. n=0
61 Extended Eichler-Shimura Theory Generalized Eichler-Shimura Original Eichler-Shimura Isomorphism
62 Extended Eichler-Shimura Theory Generalized Eichler-Shimura Original Eichler-Shimura Isomorphism Theorem (Eichler-Shimura) If W 0 W is the codim. 1 space not containing z k 2 1, then r : S k W 0 is an isomorphism.
63 Extended Eichler-Shimura Theory Generalized Eichler-Shimura New Eichler-Shimura isomorphisms
64 Extended Eichler-Shimura Theory Generalized Eichler-Shimura New Eichler-Shimura isomorphisms The following diagram is commutative: 0 0 S k S k r +ir + W 0 0 D k 1 (S! 2 k ) S! k = r W 0 0 D k 1 (S 2 k! ) D k 1 (M 2 k! ) r z k
65 Extended Eichler-Shimura Theory Hecke theory More eigenforms
66 Extended Eichler-Shimura Theory Hecke theory More eigenforms Remark The isomorphism S! k /Dk 1 (M! 2 k ) = W 0 suggests that there are more eigenforms than just those in S k.
67 Extended Eichler-Shimura Theory Hecke theory More eigenforms Remark The isomorphism S! k /Dk 1 (M! 2 k ) = W 0 suggests that there are more eigenforms than just those in S k. Definition For any positive integer m 2, let T (m) be the usual weight k index m Hecke operator.
68 Extended Eichler-Shimura Theory Hecke theory More eigenforms
69 Extended Eichler-Shimura Theory Hecke theory More eigenforms Definition We say f S k! is a Hecke eigenform if for every Hecke operator T (m) there is a complex number λ m for which ( ) (f k T (m) λ m f ) (z) D k 1 M 2 k!.
70 Extended Eichler-Shimura Theory Hecke theory More eigenforms Definition We say f S k! is a Hecke eigenform if for every Hecke operator T (m) there is a complex number λ m for which ( ) (f k T (m) λ m f ) (z) D k 1 M 2 k!. Remark This extends the usual definition of Hecke eigenform for S k.
71 Extended Eichler-Shimura Theory Hecke theory Multiplicity two theorem Theorem (Bringmann, Guerzhoy, Kent, Ono) Let d = dim S k. Then S! k /Dk 1 (M! 2 k ) = d i=1 T i where each T i is a 2-dimensional Hecke eigenspace.
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