Class Number Type Relations for Fourier Coefficients of Mock Modular Forms

Transcription

1 Class Number Type Relations for Fourier Coefficients of Mock Modular Forms Michael H. Mertens University of Cologne Lille, March 6th, 2014 M.H. Mertens (University of Cologne) Class Number Type Relations / 37

2 The Hurwitz class number Recall: Class number of discriminant d with. h(d) = #Q d / SL 2 (Z) Q d = {( ) 2a b b 2c Z 2 2 a > 0, gcd(a, b, c) = 1, b 2 4ac = d } M.H. Mertens (University of Cologne) Class Number Type Relations / 37

3 The Hurwitz class number Recall: Class number of discriminant d h(d) = #Q d / SL 2 (Z) with Q d = {( ) 2a b b 2c Z 2 2 a > 0, gcd(a, b, c) = 1, b 2 4ac = d }. Hurwitz class number of discriminant d 1 12 if d = 0 h( d/f 2 ) H(d) = if d N w( d/f 2 ), f 2 d 0 if d / N 0 3 if n = 3 w(n) = 2 if n = 4 1 otherwise M.H. Mertens (University of Cologne) Class Number Type Relations / 37

4 Class number relations L. Kronecker (1860), A. Hurwitz (1884): H(4n s 2 ) + 2λ 1 (n) = 2σ 1 (n) s Z M.H. Mertens (University of Cologne) Class Number Type Relations / 37

5 Class number relations L. Kronecker (1860), A. Hurwitz (1884): H(4n s 2 ) + 2λ 1 (n) = 2σ 1 (n) s Z M. Eichler (1955): H(n s 2 ) + λ 1 (n) = 1 3 σ 1(n) s Z M.H. Mertens (University of Cologne) Class Number Type Relations / 37

6 Class number relations L. Kronecker (1860), A. Hurwitz (1884): H(4n s 2 ) + 2λ 1 (n) = 2σ 1 (n) s Z M. Eichler (1955): H(n s 2 ) + λ 1 (n) = 1 3 σ 1(n) s Z M. Eichler, A. Selberg (1955/56): (s 2 n)h(4n s 2 ) + 2λ 3 (n) = 0 s Z (s 4 3ns 2 + n 2 )H(4n s 2 ) + 2λ 5 (n) = 0 s Z... M.H. Mertens (University of Cologne) Class Number Type Relations / 37

7 1 Mock Modular Forms 2 Cohen s conjecture 3 Holomorphic Projection 4 Fourier Coefficients of Mock Modular Forms M.H. Mertens (University of Cologne) Class Number Type Relations / 37

8 Table of Contents 1 Mock Modular Forms 2 Cohen s conjecture 3 Holomorphic Projection 4 Fourier Coefficients of Mock Modular Forms M.H. Mertens (University of Cologne) Class Number Type Relations / 37

9 Harmonic Maaß forms Definition A smooth function f : H C is called a harmonic weak Maaß form of weight k 1 2Z, level N N, and character χ modulo N (with 4 N if k / Z) if it fulfills the following properties: 1 (f k γ)(τ) = χ(d)f(τ) for all γ = ( a b c d ) Γ0 (N), M.H. Mertens (University of Cologne) Class Number Type Relations / 37

10 Harmonic Maaß forms Definition A smooth function f : H C is called a harmonic weak Maaß form of weight k 1 2Z, level N N, and character χ modulo N (with 4 N if k / Z) if it fulfills the following properties: 1 (f k γ)(τ) = χ(d)f(τ) for all γ = ( ) a b c d Γ0 (N), [ ( ) ( )] 2 k f = y iky x 2 y 2 x + i y f 0, M.H. Mertens (University of Cologne) Class Number Type Relations / 37

11 Harmonic Maaß forms Definition A smooth function f : H C is called a harmonic weak Maaß form of weight k 1 2Z, level N N, and character χ modulo N (with 4 N if k / Z) if it fulfills the following properties: 1 (f k γ)(τ) = χ(d)f(τ) for all γ = ( ) a b c d Γ0 (N), [ ( ) ( )] 2 k f = y iky x 2 y 2 x + i y f 0, 3 f grows at most exponentially approaching the cusps of Γ 0 (N). M.H. Mertens (University of Cologne) Class Number Type Relations / 37

12 Harmonic Maaß forms Definition A smooth function f : H C is called a harmonic weak Maaß form of weight k 1 2Z, level N N, and character χ modulo N (with 4 N if k / Z) if it fulfills the following properties: 1 (f k γ)(τ) = χ(d)f(τ) for all γ = ( ) a b c d Γ0 (N), [ ( ) ( )] 2 k f = y iky x 2 y 2 x + i y f 0, 3 f grows at most exponentially approaching the cusps of Γ 0 (N). The vector space of harmonic weak Maaß forms of weight k, level N, and character χ is denoted by H k (N, χ). M.H. Mertens (University of Cologne) Class Number Type Relations / 37

13 Splitting Lemma Let f be a harmonic weak Maaß form of weight k 1. Then f has a canonical splitting into f(τ) = f + (τ) + (4πy)1 k k 1 c f (0) + f (τ), where for some m 0, n 0 Z we have the Fourier expansions f + (τ) = m=m 0 c + f (m)qm and f (τ) = c f (n)nk 1 Γ(1 k; 4πny)q n. n=n 0 n 0 M.H. Mertens (University of Cologne) Class Number Type Relations / 37

14 Mock modular forms Definition Let f be a harmonic weak Maaß form as above. f + (τ) is called the holomorphic part of f. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

15 Mock modular forms Definition Let f be a harmonic weak Maaß form as above. f + (τ) is called the holomorphic part of f. (4πy) 1 k k 1 c f (0) + f (τ) is called the non-holomorphic part of f. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

16 Mock modular forms Definition Let f be a harmonic weak Maaß form as above. f + (τ) is called the holomorphic part of f. (4πy) 1 k k 1 c f (0) + f (τ) is called the non-holomorphic part of f. A mock modular form of weight k 1 is the holomorphic part of a harmonic weak Maaß form of weight k. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

17 Mock modular forms Definition Let f be a harmonic weak Maaß form as above. f + (τ) is called the holomorphic part of f. (4πy) 1 k k 1 c f (0) + f (τ) is called the non-holomorphic part of f. A mock modular form of weight k 1 is the holomorphic part of a harmonic weak Maaß form of weight k. Let f be a mock modular form of weight k and g a modular form of weight l. Then f g is called a mixed mock modular form of weight (k, l). M.H. Mertens (University of Cologne) Class Number Type Relations / 37

18 Mock modular forms Definition Let f be a harmonic weak Maaß form as above. f + (τ) is called the holomorphic part of f. (4πy) 1 k k 1 c f (0) + f (τ) is called the non-holomorphic part of f. A mock modular form of weight k 1 is the holomorphic part of a harmonic weak Maaß form of weight k. Let f be a mock modular form of weight k and g a modular form of weight l. Then f g is called a mixed mock modular form of weight (k, l). The νth Rankin-Cohen bracket of f and g is called a mixed mock modular form of degree ν. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

19 The shadow Proposition For k 1, the mapping ξ k : H k (N, χ) M 2 k! (N, χ), f 2iyk τ f is welldefined and surjective with kernel M k! (N, χ). Moreover, for f H k (N, χ), we have that (ξ k f)(τ) = (4π) 1 k n=n 0 c f (n)qn. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

20 The shadow Definition and Notation (ξ k f)(τ) is called the shadow of f (resp. f + ). M.H. Mertens (University of Cologne) Class Number Type Relations / 37

21 The shadow Definition and Notation (ξ k f)(τ) is called the shadow of f (resp. f + ). A mock theta function is the holomorphic part of a harmonic weak Maaß form of weight 1 2 whose shadow is a linear combination of weight 3 2 unary theta functions. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

22 The shadow Definition and Notation (ξ k f)(τ) is called the shadow of f (resp. f + ). A mock theta function is the holomorphic part of a harmonic weak Maaß form of weight 1 2 whose shadow is a linear combination of weight 3 2 unary theta functions. M k (Γ, χ) = ξ 1 k (M 2 k(γ, χ)) S k (Γ, χ) = ξ 1 k (S 2 k(γ, χ)). M.H. Mertens (University of Cologne) Class Number Type Relations / 37

23 An example Theorem (D. Zagier, 1976) Let H (τ) = H(n)q n and R(τ) := 1 + i 16π n=0 i τ ϑ(z) (z + τ) 3 2 dz = 1 8π y π n=1 ( nγ 1 ) 2 ; 4πn2 y q n2. Then the function Ĥ = H + R is a harmonic Maaß form of weight 3 2 on Γ 0 (4) with shadow ξĥ = 1 8 ϑ. In particular, H is a mock modular π form of weight 3 2. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

24 Appell-Lerch sums Definition Let τ H, u, v C \ (Z Zτ). The Appell-Lerch sum of level 1 is then the expression A 1 (u, v; τ) = e πiu n Z q n(n+1) 2 e 2πinv 1 e 2πiu q n. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

25 Appell-Lerch sums Definition Let τ H, u, v C \ (Z Zτ). The Appell-Lerch sum of level 1 is then the expression A 1 (u, v; τ) = e πiu n Z q n(n+1) 2 e 2πinv 1 e 2πiu q n. Zweger s thesis: Appell-Lerch sums are one of three ways to realize Ramanujan s mock theta functions M.H. Mertens (University of Cologne) Class Number Type Relations / 37

26 Appell-Lerch sums Definition Let τ H, u, v C \ (Z Zτ). The Appell-Lerch sum of level 1 is then the expression A 1 (u, v; τ) = e πiu n Z q n(n+1) 2 e 2πinv 1 e 2πiu q n. Zweger s thesis: Appell-Lerch sums are one of three ways to realize Ramanujan s mock theta functions the others: indefinite theta functions and Fourier coefficients of meromorphic Jacobi forms M.H. Mertens (University of Cologne) Class Number Type Relations / 37

27 Appell-Lerch sums Definition The completion of the Appell-Lerch sum A 1 (u, v; τ) is given by with R(u; τ) := Â 1 (u, v; τ) = A 1 (u, v; τ) + i Θ(v; τ)r(u v; τ), 2 ν 1 2 +Z t E(t) := 2 0 { (( sgn(ν) E ν + Im u ) )} 2y y ( 1) ν 1 2 q ν2 2 e 2πiνu e πu2 du, Θ(v; τ) := ν q 2 2 e 2πiν(v+ 1 2 ) ν 1 2 +Z M.H. Mertens (University of Cologne) Class Number Type Relations / 37

28 A real-analytic Jacobi form Theorem (S. Zwegers, 2002) Â 1 transforms like a Jacobi form of weight 1 and index 1 Â 1 ( u, v) = Â1(u, v). ( ) 1 1 : 1 0 M.H. Mertens (University of Cologne) Class Number Type Relations / 37

29 A real-analytic Jacobi form Theorem (S. Zwegers, 2002) Â 1 transforms like a Jacobi form of weight 1 and index 1 Â 1 ( u, v) = Â1(u, v). 2 For λ i, µ i Z: Â 1 (u + λ 1 τ + µ 1, v + λ 2 τ + µ 2 ) ( ) 1 1 : 1 0 = ( 1) (λ 1+µ 1 ) e 2πi((λ 1 λ 2 )u λ 1 v) q λ λ 1λ 2 Â 1 (u, v) M.H. Mertens (University of Cologne) Class Number Type Relations / 37

30 A real-analytic Jacobi form Theorem (S. Zwegers, 2002)  1 transforms like a Jacobi form of weight 1 and index 1  1 ( u, v) = Â1(u, v). 2 For λ i, µ i Z:  1 (u + λ 1 τ + µ 1, v + λ 2 τ + µ 2 ) ( ) 3 a b For γ = SL c d 2 (Z): ( u  1 cτ + d, ( ) 1 1 : 1 0 = ( 1) (λ 1+µ 1 ) e 2πi((λ 1 λ 2 )u λ 1 v) q λ λ 1λ 2  1 (u, v) v cτ + d ; aτ + b ) cτ + d = (cτ + d)e πic u2 +2uv cτ+d  1 (u, v; τ). M.H. Mertens (University of Cologne) Class Number Type Relations / 37

31 Table of Contents 1 Mock Modular Forms 2 Cohen s conjecture 3 Holomorphic Projection 4 Fourier Coefficients of Mock Modular Forms M.H. Mertens (University of Cologne) Class Number Type Relations / 37

32 A generalized function From Dirichlet s class number formula we generalize for r N ( 1) r/2 (r 1)!n r 1/2 L(r, χ 2 r 1 π r ( 1) r n) if n > 0 and h r (n) = ( 1) r n 0, 1 (4) 0 otherwise M.H. Mertens (University of Cologne) Class Number Type Relations / 37

33 A generalized function From Dirichlet s class number formula we generalize for r N ( 1) r/2 (r 1)!n r 1/2 L(r, χ 2 r 1 π r ( 1) r n) if n > 0 and h r (n) = ( 1) r n 0, 1 (4) 0 otherwise h r (n/f 2 ) if n > 0 and ( 1) r n 0, 1 (4) f 2 n H r (n) = ζ(1 2r) if n = 0 0 otherwise M.H. Mertens (University of Cologne) Class Number Type Relations / 37

34 A generalized function From Dirichlet s class number formula we generalize for r N ( 1) r/2 (r 1)!n r 1/2 L(r, χ 2 r 1 π r ( 1) r n) if n > 0 and h r (n) = ( 1) r n 0, 1 (4) 0 otherwise h r (n/f 2 ) if n > 0 and ( 1) r n 0, 1 (4) f 2 n H r (n) = ζ(1 2r) if n = 0 0 otherwise H r (τ) = H r (n)q n n=0 M.H. Mertens (University of Cologne) Class Number Type Relations / 37

35 A generalized function From Dirichlet s class number formula we generalize for r N ( 1) r/2 (r 1)!n r 1/2 L(r, χ 2 r 1 π r ( 1) r n) if n > 0 and h r (n) = ( 1) r n 0, 1 (4) 0 otherwise h r (n/f 2 ) if n > 0 and ( 1) r n 0, 1 (4) f 2 n H r (n) = ζ(1 2r) if n = 0 0 otherwise H r (τ) = H r (n)q n n=0 N.B.: h 1 (n) = h( n), H 1 (n) = H(n) and H 1 (n) = H (n) M.H. Mertens (University of Cologne) Class Number Type Relations / 37

36 Theorem (H. Cohen, 1975) For r 2, the function H r is a (holomorphic) modular form of weight r on Γ 0(4). M.H. Mertens (University of Cologne) Class Number Type Relations / 37

37 Theorem (H. Cohen, 1975) For r 2, the function H r is a (holomorphic) modular form of weight r on Γ 0(4). Idea of proof H r is a linear combination of certain well-known Eisenstein series. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

38 Theorem (H. Cohen, 1975) For r 2, the function H r is a (holomorphic) modular form of weight r on Γ 0(4). Idea of proof H r is a linear combination of certain well-known Eisenstein series. Results into many interesting identities for special values of L-functions. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

39 Theorem (H. Cohen, 1975) For r 2, the function H r is a (holomorphic) modular form of weight r on Γ 0(4). Idea of proof H r is a linear combination of certain well-known Eisenstein series. Results into many interesting identities for special values of L-functions. For r = 1 the Eisenstein series don t converge Hecke s trick and analytic continuation yield proof af Zagier s theorem. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

40 Conjecture Conjecture (H. Cohen, 1975 The coefficient of X ν in [ S4(τ, 1 H(n s 2 ) ] X) := 1 2sX + nx 2 + λ 2k+1 (n)x 2k n=0 n odd s Z k=0 is a (holomorphic) modular form of weight 2ν + 2 on Γ 0 (4). q n M.H. Mertens (University of Cologne) Class Number Type Relations / 37

41 Conjecture Conjecture (H. Cohen, 1975 The coefficient of X ν in [ S4(τ, 1 H(n s 2 ) ] X) := 1 2sX + nx 2 + λ 2k+1 (n)x 2k n=0 n odd s Z k=0 is a (holomorphic) modular form of weight 2ν + 2 on Γ 0 (4). Remarks For odd ν, coeff X ν (S4 1 (τ; X)) = 0. q n M.H. Mertens (University of Cologne) Class Number Type Relations / 37

42 Conjecture Conjecture (H. Cohen, 1975 The coefficient of X ν in [ S4(τ, 1 H(n s 2 ) ] X) := 1 2sX + nx 2 + λ 2k+1 (n)x 2k n=0 n odd s Z k=0 is a (holomorphic) modular form of weight 2ν + 2 on Γ 0 (4). Remarks For odd ν, coeff X ν (S4 1 (τ; X)) = 0. ν = 0 yields Eichler s class number relation. q n M.H. Mertens (University of Cologne) Class Number Type Relations / 37

43 New class number relations Corollary ( 4s 2 n ) H ( n s 2) + λ 3 (n) = 0, s Z M.H. Mertens (University of Cologne) Class Number Type Relations / 37

44 New class number relations Corollary ( 4s 2 n ) H ( n s 2) + λ 3 (n) = 0, s Z ( 16s 4 12ns 2 + n 2) H ( n s 2) + λ 5 (n) s Z = 1 12 n=x 2 +y 2 +z 2 +t 2 ( x 4 6x 2 y 2 + y 4) M.H. Mertens (University of Cologne) Class Number Type Relations / 37

45 New class number relations Corollary ( 4s 2 n ) H ( n s 2) + λ 3 (n) = 0, s Z ( 16s 4 12ns 2 + n 2) H ( n s 2) + λ 5 (n) s Z = 1 ( x 4 6x 2 y 2 + y 4) 12 n=x 2 +y 2 +z 2 +t 2 ( 64s 6 80s 4 n + 24s 2 n 2 n 3) H ( n s 2) + λ 7 (n) s Z = 1 3 n=x 2 +y 2 +z 2 +t 2 ( x 6 5x 4 y 2 10x 4 z x 2 y 2 z 2 + 5x 2 z 4 5y 2 z 4) M.H. Mertens (University of Cologne) Class Number Type Relations / 37

46 New class number relations Corollary ( 4s 2 n ) H ( n s 2) + λ 3 (n) = 0, s Z ( 16s 4 12ns 2 + n 2) H ( n s 2) + λ 5 (n) s Z = 1 ( x 4 6x 2 y 2 + y 4) 12 n=x 2 +y 2 +z 2 +t 2 ( 64s 6 80s 4 n + 24s 2 n 2 n 3) H ( n s 2) + λ 7 (n) s Z = n=x 2 +y 2 +z 2 +t 2 ( x 6 5x 4 y 2 10x 4 z x 2 y 2 z 2 + 5x 2 z 4 5y 2 z 4) M.H. Mertens (University of Cologne) Class Number Type Relations / 37

47 Solution Theorem 1 (M., 2013) Cohen s conjecture is true. Moreover, coeff X 2ν (S4 1 (τ; X)) is a cusp form for ν > 0. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

48 Solution Theorem 1 (M., 2013) Cohen s conjecture is true. Moreover, coeff X 2ν (S4 1 (τ; X)) is a cusp form for ν > 0. Remark Recall Rankin-Cohen brackets: f, g modular of weights k, l [f, g] ν := ν ( )( ) k + ν 1 l + ν 1 ( 1) µ D µ fd ν µ g. ν µ µ µ=0 M.H. Mertens (University of Cologne) Class Number Type Relations / 37

49 Solution Theorem 1 (M., 2013) Cohen s conjecture is true. Moreover, coeff X 2ν (S4 1 (τ; X)) is a cusp form for ν > 0. Remark Recall Rankin-Cohen brackets: f, g modular of weights k, l [f, g] ν := ν ( )( ) k + ν 1 l + ν 1 ( 1) µ D µ fd ν µ g. ν µ µ µ=0 coeff X 2ν (S4(τ; 1 X)) = c ν ( ( [H, ϑ]ν (τ) [H, ϑ] ν τ )) + Λ2ν+1,odd (τ), π c ν = ν! Γ(ν ) M.H. Mertens (University of Cologne) Class Number Type Relations / 37

50 Some Lemmas Lemma where Λ 2ν+1,odd = 1 2 ( D 2ν+1 v ) (0, A odd 1 τ ; 2τ), A odd 1 (u, v; τ) = 1 (A 1 (u, v; τ) A 1 (u, v + 12 )) 2 ; τ. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

51 Some Lemmas Lemma where Λ 2ν+1,odd = 1 2 ( D 2ν+1 v ) (0, A odd 1 τ ; 2τ), A odd 1 (u, v; τ) = 1 (A 1 (u, v; τ) A 1 (u, v + 12 )) 2 ; τ. Lemma The R-function completing the Appell-Lerch sums is in the heat kernel, i.e. D 2 ur = 2D τ R. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

52 Outline of the proof Complete [H, ϑ] ν and Λ 2ν+1,odd to transform like modular forms M.H. Mertens (University of Cologne) Class Number Type Relations / 37

53 Outline of the proof Complete [H, ϑ] ν and Λ 2ν+1,odd to transform like modular forms prove that non-holomorphic terms cancel M.H. Mertens (University of Cologne) Class Number Type Relations / 37

54 Outline of the proof Complete [H, ϑ] ν and Λ 2ν+1,odd to transform like modular forms prove that non-holomorphic terms cancel induction on ν: M.H. Mertens (University of Cologne) Class Number Type Relations / 37

55 Outline of the proof Complete [H, ϑ] ν and Λ 2ν+1,odd to transform like modular forms prove that non-holomorphic terms cancel induction on ν: ν = 0: rather direct calculation M.H. Mertens (University of Cologne) Class Number Type Relations / 37

56 Outline of the proof Complete [H, ϑ] ν and Λ 2ν+1,odd to transform like modular forms prove that non-holomorphic terms cancel induction on ν: ν = 0: rather direct calculation ν ν + 1: Use heat kernel property of R Remark Let T (k) n be the nth Hecke operator on S k (Γ 0 (4)). Then coeff X 2ν (S4(τ; 1 X)) = 3 ( trace n odd T (2ν+2) n ) q n (not proved by this method) M.H. Mertens (University of Cologne) Class Number Type Relations / 37

57 Table of Contents 1 Mock Modular Forms 2 Cohen s conjecture 3 Holomorphic Projection 4 Fourier Coefficients of Mock Modular Forms M.H. Mertens (University of Cologne) Class Number Type Relations / 37

58 Definition Definition f : H C a continuous function with Fourier expansion f(τ) = n Z a f (n, y)q n, M.H. Mertens (University of Cologne) Class Number Type Relations / 37

59 Definition Definition f : H C a continuous function with Fourier expansion f(τ) = n Z a f (n, y)q n, 1 (f k γ)(τ) = O(y δ ) as y for all γ SL 2 (Z), M.H. Mertens (University of Cologne) Class Number Type Relations / 37

60 Definition Definition f : H C a continuous function with Fourier expansion f(τ) = n Z a f (n, y)q n, 1 (f k γ)(τ) = O(y δ ) as y for all γ SL 2 (Z), 2 a f (n, y) = c 0 + O(y 1 k+ε ) as y 0 for all n > 0, where δ, ε > 0, and 2 < k 1 2 Z. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

61 Definition Definition f : H C a continuous function with Fourier expansion f(τ) = n Z a f (n, y)q n, 1 (f k γ)(τ) = O(y δ ) as y for all γ SL 2 (Z), 2 a f (n, y) = c 0 + O(y 1 k+ε ) as y 0 for all n > 0, where δ, ε > 0, and 2 < k 1 2 Z. Then we define the holomorphic projection of f by with (π hol f)(τ) := (πhol k f)(τ) := c(n)q n, c(n) = (4πn)k 1 (k 2)! M.H. Mertens (University of Cologne) 0Class Number Type Relations / 37 n=0 a f (n, y)e 4πny y k 2 dy, n > 0.

62 Properties Proposition If f is holomorphic, then π hol f = f. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

63 Properties Proposition If f is holomorphic, then π hol f = f. If f transforms like a modular form of weight k 1 2Z, k > 2, on some group Γ SL 2 (Z), then π hol f M k (Γ). M.H. Mertens (University of Cologne) Class Number Type Relations / 37

64 Properties Proposition If f is holomorphic, then π hol f = f. If f transforms like a modular form of weight k 1 2Z, k > 2, on some group Γ SL 2 (Z), then π hol f M k (Γ). The operator π hol commutes with all the operators U(N), V (N), and S N,r (sieving operator). M.H. Mertens (University of Cologne) Class Number Type Relations / 37

65 Properties Proposition If f is holomorphic, then π hol f = f. If f transforms like a modular form of weight k 1 2Z, k > 2, on some group Γ SL 2 (Z), then π hol f M k (Γ). The operator π hol commutes with all the operators U(N), V (N), and S N,r (sieving operator). If f is modular of weight k > 2 on Γ SL 2 (Z) then we have f, g = π hol (f), g, where, denotes the Petersson scalar product, for every cusp form g S k (Γ). M.H. Mertens (University of Cologne) Class Number Type Relations / 37

66 Rankin-Cohen brackets From now on: f + (τ) + (4πy)1 k k 1 c f (0) + f (τ) M k (Γ), g M l (Γ), ν > 0 s.t. [f, g] ν satisfies conditions in definition. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

67 Rankin-Cohen brackets From now on: f + (τ) + (4πy)1 k k 1 c f (0) + f (τ) M k (Γ), g M l (Γ), ν > 0 s.t. [f, g] ν satisfies conditions in definition. Lemma (4π) 1 k k 1 c f (0)π hol([y 1 k, g] ν ) = κc f (0) n k+ν 1 a g (n)q n, with 1 ν [ Γ(2 k)γ(l + 2ν µ) κ = κ(k, l, ν) = (k + l + 2ν 2)!(k 1) Γ(2 k µ) µ=0 ( )( )] k + ν 1 l + ν 1. ν µ µ n=0 M.H. Mertens (University of Cologne) Class Number Type Relations / 37

68 Rankin-Cohen brackets Let a 2 ( ) j + b 2 P a,b (X, Y ) := X j (X + Y ) a j 2. j j=0 M.H. Mertens (University of Cologne) Class Number Type Relations / 37

69 Rankin-Cohen brackets Let a 2 ( ) j + b 2 P a,b (X, Y ) := X j (X + Y ) a j 2. j j=0 Theorem π hol ([f, g] ν ) = b(r)q r, where ν ( )( ) k + ν 1 l + ν 1 b(r) = Γ(1 k) m ν µ a g (m)c ν µ µ f (n) m n=r µ=0 (m µ 2ν l+1 P k+l+2ν,2 k µ (r, n) n k+µ 1) r=1 M.H. Mertens (University of Cologne) Class Number Type Relations / 37

70 Properities of P a,b (X, Y ) Lemma For b 1, 2, the polynomial P a,b (X, Y ) satisfies a 2 ( ) a + b 3 P a,b (X, Y ) = X j Y a 2 j j j=0 a 2 ( a + b 3 = a 2 j j=0 )( j + b 2 j ) (X + Y ) a 2 j ( Y ) j. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

71 Table of Contents 1 Mock Modular Forms 2 Cohen s conjecture 3 Holomorphic Projection 4 Fourier Coefficients of Mock Modular Forms M.H. Mertens (University of Cologne) Class Number Type Relations / 37

72 The Theorem of Serre-Stark Theorem (J.-P. Serre, H. Stark, 1977) Let ϕ be a modular form of weight 1 2 on Γ 1(N). Then ϕ is a unique linear combination of the theta series ϑ χ,t (τ) = n Z χ(n)q tn2 with χ a primitive even character with conductor F (χ) and t N such that 4F (χ) 2 t N. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

73 Proposition Let r = m n. Then it holds that ν µ=0 ( ν ν µ )( ν 1 2 µ ) ( m 1 2 ν P 2ν+2, 1 2 µ(r, n) n 1 2 +µ m ν µ) M.H. Mertens (University of Cologne) Class Number Type Relations / 37

74 Proposition Let r = m n. Then it holds that ν µ=0 ( ν ν µ =2 2ν ( 2ν ν )( ν 1 2 µ ) (m 1 2 n 1 2 ) 2ν+1. ) ( m 1 2 ν P 2ν+2, 1 2 µ(r, n) n 1 2 +µ m ν µ) M.H. Mertens (University of Cologne) Class Number Type Relations / 37

75 Proposition Let r = m n. Then it holds that ν µ=0 ( ν ν µ =2 2ν ( 2ν ν Proposition )( ν 1 2 µ ) (m 1 2 n 1 2 ) 2ν+1. Let r = m n. Then it holds that ν µ=0 ( ν 1 2 ν µ )( ν µ ) ( m 1 2 ν P 2ν+2, 1 2 µ(r, n) n 1 2 +µ m ν µ) ) ( m ν 1 2 P2ν+2, 3 2 µ(r, n) nµ 1 2 m ν µ ) M.H. Mertens (University of Cologne) Class Number Type Relations / 37

76 Proposition Let r = m n. Then it holds that ν µ=0 ( ν ν µ =2 2ν ( 2ν ν Proposition )( ν 1 2 µ ) (m 1 2 n 1 2 ) 2ν+1. Let r = m n. Then it holds that ν µ=0 ( ν 1 2 ν µ = 2 2ν ( 2ν ν )( ν µ ) ( m 1 2 ν P 2ν+2, 1 2 µ(r, n) n 1 2 +µ m ν µ) ) ( m ν 1 2 P2ν+2, 3 2 µ(r, n) nµ 1 2 m ν µ ) ) ) (mn) 1 2 (m 1 1 2ν+1 2 n 2. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

77 Result in weight ( 3 2, 1 2 ) Theorem 2 (M., 2013) Let f M 3 (Γ) and g M 1 (Γ) with Γ = Γ 1 (4N) for some N N and 2 2 fix ν N. Then there is a finite linear combination L f,g ν of functions of the form s,t (τ; ν) = 2 Λ χ,ψ r=1 sm 2 tn 2 =r m,n 1 +ψ(0) χ(m)ψ(n)( sm tn) 2ν+1 χ(r)( sr) 2ν+1 q sr2 r=1 qr with s, t N and χ, ψ are even characters of conductors F (χ) and F (ψ) respectively with sf (χ) 2, tf (ψ) 2 N, such that [f, g] ν + L f,g ν is a holomorphic cusp form of weight 2ν + 2. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

78 Result for mock theta functions Theorem 3 (M., 2013) Let f S 1 (Γ) be a completed mock theta function and g S θ 3 (Γ), where 2 2 Γ = Γ 1 (4N) for some N N and let ν be a fixed non-negative integer. Then there is a finite linear combination Dν f,g of functions of the form s,t (τ; ν) =2 χ,ψ r=1 sm 2 tn 2 =r m,n 1 χ(m)ψ(n)( sm tn) 2ν+1 qr, where s, t N and χ, ψ are odd characters of conductors F (χ) and F (ψ) respectively with sf (χ) 2, tf (ψ) 2 N, such that [f, g] ν + Dν f,g is a holomorphic cusp form of weight 2ν + 2. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

79 Generating systems Corollary With the notation from the theorem the following is true. The equivalence classes Λ χ,ψ s,t + M 2ν+2! (Γ 1(N)) generate the C-vector space (Γ), M 1 (Γ)] ν /M 2ν+2(Γ).! 2 [M mock 3 2 M.H. Mertens (University of Cologne) Class Number Type Relations / 37

80 Generating systems Corollary With the notation from the theorem the following is true. The equivalence classes Λ χ,ψ s,t + M 2ν+2! (Γ 1(N)) generate the C-vector space (Γ), M 1 (Γ)] ν /M 2ν+2(Γ).! 2 Corollary [M mock 3 2 The equivalence classes χ,ψ s,t space [S mock ϑ M! 2ν+2 (Γ 1(N)) generate the C-vector (Γ), S θ 3 (Γ)] ν /M 2ν+2(Γ).! 2 M.H. Mertens (University of Cologne) Class Number Type Relations / 37

81 Remarks Λ χ,ψ s,t functions and χ,ψ s,t are (up to a polynomial factor) indefinite theta M.H. Mertens (University of Cologne) Class Number Type Relations / 37

82 Remarks Λ χ,ψ s,t functions and χ,ψ s,t are (up to a polynomial factor) indefinite theta if s, t =, the coefficients of Λ s, t and s,t are essentially minimal divisor power sums ( derivatives of Appell-Lerch sums) M.H. Mertens (University of Cologne) Class Number Type Relations / 37

83 Remarks Λ χ,ψ s,t functions and χ,ψ s,t are (up to a polynomial factor) indefinite theta if s, t =, the coefficients of Λ s, t and s,t are essentially minimal divisor power sums ( derivatives of Appell-Lerch sums) in general, minimal divisor power sums in a real-(bi)quadratic number field M.H. Mertens (University of Cologne) Class Number Type Relations / 37

84 Remarks Λ χ,ψ s,t functions and χ,ψ s,t are (up to a polynomial factor) indefinite theta if s, t =, the coefficients of Λ s, t and s,t are essentially minimal divisor power sums ( derivatives of Appell-Lerch sums) in general, minimal divisor power sums in a real-(bi)quadratic number field ( generalized Appell-Lerch sums?) M.H. Mertens (University of Cologne) Class Number Type Relations / 37

85 Example: Eichler-Selberg and Cohen s conjecture Recall: H (τ) = n=0 H(n)qn M mock 3 (Γ 0 (4)) with shadow 1 8 π ϑ. 2 M.H. Mertens (University of Cologne) Class Number Type Relations / 37

86 Example: Eichler-Selberg and Cohen s conjecture Recall: H (τ) = n=0 H(n)qn M mock 3 (Γ 0 (4)) with shadow 1 8 π ϑ. From Theorem 2: For ν > 0, π hol ([Ĥ, ϑ] ν)(τ) = [H, ϑ] ν (τ) + 2 2ν 1 ( 2ν ν where Λ = Λ 1,1 1,1. 2 ) Λ (τ) S 2ν+2 (Γ 0 (4)), M.H. Mertens (University of Cologne) Class Number Type Relations / 37

87 Example: Eichler-Selberg and Cohen s conjecture Recall: H (τ) = n=0 H(n)qn M mock 3 (Γ 0 (4)) with shadow 1 8 π ϑ. From Theorem 2: For ν > 0, π hol ([Ĥ, ϑ] ν)(τ) = [H, ϑ] ν (τ) + 2 2ν 1 ( 2ν ν where Λ = Λ 1,1 1,1. easy to see (Λ U(4))(τ; ν) = 2 2ν+1 (Λ S 2,1 )(τ; ν) = 2 2 n odd n=1 ) Λ (τ) S 2ν+2 (Γ 0 (4)), 2λ 2ν+1 (n)q n λ 2ν+1 (n)q n. M.H. Mertens (University of Cologne) Class Number Type Relations / 37

88 Example: Eichler-Selberg and Cohen s conjecture Rewrite: ( ) g ν (1) (s, n)h(4n s 2 ) q n + 2 λ 2ν+1 (n)q n S 2ν+2 (Γ 0 (1)) n=1 s Z n=0, M.H. Mertens (University of Cologne) Class Number Type Relations / 37

89 Example: Eichler-Selberg and Cohen s conjecture Rewrite: ( ) g ν (1) (s, n)h(4n s 2 ) q n + 2 λ 2ν+1 (n)q n S 2ν+2 (Γ 0 (1)) n=1 n odd s Z ( s Z g (4) ν (s, n)h(n s 2 ) ) n=0 q n + n odd λ 2ν+1 (n)q n S 2ν+2 (Γ 0 (4)), M.H. Mertens (University of Cologne) Class Number Type Relations / 37

90 Example: Eichler-Selberg and Cohen s conjecture Rewrite: ( ) g ν (1) (s, n)h(4n s 2 ) q n + 2 λ 2ν+1 (n)q n S 2ν+2 (Γ 0 (1)) n=1 n odd s Z ( s Z g (4) ν (s, n)h(n s 2 ) ) n=0 q n + n odd g (1) ν (s, n) = coeff X 2ν ((1 sx + nx 2 ) 1 ), g (4) ν (s, n) = coeff X 2ν ((1 2sX + nx 2 ) 1 ) λ 2ν+1 (n)q n S 2ν+2 (Γ 0 (4)), M.H. Mertens (University of Cologne) Class Number Type Relations / 37

91 Example: Eichler-Selberg and Cohen s conjecture Rewrite: ( ) g ν (1) (s, n)h(4n s 2 ) q n + 2 λ 2ν+1 (n)q n S 2ν+2 (Γ 0 (1)) n=1 n odd s Z ( s Z g (4) ν (s, n)h(n s 2 ) ) n=0 q n + n odd λ 2ν+1 (n)q n S 2ν+2 (Γ 0 (4)), g (1) ν (s, n) = coeff X 2ν ((1 sx + nx 2 ) 1 ), g (4) ν (s, n) = coeff X 2ν ((1 2sX + nx 2 ) 1 ) Rankin-Selberg unfolding trick (f normalized Hecke eigenform): [Ĥ, ϑ] ν, f = π hol ([Ĥ, ϑ] ν), f. = f, f trace formulae for SL 2 (Z) and Γ 0 (4). M.H. Mertens (University of Cologne) Class Number Type Relations / 37

92 Thank you for your attention. M.H. Mertens (University of Cologne) Class Number Type Relations / 37