Mock Modular Forms and Class Number Relations

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 05.13.2014 1 / 30

Notation H(n) := Hurwitz class number of n, M.H. Mertens (University of Cologne) Class Number Relations 05.13.2014 2 / 30

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 05.13.2014 2 / 30

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 05.13.2014 2 / 30

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 05.13.2014 3 / 30

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 05.13.2014 3 / 30

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 05.13.2014 3 / 30

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 05.13.2014 4 / 30

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 05.13.2014 4 / 30

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 2 2 + 2 + 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 05.13.2014 5 / 30

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 2 2 + 2 + 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 05.13.2014 5 / 30

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 05.13.2014 6 / 30

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 05.13.2014 7 / 30

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 05.13.2014 7 / 30

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 + 1 4 π n=1 ( nγ 1 ) 2 ; 4πn2 y q n2. M.H. Mertens (University of Cologne) Class Number Relations 05.13.2014 8 / 30

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 + 1 4 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 05.13.2014 8 / 30

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 + 1 4 π n=1 ( nγ 1 ) 2 ; 4πn2 y q n2. Ĥ = H + R H 3 (4) 2 ξ 3 Ĥ = 1 2 8 π ϑ M.H. Mertens (University of Cologne) Class Number Relations 05.13.2014 8 / 30

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 05.13.2014 9 / 30

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 05.13.2014 9 / 30

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 05.13.2014 9 / 30

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 05.13.2014 10 / 30

A real-analytic Jacobi form Theorem (S. Zwegers, 2002) ( ) 1 1 Â 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 Â 1 cτ + d, v cτ + d ; aτ + b ) = (cτ + d)e πic u2 +2uv cτ+d Â 1 (u, v; τ). cτ + d M.H. Mertens (University of Cologne) Class Number Relations 05.13.2014 11 / 30

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 05.13.2014 13 / 30

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 05.13.2014 13 / 30

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 05.13.2014 13 / 30

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 05.13.2014 14 / 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 05.13.2014 14 / 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 ( 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 2 + 30x 2 y 2 z 2 + 5x 2 z 4 5y 2 z 4) M.H. Mertens (University of Cologne) Class Number Relations 05.13.2014 14 / 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 ( 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 2 + 30x 2 y 2 z 2 + 5x 2 z 4 5y 2 z 4) M.H. Mertens (University of Cologne) Class Number Relations 05.13.2014 14 / 30

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 05.13.2014 15 / 30

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 + 1 2 2)) + Λ2ν+1,odd (τ), π c ν = ν! Γ(ν + 1 2 ), Λ 2ν+1 (τ) := λ k (n)q n n=1 M.H. Mertens (University of Cologne) Class Number Relations 05.13.2014 15 / 30

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 + 1 2 2)) + Λ2ν+1,odd (τ), π c ν = ν! Γ(ν + 1 2 ), 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 05.13.2014 15 / 30

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 05.13.2014 16 / 30

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 05.13.2014 16 / 30

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 05.13.2014 16 / 30

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 05.13.2014 16 / 30

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 05.13.2014 16 / 30

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 05.13.2014 18 / 30

Properties Proposition If f is holomorphic, then π hol f = f. M.H. Mertens (University of Cologne) Class Number Relations 05.13.2014 19 / 30

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 05.13.2014 19 / 30

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 05.13.2014 19 / 30

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 05.13.2014 19 / 30

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 05.13.2014 20 / 30

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 05.13.2014 20 / 30

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 05.13.2014 21 / 30

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 05.13.2014 21 / 30

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 05.13.2014 23 / 30

Proposition Let r = m n. Then it holds that ν µ=0 ( ν + 1 2 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 05.13.2014 24 / 30

Proposition Let r = m n. Then it holds that ν µ=0 ( ν + 1 2 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 05.13.2014 24 / 30

Proposition Let r = m n. Then it holds that ν µ=0 ( ν + 1 2 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 + 1 2 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 05.13.2014 24 / 30

Proposition Let r = m n. Then it holds that ν µ=0 ( ν + 1 2 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 + 1 2 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 05.13.2014 24 / 30

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 05.13.2014 25 / 30

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 05.13.2014 26 / 30

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 05.13.2014 27 / 30

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 05.13.2014 27 / 30

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 05.13.2014 27 / 30

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 05.13.2014 27 / 30

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 05.13.2014 28 / 30

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 05.13.2014 28 / 30

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 05.13.2014 28 / 30

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 05.13.2014 29 / 30

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 05.13.2014 29 / 30

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 05.13.2014 29 / 30

Thank you for your attention. M.H. Mertens (University of Cologne) Class Number Relations 05.13.2014 30 / 30