Mock Modular Forms and Class Number Relations

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1 Mock Modular Forms and Class Number Relations Michael H. Mertens University of Cologne 28th Automorphic Forms Workshop, Moab, May 13th, 2014 M.H. Mertens (University of Cologne) Class Number Relations / 30

2 Notation H(n) := Hurwitz class number of n, M.H. Mertens (University of Cologne) Class Number Relations / 30

3 Notation H(n) := Hurwitz class number of n, λ k (n) := 1 2 d n ( min d, n ) k, d M.H. Mertens (University of Cologne) Class Number Relations / 30

4 Notation H(n) := Hurwitz class number of n, λ k (n) := 1 2 d n ( min d, n ) k, d σ k (n) := d n d k. M.H. Mertens (University of Cologne) Class Number Relations / 30

5 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 Relations / 30

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.H. Mertens (University of Cologne) Class Number Relations / 30

7 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 Relations / 30

8 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 Relations / 30

9 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 Relations / 30

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), M.H. Mertens (University of Cologne) Class Number Relations / 30

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, M.H. Mertens (University of Cologne) Class Number Relations / 30

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 linearly exponentially approaching the cusps of Γ 0 (N). M.H. Mertens (University of Cologne) Class Number Relations / 30

13 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 linearly 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 Relations / 30

14 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 Relations / 30

15 The shadow Proposition For k 1, the mapping ξ k : H k (N, χ) M 2 k! (N, χ), f 2iyk τ f is well-defined and surjective with kernel M k! (N, χ). Moreover, for f H k (N, χ), we have that the shadow of f is given by (ξ k f)(τ) = (4π) 1 k n=n 0 c f (n)qn. M.H. Mertens (University of Cologne) Class Number Relations / 30

16 The shadow Proposition For k 1, the mapping ξ k : H k (N, χ) M 2 k! (N, χ), f 2iyk τ f is well-defined and surjective with kernel M k! (N, χ). Moreover, for f H k (N, χ), we have that the shadow of f is given by (ξ k f)(τ) = (4π) 1 k n=n 0 c f (n)qn. 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 Relations / 30

17 An example Theorem (D. Zagier, 1976) Let H (τ) = n=0 H(n)q n, ϑ(τ) := n Z q n2 and R(τ) := 1 + i 16π i τ ϑ(z) (z + τ) 3 2 dz = 1 8π y π n=1 ( nγ 1 ) 2 ; 4πn2 y q n2. M.H. Mertens (University of Cologne) Class Number Relations / 30

18 An example Theorem (D. Zagier, 1976) Let H (τ) = n=0 H(n)q n, ϑ(τ) := n Z q n2 and R(τ) := 1 + i 16π i τ ϑ(z) (z + τ) 3 2 dz = 1 8π y π n=1 ( nγ 1 ) 2 ; 4πn2 y q n2. Ĥ = H + R H 3 (4) 2 M.H. Mertens (University of Cologne) Class Number Relations / 30

19 An example Theorem (D. Zagier, 1976) Let H (τ) = n=0 H(n)q n, ϑ(τ) := n Z q n2 and R(τ) := 1 + i 16π i τ ϑ(z) (z + τ) 3 2 dz = 1 8π y π n=1 ( nγ 1 ) 2 ; 4πn2 y q n2. Ĥ = H + R H 3 (4) 2 ξ 3 Ĥ = π ϑ M.H. Mertens (University of Cologne) Class Number Relations / 30

20 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 Relations / 30

21 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 Relations / 30

22 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 Relations / 30

23 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 Relations / 30

24 A real-analytic Jacobi form Theorem (S. Zwegers, 2002) ( ) 1 1  1 transforms like a Jacobi form of weight 1 and index. In 1 0 particular we have for γ = ( ) a b c d SL2 (Z) that ( u  1 cτ + d, v cτ + d ; aτ + b ) = (cτ + d)e πic u2 +2uv cτ+d  1 (u, v; τ). cτ + d M.H. Mertens (University of Cologne) Class Number Relations / 30

25 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 Relations / 30

26 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 Relations / 30

27 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 Relations / 30

28 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 Relations / 30

29 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 Relations / 30

30 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 Relations / 30

31 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 Relations / 30

32 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 Relations / 30

33 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 Relations / 30

34 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 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 ), Λ 2ν+1 (τ) := λ k (n)q n n=1 M.H. Mertens (University of Cologne) Class Number Relations / 30

35 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 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 ), Λ 2ν+1 (τ) := λ k (n)q n = 1 2 (D2ν+1 v A 1 ) (0, τ + 12 ) ; 2τ n=1 M.H. Mertens (University of Cologne) Class Number Relations / 30

36 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 Relations / 30

37 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 Relations / 30

38 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 Relations / 30

39 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 Relations / 30

40 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 (D 2 ur = 2D τ R) M.H. Mertens (University of Cologne) Class Number Relations / 30

41 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 Relations / 30

42 Definition Definition f : H C a continuous function with Fourier expansion f(τ) = n Z a f (n, y)q n, which behaves nicely at and the rationals and whose Fourier-coefficients don t grow too fast. M.H. Mertens (University of Cologne) Class Number Relations / 30

43 Definition Definition f : H C a continuous function with Fourier expansion f(τ) = n Z a f (n, y)q n, which behaves nicely at and the rationals and whose Fourier-coefficients don t grow too fast. 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)! 0 n=0 a f (n, y)e 4πny y k 2 dy, n > 0. M.H. Mertens (University of Cologne) Class Number Relations / 30

44 Properties Proposition If f is holomorphic, then π hol f = f. M.H. Mertens (University of Cologne) Class Number Relations / 30

45 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 Relations / 30

46 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 Relations / 30

47 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 Relations / 30

48 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 Relations / 30

49 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 π hol([y 1 k, g] ν ) = κ 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 Relations / 30

50 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 Relations / 30

51 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 Relations / 30

52 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 Relations / 30

53 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 Relations / 30

54 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 Relations / 30

55 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 Relations / 30

56 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 Relations / 30

57 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 Relations / 30

58 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 modular form of weight 2ν + 2. M.H. Mertens (University of Cologne) Class Number Relations / 30

59 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 modular form of weight 2ν + 2. M.H. Mertens (University of Cologne) Class Number Relations / 30

60 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 Relations / 30

61 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 Relations / 30

62 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 Relations / 30

63 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 Relations / 30

64 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 Relations / 30

65 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 Relations / 30

66 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 2λ 2ν+1 (n)q n (Λ S 2,1 )(τ; ν) = 2 2 n odd n=1 λ 2ν+1 (n)q n. ) Λ (τ) S 2ν+2 (Γ 0 (4)), M.H. Mertens (University of Cologne) Class Number Relations / 30

67 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 Relations / 30

68 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 Relations / 30

69 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 Relations / 30

70 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 Relations / 30

71 Thank you for your attention. M.H. Mertens (University of Cologne) Class Number Relations / 30

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