Arithmetic properties of harmonic weak Maass forms for some small half integral weights

Transcription

1 Arithmetic properties of harmonic weak Maass forms for some small half integral weights Soon-Yi Kang (Joint work with Jeon and Kim) Kangwon National University Pure and Applied Number Theory School NIMS, Daejeon Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 1 / 38

2 Table of contents 1 Introduction Classical meromorphic modular form Half integral weight modular form 2 Zagier Lift Traces of Singular Moduli Harmonic Weak Maass form Mock Modular forms and shadows Bases for H! 1/2 and H! 3/2 Cycle integral 3 Construction of Weak Maass Forms Poincaré series Bases for H! for small λ λ+ 1 2 generators of H! 2 k for k = λ Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 2 / 38

3 Classical meromorphic modular form H := {τ C Im (τ) > 0} the upper half of the complex plane. Γ := SL 2 (Z). Definition Let k Z. A meromorphic modular form of weight k for Γ is a meromorphic function on H satisfying 1 For all γ = ( a b c d ) Γ and τ H, we have f (γτ) = f ( aτ + b cτ + d ) = (cτ + d)k f (τ). 2 f is meromorphic at the cusp ; has Fourier expansion f (τ) = a(n)q n which converges for Im (τ) > 0. n Here, q = e 2πiτ. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 3 / 38

4 Classical meromorphic modular form If f is holomorphic in H and meromorphic at, then f is a weakly holomorphic modular form. If f is holomorphic in H and at, then f is (holomorphic) modular form; f has the Fourier expansion f (τ) = a(n)q n. n=0 If f is a modular form and a(0) = 0, then f is a cusp form. Notation M k! := vector space of all weakly holomorphic modular forms of weight k. M k M k! := subspace of holomorphic forms. S k M k := subspace of cusp forms. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 4 / 38

5 Classical meromorphic modular form If f is holomorphic in H and meromorphic at, then f is a weakly holomorphic modular form. If f is holomorphic in H and at, then f is (holomorphic) modular form; f has the Fourier expansion f (τ) = a(n)q n. n=0 If f is a modular form and a(0) = 0, then f is a cusp form. Notation M k! := vector space of all weakly holomorphic modular forms of weight k. M k M k! := subspace of holomorphic forms. S k M k := subspace of cusp forms. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 4 / 38

6 Classical meromorphic modular form Example 1 k:odd, 1 = ( 1 0 Hence M! k = {0}. f 0 1 ) Γ ( ) τ = f (τ) = ( 1) k f (τ) f (τ) = k > 2 even. Eisenstein series of weight k: G k (τ) = 1 (mτ + n) k M k m,n Normalized Eisenstein series E k (τ) = 1 2ζ(k) G k(τ) = 1 2k σ k 1 (n)q n M k. B k 3 The discriminant modular form: n 1 0 (τ) = E 3 4 E = q n 1(1 q n ) 24 = q 24q 4 + S 12. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 5 / 38

7 Classical meromorphic modular form Example 1 k:odd, 1 = ( 1 0 Hence M! k = {0}. f 0 1 ) Γ ( ) τ = f (τ) = ( 1) k f (τ) f (τ) = k > 2 even. Eisenstein series of weight k: G k (τ) = 1 (mτ + n) k M k m,n Normalized Eisenstein series E k (τ) = 1 2ζ(k) G k(τ) = 1 2k σ k 1 (n)q n M k. B k 3 The discriminant modular form: n 1 0 (τ) = E 3 4 E = q n 1(1 q n ) 24 = q 24q 4 + S 12. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 5 / 38

8 Classical meromorphic modular form Example 1 k:odd, 1 = ( 1 0 Hence M! k = {0}. f 0 1 ) Γ ( ) τ = f (τ) = ( 1) k f (τ) f (τ) = k > 2 even. Eisenstein series of weight k: G k (τ) = 1 (mτ + n) k M k m,n Normalized Eisenstein series E k (τ) = 1 2ζ(k) G k(τ) = 1 2k σ k 1 (n)q n M k. B k 3 The discriminant modular form: n 1 0 (τ) = E 3 4 E = q n 1(1 q n ) 24 = q 24q 4 + S 12. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 5 / 38

9 Classical meromorphic modular form M = j Z M j = C[E 4, E 6 ]. j-invariant: j(τ) = E 3 4 (τ) (τ) = q q q 2 + M! 0 M! 0 = C[j]. j 1 := J := j 744 = q 1 + O(q) j 2 = q 2 + O(q), j 3 = q 3 + O(q),..., j m = q m + O(q) form a unique basis of M! 0. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 6 / 38

10 Classical meromorphic modular form M = j Z M j = C[E 4, E 6 ]. j-invariant: j(τ) = E 3 4 (τ) (τ) = q q q 2 + M! 0 M! 0 = C[j]. j 1 := J := j 744 = q 1 + O(q) j 2 = q 2 + O(q), j 3 = q 3 + O(q),..., j m = q m + O(q) form a unique basis of M! 0. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 6 / 38

11 Classical meromorphic modular form M = j Z M j = C[E 4, E 6 ]. j-invariant: j(τ) = E 3 4 (τ) (τ) = q q q 2 + M! 0 M! 0 = C[j]. j 1 := J := j 744 = q 1 + O(q) j 2 = q 2 + O(q), j 3 = q 3 + O(q),..., j m = q m + O(q) form a unique basis of M! 0. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 6 / 38

12 Half integral weight modular form Let k = λ + 1/2 for an integer λ. For γ = ( ) a b c d Γ0 (4), ( c ) ( ) 1 1 j(γ, τ) := cτ + d, d d the automorphy factor for the classical theta function θ(τ) = n Z q n2. Definition A weakly holomorphic modular form of weight k for Γ 0 (4) is a holomorphic function on H that satisfies f (γτ) = j(γ, τ) 2k f (τ) for all γ Γ 0 (4) and that may have poles at the cusps, and have a q-expansion supported on integers n with ( 1) λ n 0, 1 (mod 4). Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 7 / 38

13 Borcherds-Zagier bases for M! 1/2 and M! 3/2 The basis {f d } d 0 for M! 1/2 : f 0 = 1 + 2q + 2q 4 + 0q 5 + 0q f 3 = q 3 248q q q q 8... f 4 = q q q q q f 7 = q q q q q The basis {g D } D>0 for M! 3/2 : g 1 = q q 3 492q q 7... g 4 = q q q q 7... g 5 = q q q q 7... g 8 = q q q q Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 8 / 38

14 Singular Moduli Let d be an integer with d 0, 1 (mod 4). Q d := {Q(x, y) = [A, B, C] Ax 2 + Bxy + Cy 2, B 2 4AC = d}. Assume d < 0. Then the group Γ acts on Q d with finitely many orbits. Let α Q be the CM point associated to Q, that is, the unique root of Q(x, 1) = 0 in H. According to the theory of complex multiplication, j(α Q ) is an algebraic integer, singular moduli. If d is fundamental and K = Q( d), then for each Q Q d /Γ, j(α Q ) generates the Hilbert class field of K with degree [K(j(α Q )) : K] = h(d). Q Q d /Γ (X j(α Q)) is its minimal polynomial. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral weights 9 / 38

15 Twisted Trace of Singular Moduli Let d, D 0, 1 (mod 4) with D fundamental. If w Q is defined by 2, if Q Γ [a, 0, a]; w Q = 3, if Q Γ [a, a, a]; 1, otherwise, then the Hurwitz-Kronecker class number is given by H(d) = 1. w Q Q Q d /Γ For each Q = [a, b, c] Q dd, let the genus character χ be given by { ( D χ(q) := χ D (Q) = r ), if (a, b, c, D) = 1 and Q represents r, (r, D) = 1 0, if (a, b, c, D) > 1. For dd < 0, define the twisted trace of singular moduli for a modular function f by Tr d,d (f ) := 1 χ d (Q)f (α Q ). D w Q Q Q dd /Γ Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 10 weights / 38

16 Twisted Trace of Singular Moduli Let d, D 0, 1 (mod 4) with D fundamental. If w Q is defined by 2, if Q Γ [a, 0, a]; w Q = 3, if Q Γ [a, a, a]; 1, otherwise, then the Hurwitz-Kronecker class number is given by H(d) = 1. w Q Q Q d /Γ For each Q = [a, b, c] Q dd, let the genus character χ be given by { ( D χ(q) := χ D (Q) = r ), if (a, b, c, D) = 1 and Q represents r, (r, D) = 1 0, if (a, b, c, D) > 1. For dd < 0, define the twisted trace of singular moduli for a modular function f by Tr d,d (f ) := 1 χ d (Q)f (α Q ). D w Q Q Q dd /Γ Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 10 weights / 38

17 Zagier s basis for M! 3/2 Theorem (Zagier, 2002) Let D, d 0, 1 (mod 4) with (D, d) = 1. If D is a fundamental discriminant and D > 0, d < 0, then g D (τ) := q D 2δ D, d<0 Tr d,d (J)q d M! 3/2. The space of weakly holomorphic modular forms of weight 3/2, M! 3/2 is generated by g 1 = q q 3 492q q 7... g 4 = q q q q 7... g 5 = q q q q 7. g 8 = q q q q 7. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 11 weights / 38

18 Borcherds basis for M! 1/2 The space M 1/2! is generated by f 0 = 1 + 2q + 2q 4 + 0q 5 + 0q 8 + f 3 = q 3 248q q q f 4 = q q q q f 7 = q q q q Theorem (Zagier, 2002) Let D, d 0, 1 (mod 4) with (D, d) = 1. If D is a fundamental discriminant and D > 0, d < 0, then g D (τ) := q D 2δ D, d<0 Tr d,d (J)q d M! 3/2. f d (τ) := q d + D>0 Tr d,d (J)q D M! 1/2. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 12 weights / 38

19 Zagier lift For D > 0 and d < 0, Z d (J) = f d(τ) = q d + D;dD<0 Z D (J) = g D(τ) = q D 2δ D, Tr d,d (J)q D M! 1/2 d;dd<0 Tr d,d (J)q d M! 3/2. Generalization 1 Bruinier and Funke (2006) showed that a weakly holomorphic modular function of arbitrary level is lifted to a harmonic weak Maass form of weight 3/2. Bringmann and Ono (2007) lifted a non-holomorphic Poincaré series of weight 0 to a half- integral weight Poincaré series More generalization of Zagier-lift to a weakly holomorphic or non-holomorphic modular form of weight 0 Bringmann, Bruinier, D. Choi, Duke, Imamoḡlu, Jenkins, D. Jeon, S.-Y. K. C. H. Kim, Miller, Ono, Pixton, Rouse, Tóth and so on. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 13 weights / 38

20 Zagier lift For D > 0 and d < 0, Z d (J) = f d(τ) = q d + D;dD<0 Z D (J) = g D(τ) = q D 2δ D, Tr d,d (J)q D M! 1/2 d;dd<0 Tr d,d (J)q d M! 3/2. Generalization 1 Bruinier and Funke (2006) showed that a weakly holomorphic modular function of arbitrary level is lifted to a harmonic weak Maass form of weight 3/2. Bringmann and Ono (2007) lifted a non-holomorphic Poincaré series of weight 0 to a half- integral weight Poincaré series More generalization of Zagier-lift to a weakly holomorphic or non-holomorphic modular form of weight 0 Bringmann, Bruinier, D. Choi, Duke, Imamoḡlu, Jenkins, D. Jeon, S.-Y. K. C. H. Kim, Miller, Ono, Pixton, Rouse, Tóth and so on. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 13 weights / 38

21 Harmonic Weak Maass Form Definition A weak Maass form of weight k for Γ is any smooth function F : H C satisfying: ) Γ and τ H, we have 1 For all γ = ( a b c d F (γτ) = { (cτ + d) k F (τ) if k Z, ( c 2kɛ 2k d) d (cτ + d) k F (τ) if k 1 2 Z \ Z. ( ) ( 2 k F := [ y iky x 2 y 2 x + i y ) ]F = sf. 3 The function F (τ) has at most linear exponential growth at all cusps. A weak Maass form F is called harmonic if s = 0. H k := the space of harmonic weak Maass forms of weight k. S k M k M! k H k. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 14 weights / 38

22 Harmonic Weak Maass form Example Recall Hurwitz-Kronecker class number H(d) = Set H(0) = Zagier s weight 3/2 Eisenstein series E(τ) = d 0 H(d)q d + Q Q d /Γ 1 16π y 1 w Q. n= is a harmonic weak Maass form of weight 3/2. β(4πn 2 y)q n2 If ξ k := 2iy k τ, then ξ 3/2(E(τ)) = 1 16π θ(τ). Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 15 weights / 38

23 Mock Modular Forms and Shadows If F is a harmonic weak Maass form, then F = F h + F nh. Mock modular form F h (z) = n Shadow of the mock modular form F h a(n)q n, q := e 2πiτ. ξ k (F ) = ξ k (F nh ), where ξ k := 2iy k τ. Theorem (Bruinier-Funke, 2004) ξ k : H k M! 2 k. M k := the space of mock modular forms of weight k. H k M k Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 16 weights / 38

24 Fourier expansion of a harmonic weak Maass form h(z) = c + h (n)qn +c h (0)y 1 k + c h (n)γ(1 k, 4πny)qn H k n 0 n ξ k (h) = (1 k)c h (0) c h (n)( 4πn)1 k q n. 0 n k h = ( ξ 2 k ξ k )(h) = 0. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 17 weights / 38

25 Generalization of Zagier lift Generalization 2 Duke and Jenkins (2008) proved for f M 2 2ν!, ν 2 an integer and d, D fundamental discriminants with dd < 0 that Z d (f ) M! 3/2 ν, if ( 1)ν d > 0, Z D (f ) M! ν+1/2, if ( 1)ν D < 0. The original Zagier lift is the case when ν = 1. Z D (( 1)ν D < 0) M! ν+ 1 2,d M 2 2ν! (Dd < 0) 3 Z d (( 1)ν d > 0) M! 3 2 ν,d Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 18 weights / 38

26 Basis for H! 1/2 Duke, Imamoḡlu and Tóth (2011) extended the basis {f d } d 0 for M! 1 to a 2 basis {h d d 0, 1 (mod 4)} for H! 1 : 2 { fd, if d 0, h d = D>0 a(d, d)qd + 2 dgd, if d > 0, where ξ 1/2 (h d ) = 2 dg d. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 19 weights / 38

27 Basis for H! 3/2 Jeon, K., Kim (2013) extended the basis {g D } D>0 for M! 3 2 {k D D 0, 1 (mod 4)} for H! 3 : 2 k D = where ξ 3/2 (k D ) = 1 2 πd f D. g D, if D > 0, d 0 b(d, d)q d f πd D, if D < 0, d 0 b(0, d)q d + 3, if D = 0, 4π f 0 to a basis Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 20 weights / 38

28 Real quadratic analogue of trace of singular moduli Let d > 0 be not a square. For Q = [A, B, C] Q d, define the oriented semi-circle S Q : A τ 2 + BReτ + C = 0, directed counterclockwise if A > 0 and clockwise if A < 0. The group Γ acts on Q d. Γ Q = {γ Γ γq = Q}, the group of automorphs of Q is infinite cyclic. dz Q(τ, 1) = 2 log ɛ d, d Γ Q \S Q where ɛ d is the smallest unit (> 1 of norm 1) of Q( d). For each Γ-invariant function f, the cycle integral defined by dτ f (τ) Γ Q \S Q Q(τ, 1) is also a class invariant. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 21 weights / 38

29 Zagier lift via cycle integral Let D, d 0, 1 (mod 4) with (D, d) = 1. Also, suppose D is a fundamental discriminant with Dd > 0 not a square. Define a general twisted trace by Tr d,d (f ) = dτ f (τ) Γ Q \S Q Q(τ, 1). Q Γ\Q dd χ(q) Theorem (Duke-Imamoḡlu-Tóth (2011)) For d > 0, Tr D,d (J)q D D>0 is a mock modular form of weight 1/2 with shadow a constant multiple of g d M! 3/2. In fact, Z + d (J) = h d(τ) = D>0 Tr D,d (J)q D + 2 dg d H! 1/2,D. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 22 weights / 38

30 Zagier lift via cycle integral Theorem (Jeon-K.-Kim (2014)) For D < 0, if g D (τ) = b(d, d)q d d<0 is a mock modular form of weight 3/2 with shadow a constant multiple of f D M! 1/2, then b(d, d) = 192πH( d )H( D ) 8 ( ) dd Tr d,d J(τ) ˆ. Here ˆ J(τ) is a sesqui-harmonic Maass form satisfying 0 ( ˆ J) = j 24. Z + D (J) = k D(τ) = D>0 {192πH( d )H( D ) 8 dd Tr d,d (J(τ))} πd f D H! 3/2,d. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 23 weights / 38

31 Zagier-lift for harmonic weak Maass forms Jeon-K.-Kim (?) Z + D (( 1)ν D > 0) H! ν+ 1 2,d H 2 2ν! (Dd > 0) 3 Z + d (( 1)ν d < 0) H! 3 2 ν,d Duke-Jenkins (2008) Z D (( 1)ν D < 0) M! ν+ 1 2,d M 2 2ν! (Dd < 0) 3 Z d (( 1)ν d > 0) M! 3 2 ν,d Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 24 weights / 38

32 Zagier-lift for harmonic weak Maass forms Jeon-K.-Kim (?) Z + D (( 1)ν D > 0) H! ν+ 1 2,d H 2 2ν! (Dd > 0) 3 Z + d (( 1)ν d < 0) H! 3 2 ν,d Duke-Jenkins (2008) Z D (( 1)ν D < 0) M! ν+ 1 2,d M 2 2ν! (Dd < 0) 3 Z d (( 1)ν d > 0) M! 3 2 ν,d Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 24 weights / 38

33 Zagier-lift for harmonic weak Maass forms Jeon-K.-Kim (?) Z + D (( 1)ν D > 0) H! ν+ 1 2,d H 2 2ν! (Dd > 0) 3 Z + d (( 1)ν d < 0) H! 3 2 ν,d Bringmann-Guerzhoy-Kane (2014) 2 2ν (Dd < 0) H cusp Z D (( 1)ν D < 0) H cusp ν+ 1 2,d 3 Z d (( 1)ν d > 0) H cusp 3 2 ν,d Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 25 weights / 38

34 Zagier-lift for harmonic weak Maass forms Jeon-K.-Kim (?) Z + D (( 1)ν D > 0) H! ν+ 1 2,d H 2 2ν! (Dd > 0) 3 Z + d (( 1)ν d < 0) H! 3 2 ν,d Bringmann-Guerzhoy-Kane (2014) 2 2ν (Dd < 0) H cusp Z D (( 1)ν D < 0) H cusp ν+ 1 2,d 3 Z d (( 1)ν d > 0) H cusp 3 2 ν,d Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 25 weights / 38

35 Poincaré series For γ = ( a b c d where ε v = ) Γ, we define { cτ + d, if k Z, j(γ, τ) := ( c d )ε 1 d cτ + d, if k Z, { 1, if v 1 (mod 4), i, if v 3 (mod 4). For any complex-valued function f defined on H, the weight k slash operator is defined by f k γ(τ) := j(γ, τ) 2k f (γτ). Let φ : R + C a smooth function satisfying φ(y) = O(y 1+ε ) ε > 0 Γ : the subgroup of translations of Γ. Set φ m (τ) := e 2πimx φ(y). The general Poincaré series P m,k (τ, φ) = φ m k γ(τ) γ Γ \Γ is a smooth function on H satisfying the weight k modularity: Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 26 weights / 38

36 Harmonic Weak Maass Poincaré Series M µ,ν (τ) and W µ,ν (τ) be the usual Whittaker functions. We define (4π n y) k/2 M k2 sgn(n),s 1/2 (4π n y) M n (k, y, s) = Γ(2s), if n 0; y s k/2, if n = 0, Let W n (k, y, s) = where e(x) := exp(2πix). n k/2 1 (4πy) k/2 W k2 sgn(n),s 1/2 (4π n y) Γ(s+ k 2 sgn(n)), if n 0; (4π) 1 k y 1 s k/2 (2s 1)Γ(s k/2)γ(s+k/2), if n = 0. ϕ m,k (s, τ) := M m (k, y, s)e(mx), Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 27 weights / 38

37 For m Z, k ϕ m,k (s, τ) = ( ) s(1 s) + k2 2k ϕ m (s, τ). 4 Lemma Let ϕ m,k (s, τ) = O(y Re(s) k/2 ) as y 0 F m (k, τ; s) := P m,k (τ, ϕ m,k (s, τ)). The Poincaré series F m (k, τ; s) converges absolutely and uniformly on compacta for Re(s) > 1, and it defines a Γ-invariant eigenfunction of the Laplacian k satisfying ( k F m (k, τ, s) = s k ) (1 k2 ) 2 s F m (k, τ; s). Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 28 weights / 38

38 Theorem If m is an integer, then the Poincaré series F m (k, τ; s) has the Fourier expansion F m (k, τ; s) = M m (k, y, s)e(mx) + n Z c m,n (k, s)w n (k, y, s)e(nx), where the coefficients c m,n (k, s) are given by ( mn 1 k 2 4π ) mn J 2s 1, mn > 0; c c ( 2πi mn 1 k 2 4π ) mn I 2s 1, mn < 0; k K k (m, n, c) c c c>0 2 k 1 π s+ k 2 1 m + n s k 2 c 0(4) c 2s 1, mn = 0, m + n 0; 2 2k 2 π k 1 Γ(2s) (2c) 2s 1, m = n = 0, Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 29 weights / 38

39 K k (m, n, c) := ( c v v(c) ) 2k ɛ 2k v e ( m v + nv where the sum runs through the primitive residue classes modulo c and v v 1 (mod c). For any m, ( k F m (k, τ; s) = s k ) (1 k2 ) 2 s F m (k, τ; s). F m (k, τ; s) has an analytic continuation in s to Re(s) > 1/2 except for possibly finitely many simple poles in ( 1 2, 1). If k 1/2, then F m (k, τ; s) is holomorphic in s near s = 1 k/2. If k 3/2, then F m (k, τ; s) is holomorphic in s near s = k/2. c ), Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 30 weights / 38

40 Weak Maass Poincaré Series of half integral weight We now construct a Maass Poincaré series satisfying Kohnen s plus space condition: Fourier coefficients at are supported on integers n with ( 1) k 1/2 n 0, 1 (mod 4). Poincaré series in the plus space of weight k and index m for Γ 0 (4) is given by F + m (k, τ; s) = F m (k, τ; s) k pr, where k pr is Kohnen s projection operator. Theorem For any ( 1) k 1/2 m 0, 1 (mod 4) and Re(s) > 1 the function F m + (k, τ; s) has weight k and satisfies ( k F m + (k, τ; s) = s k ) (1 k2 ) 2 s F m + (k, τ; s). Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 31 weights / 38

41 Theorem For any ( 1) k 1/2 m 0, 1 (mod 4) and Re(s) > 1, the function F + m (k, τ; s) has Fourier expansion F m + (k, τ; s) = M m (k, y, s)e(mx) + b m,n (k, s)w n (k, y, s)e(nx), ( 1) k 1/2 n 0,1(4) where the coefficients b m,n (k, s) are given by ( ( )) 4 2πi k 1 + K k (m, n, c) c/4 0<c 0(4) mn 1 k 2 c 1 J 2s 1 (4π mn c 1), mn > 0; mn 1 k 2 c 1 I 2s 1 (4π mn c 1), mn < 0; 2 k 1 π s+ k 2 1 m + n s k 2 c 1 2s, mn = 0, m + n 0; 2 2k 2 π k 1 Γ(2s)(2c) 1 2s, m = n = 0. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 32 weights / 38

42 Harmonic weak Maass Poincaré Series 1 F m + (k, τ; 1 k/2) is a harmonic weak Maass form if k < 0 and F m + (k, τ; k/2) is a weakly holomorphic modular form if k > 2. 2 F m + (1/2, τ; s) has a simple pole at s = 3/4. Using a linear combination of F m + (1/2, τ; s) and F 0 + (1/2, τ; s), Duke, Imamoḡlu and Tóth constructed the basis for H 1/2!. 3 One can find that the harmonic weak Maass form F m + (ν + 1 2, τ; ν ) H! is in fact a cusp form when m > 0. ν+ 1 2 Since the spaces of the cusp forms of weights 2ν are trivial precisely when ν = 1, 2, 3, 4, 5, 7, F m + (ν + 1 2, τ; ν ) = 0 for these values of ν by Shimura correspondence. However, there must be non-trivial weak harmonic Maass forms of these weights, because the map ξ ν+ 1 : H! M! 2 ν+ 1 3 is surjective as claimed By Bruinier-Funke ν 2 2 (2006) and M! 3 is a non-trivial space for each of these ν. When ν 2 ν = 1, non-trivial harmonic weak Maass forms of weight 3/2 were constructed by Jeon-K.-Kim (2013) using the derivatives of F m + (3/2, τ; s) at s = 3/4 which are the basis for H 3/2!. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 33 weights / 38

43 Theorem Let λ = 2, 3, 4, 5, 7 and m be an integer satisfying ( 1) λ m 0, 1 (mod 4). If we define a function h m,λ+ 1 by 2 h m,λ+ 1 (τ) := s F m + (λ + 1 2, τ; s) s= λ 2 + 1, m > 0, 4 2 F m + (λ τ; λ ), m 0, 4 then the set {h m,λ+ 1 } forms a basis for H!. When m 0, h 2 λ+ 1 m,λ+ 1 (τ) 2 2 forms a basis for M!. In particular, λ+ 1 2 h m,λ+ 1 (τ) = Γ(λ ) 1 f λ+ 1, m(τ). 2 When m > 0, h m,λ+ 1 (τ) is a unique weakly harmonic Maass form with 2 bounded holomorphic part satisfying ( ) ξ λ+ 1 h m,λ+ 1 (τ) = (4πm) 1/2 λ f 3 λ,m(τ) M! 3 λ Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 34 weights / 38

44 Duke-Jenkins basis for M! 1 2 +λ For any k Z, let 2k 1 = 12l + k with uniquely determined l Z and k = {0, 4, 6, 8, 10, 14}. Then l is the dimension of the space of cusp forms of weight 2k 1 (l 0). If A denotes the maximal order of a non-zero f M k! at i, then by the Shimura correspondence, { 2l ( 1) k 1/2 if l is odd, A = 2l otherwise. A unique basis for M! k then consists of functions of the form f k,r (τ) = q r + n>a a k (r, n)q n, where r A satisfies ( 1) k 3/2 r 0, 1 (mod 4). Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 35 weights / 38

45 Duke-Jenkins basis for M! 1 2 +λ For k = λ. Let 2λ = 12l + k with k = {0, 4, 6, 8, 10, 14}. Let 2 k = t + 1/2 and l be defined by 2t = 12l + k. k 2 k t l k A 5/2 1/ /2 3/ /2 5/ /2 7/ /2 11/ Using the table above, we compare ξ k (h m,k (τ)) with Duke-Jenkins basis and can conclude that (4πm) k 1 ξ k (h m,k (τ)) = f m,2 k, because the only holomorphic modular form of weight k satisfying the plus space condition is zero. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 36 weights / 38

46 H! 2 k for k = λ Let λ = 1, 2, 3, 4, 5, 7. Consider Poincaré series: P m,k (τ) = γ Γ \Γ 0 (4) q m k γ and the projection g m,k (τ) of P m,k (τ) to M! k. The Fourier expansion is g m,k (τ) = q m + O(q) by Duke-Jenkins (2008). Note ξ 2 k (F m (2 k, τ, 1 k/2)) = CP m,k (τ). For m Z satisfying ( 1) λ m 0, 1 (mod 4), we have ξ 2 k (F + m (2 k, τ, 1 k/2)) = Cg m,k. As g m,k and Duke-Jenkins basis f m,k differ by a cusp form of weight k satisfying the plus space condition, g m,k = f m,k precisely when k = λ when λ = 1, 2, 3, 4, 5, 7. Hence the Maass Poincaré series F m + (2 k, τ, 1 k/2) together with elements in the canonical basis of M 2 k! spans H! 2 k when 2 k { 11/2, 7/2, 5/2, 3/2, 1/2, 1/2}. Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 37 weights / 38

47 Thank You! Soon-Yi Kang (Joint work with Jeon and Kim) Arithmetic (KNU) properties of harmonic weak Maass forms for some small half integral 38 weights / 38