aa r X i v : . [ m a t h . N T ] J a n HARMONIC MAASS FORM EIGENCURVES
IAN WAGNER
Abstract.
We construct two families of harmonic Maass Hecke eigenforms. Using these families,we construct p -adic harmonic Maass forms in the sense of Serre. The p -adic properties of theseforms answer a question of Mazur about the existence of an “eigencurve-type” object in the worldof harmonic Maass forms. Introduction and statement of results
In [18] Serre introduced the notion of a p -adic modular form and showed the power of studying a p -adic analytic family of modular eigenforms. Work of Hida [8] and Coleman [4] expanded on Serre’sinitial definition of p -adic modular form to introduce overconvergent modular forms and offered moreexamples and applications. Coleman, in particular, defined the slope of an eigenform as the p -adicvaluation of its U p eigenvalue and proved that overconvergent modular forms with small slope areclassical modular forms. In [5] Coleman and Mazur organized all of these results by constructinga geometric object called the “eigencurve.” The eigencurve is a rigid-analytic curve whose pointscorrespond to normalized finite slope p -adic overconvergent modular eigenforms.Using Kummer’s congruences, Serre was able to give the first examples of p -adic modular forms.Let v p be the p -adic valuation on Q p . If f = P a ( n ) q n ∈ Q [[ q ]] is a formal power series in q thendefine v p ( f ) := inf n v p ( a ( n )). We then say that f is a p -adic modular form if there exists a sequenceof classical modular forms f i of weights k i such that v p ( f − f i ) −→ ∞ as i −→ ∞ . The weight of a p -adic modular form is given by the limits of weights of the classical (holomorphic) modular formsin X := Z p × Z / ( p − Z . For a more in-depth discussion of weights see [18].The first examples Serre offered came from the Eisenstein series. Let σ k ( n ) := P d | n d k be thedivisor function, z = x + iy ∈ H , and q = e πiz . Then for k ≥
1, the weight 2 k Eisenstein series isgiven by(1.1) G k ( z ) := 12 ζ (1 − k ) + ∞ X n =1 σ k − ( n ) q n , where ζ ( s ) is the Riemann zeta function. For 2 k ≥ G k ( z ) is a weight 2 k holomorphic modularform on SL ( Z ). Using the Eisenstein series Serre constructed the p -adic Eisenstein series. Define(1.2) σ ( p ) k := X d | n gcd( d,p )=1 d k , and let ζ ( p ) ( s ) be the p -adic zeta function (see [12]). We now have that(1.3) G ( p )2 k ( z ) = 12 ζ ( p ) (1 − k ) + ∞ X n =1 σ ( p )2 k − ( n ) q n is a p -adic Eisenstein series of weight 2 k . Clearly there is a sequence 2 k i of positive even integersthat tends to 2 k p -adically and σ k i − ( n ) tends to σ ( p )2 k − ( n ) p -adically. The p -adic Eisenstein series Mathematics Subject Classification . 11F03, 11F37. are also classical modular forms on Γ ( p ) and can be written as G ( p )2 k ( z ) = G k ( z ) − p k − G k ( pz ) . This form is a p -stabilization of G k ( z ) so that G ( p )2 k ( z ) is an eigenform for the U p operator witheigenvalue coprime to p . The p -adic Eisenstein series satisfy incredible congruences; we have that G ( p ) k ( z ) ≡ G ( p ) k ( z ) (mod p a ) whenever k ≡ k (mod ( p − p a − ) and k and k are not divisible by p −
1. For example, 6 ≡
10 (mod 4) and 6 , G (5)6 ( z ) = 781126 + q + 33 q + 244 q + 1057 q + q + · · · , and G (5)10 ( z ) = 48828166 + q + 513 q + 19684 q + 262657 q + q + · · · are congruent modulo 5. The congruences can be explained using Kummer’s congruences and Euler’stheorem.Mazur recently raised the question of whether or not an eigencurve-like object exists in theworld of harmonic Maass forms. Harmonic Maass forms are traditionally built using methodswhich rarely lead to forms which are eigenforms (for background see [1]). Namely, the most wellknown constructions involve Poincar´e series, indefinite theta functions, and Ramanujan’s mock thetafunctions. These methods do not generally offer Hecke eigenforms. To this end, the first goal is toconstruct canonical families of harmonic Hecke eigenforms, out of which one hopes to be able toconstruct an eigencurve.Here we construct two families, one integer weight and one half-integer weight, of harmonic Maassforms which are eigenforms for the Hecke operators (see Section 2 for the definition of the relevantHecke operators). We define the weight k differential operator ξ k by ξ k := 2 iy k ∂∂z . The ξ -operator defines a surjective map from the space of weight 2 − k harmonic Maass forms on Γto the space of weight k weakly holomorphic modular forms on Γ (see [14]). A natural place to lookfor a suitable family of harmonic Maass forms is the pullback under the ξ -operator of the classicalEisenstein series that Serre used. The pullback, however, is infinite dimensional. For example, the ξ -operator annihilates weakly holomorphic modular forms. Therefore, the problem is to constructforms that are the pullback under the ξ -operator, and are also Hecke eigenforms and have p -adicproperties. Our first family will be a pullback of the classical Eisenstein series that saitsfies theseproperties. For Re(s) > >
0, let Γ( s, z ) := R ∞ z t s − e − t dt be the incomplete gammafunction . For k >
0, define G ( z, − k ) := (2 k )! ζ (2 k + 1)(2 π ) k + ( − k +1 y k k πζ ( − k − k + 1+ ( − k (2 π ) − k (2 k )! ∞ X n =1 σ k +1 ( n ) n k +1 q n + ( − k (2 π ) − k ∞ X n =1 σ k +1 ( n ) n k +1 Γ(1 + 2 k, πny ) q − n . (1.4)For half-integral weights, the analogue of the classical Eisenstein series are the Cohen-Eisensteinseries [3]. For more information on half-integral weight modular forms see [14]. Our family of formswill be a pullback of the Cohen-Eisenstein series under the ξ -operator. Define T χr ( v ) := X a | v µ ( a ) χ ( a ) a r − σ r − ( v/a ) , ARMONIC MAASS FORM EIGENCURVES 3 where µ ( a ) is the M¨obius function and r is an integer. Set ( − r N = Dv with D the discriminantof Q ( √ D ) and let χ D = (cid:0) D · (cid:1) be the associated character. Let(1.5) c r ( N ) = i r +1 L (1 + r, χ D ) v r +1 T χ D r +1 ( v ) N > i r − ζ (1 + 2 r ) + r +4 iπ r +1 y r + 12 ζ ( − − r )(2 r − r +1) N = 0 π / L ( − r,χ D ) T χDr +1 ( v ) N r + 12 Γ ( r + a ) Γ ( r +1+ a ) Γ ( r + ) Γ (cid:0) r + , − πN y (cid:1) N < , where a = 0 if r is odd and a = 1 if r is even. Then, for r ≥
1, define(1.6) H (cid:18) z, − r + 12 (cid:19) := X N ∈ Z c r ( N ) q N . Remark.
The coefficients for
N >
N < H (cid:0) z, − r + (cid:1) alternate between L -functions forreal and imaginary quadratic fields as r changes parity. The L -functions for real quadratic fieldsare known to encode information about the torsion groups of K -groups for real quadratic fields.Therefore, the functions H (cid:0) z, − r + (cid:1) create a grid that encodes this information for K n ( Q ( √ D ))as both n and D vary.For k ∈ R , the weight k hyperbolic Laplacian operator on H is defined by(1.7) ∆ k := − y (cid:18) ∂ ∂x + ∂ ∂y (cid:19) + iky (cid:18) ∂∂x + i ∂∂y (cid:19) = − y ∂∂z ∂∂z + 2 iky ∂∂z . A weight k harmonic Maass form on a subgroup Γ of SL ( Z ) is a smooth function f : H −→ C such that it transforms like a modular form of weight k on Γ, it is annihilated by the weight k hyperbolic Laplacian operator, and it satisfies suitable growth conditions at all cusps. In particular,we consider harmonic Maass forms with manageable growth, which are defined in Section 2 . ( N ) := ( (cid:18) a bc d (cid:19) ∈ SL ( Z ) : c ≡ N ) ) . For a more thorough background on harmonic Maass forms see [1]. We now have the followingtheorem.
Theorem 1.1.
Assuming the notation above, the following are true.(1) For positive integers k , we have that G ( z, − k ) is a weight − k harmonic Maass form on SL ( Z ) . Furthermore, G ( z, − k ) has eigenvalue p k +1 under the Hecke operator T ( p ) .(2) For positive integers r , we have that H (cid:0) z, − r + (cid:1) is a weight − r + harmonic Maass formon Γ (4) . Furthermore, H (cid:0) z, − r + (cid:1) has eigenvalue p r +1 under the Hecke operator T ( p ) .Remark. The proof of Theorem 1.1 will show that these forms can be viewed as two parameterfunctions in z and w where w is the weight of the form. Specializing w to − k for the integer weightcase and to − r + in the half-integral weight case produces two families of harmonic Maass Heckeeigenforms which define lines on two Hecke eigencurves. Remark.
Just as the weight 2 Eisenstein series is not a modular form, the weight 0 form here is nota harmonic Maass form. However, we will see that there is a weight 0 p -adic harmonic Maass formin the same way that there is a weight 2 p -adic Eisenstein series. Remark.
Integer weight non-holomorphic Eisenstein series have been studied before. For example,in [21] Zagier considers the form e G ( z, s ) = 12 X ( m,n ) ∈ Z ( m,n ) =(0 , y s | nz + m | s , WAGNER which transforms as a weight 0 modular form with respect to z is an eigenform of ∆ with eigenvalue s (1 − s ). This form plays an important role in the Rankin-Selberg method [15], [17]. Zagier shows thatit has a meromorphic continuation so that e G ∗ ( z, s ) = π − s Γ( s ) e G ( z, s ) satisfies e G ∗ ( z, s ) = e G ∗ ( z, − s ).The Maass lowering operator L = − iy ∂∂z takes a function that transforms like a modular form ofweight k to a function that transforms like a modular form of weight k −
2. Furthermore, if f is aneigenform for ∆ k with eigenvalue λ , then L ( f ) is an eigenform for ∆ k − with eigenvalue λ − k + 2(Chapter 5 of [1]). In particular, we can see that L k ( e G ( z, s )) = Γ( s + k )2Γ( s ) X ( m,n ) ∈ Z ( m,n ) =(0 , y s + k ( nz + m ) k | nz + m | s is an eigenform of ∆ − k with eigenvalue − ( s + k )( s − k − s = k + 1 makes theform harmonic and gives the same forms as the ones in Theorem 1.1 (1). Remark.
The case of r = 0 has been constructed by Rhoades and Waldherr in [16] using a slightlydifferent method. Their result can be recovered using the same method as in this paper and thensieving to suitably modify the Fourier expansion. The work of Rhoades and Waldherr follows up onwork of Duke and Imamoglu ( [6]) and Duke, Imamoglu, and T´oth ( [7]). In [6] Duke and Imamogluuse the Kronecker limit formula to construct a function which has values of L -functions at s = 1 forits Fourier coefficients. This function was the first example and the motivation for the work in [7]where Duke, Imamoglu, and T´oth construct forms of weight on Γ (4) whose Fourier coefficientsare given in terms of cycle integrals of the modular j -function. Remark.
The forms in part 1 of Theorem 1.1 behave nicely under the flipping operator (see [1]).Similar functions are studied by Bringmann, Kane, and Rhoades in [2].Serre used the classical Eisenstein series to build p -adic modular forms. In a similar way we canuse these harmonic Maass forms to build p -adic harmonic Maass forms. Definition. A weight k p -adic harmonic Maass form is a formal power series f ( z ) = X n ≫−∞ c + f ( n ) q n + c − f (0) y − k + X = n ≪∞ c − f ( n )Γ (1 − k, − πny ) q n , where Γ(1 − k, − πny ) is taken as a formal symbol and where the coefficients c ± f ( n ) are in C p ,such that there exists a series of harmonic Maass forms f i ( z ), of weights k i , such that the followingproperties are satisfied:(1) lim i −→∞ n − k i c ± f i ( n ) = n − k c ± f ( n ) for n = 0.(2) lim i −→∞ c ± f i (0) = c ± f (0). Remark.
Here lim i −→∞ n − k i c ± f i ( n ) = n − k c ± f ( n ) means v p ( n − k i c ± f i ( n ) − n − k c ± f ( n )) tends to ∞ andwe have that k is the limit of the k i in X .We will need a few definitions before describing our p -adic harmonic Maass forms. Let L p ( s, χ )be the p -adic L -function (see [10]) and define(1.8) T χ, ( p ) r ( v ) := X a | v gcd( a,p )=1 µ ( a ) χ ( a ) a r − σ ( p )2 r − ( v/a ) . Also define the usual p -adic Gamma function (see [13]) byΓ ( p ) ( n ) := ( − n Y Suppose p is prime and let Γ( · , · ) be a formal symbol. Then the following are true.(1) For each k ∈ X , we have that G ( p ) ( z, − k ) := Γ ( p ) (2 k + 1) ζ ( p ) (2 k + 1)(2 π ) k + ( − k +1 y k k πζ ( p ) ( − k − k + 1+ ( − k (2 π ) − k Γ ( p ) (2 k + 1) ∞ X n =1 σ ( p )2 k +1 ( n ) n k +1 q n + ( − k (2 π ) − k ∞ X n =1 σ ( p )2 k +1 ( n ) n k +1 Γ(1 + 2 k, πny ) q − n is a weight − k p -adic harmonic Maass form.(2) For each − r + ∈ X , let c ( p ) r ( N ) := i r +1 L p (1 + r, χ D ) v r +1 T χ D , ( p ) r +1 ( v ) N > i r − ζ ( p ) (1 + 2 r ) + r +4 iπ r +1 y r + 12 ζ ( p ) ( − − r )(2 r − ( p ) (2 r +1) N = 0 π / L p ( − r,χ D ) T χD, ( p ) r +1 ( v ) N r + 12 Γ ( p ) ( r + a ) Γ ( p ) ( r +1+ a ) Γ ( p ) ( r + ) Γ (cid:0) r + , − πN y (cid:1) N < . Then H ( p ) (cid:0) z, − r + (cid:1) = P N ∈ Z c ( p ) r q N is a weight − r + p -adic harmonic Maass form.Remark. Note that these forms enjoy congruences similar to the ones for p -adic modular formsbecause of the generalized Bernoulli number congruences. Congruences for the holomorphic partsrely on the existence of a p -adic regulator for L -functions. Remark. When k is an integer, the forms G ( p ) ( z, − k ) satisfy G ( p ) ( z, − k ) = G ( z, − k ) − G ( pz, − k ) , which is the analogue of the equation above that the classical p -adic Eisenstein series satisfy. Thisimplies that G ( p ) ( z, − k ) is a standard harmonic Maass form on Γ ( p ). We do not know a similarformula for the half-integral weight forms. Remark. Suppose p is a prime and consider an infinite sequence of even integers which p -adicallygo to zero (i.e. { p t } ∞ t =1 ). By the proof of Theorem 1.2, taking the p -adic limit of a series of formswith these weights defines a p -adic harmonic Maass form of weight 0. As noted above, this is theanalogue to the quasimodular form E which is not quite a modular form, but Serre showed leadsto a weight 2 p -adic modular form. In fact, the weight 0 p -adic harmonic Maass form constructedhere is the preimage of Serre’s weight 2 p -adic Eisenstein series under the ξ -operator. Remark. Theorem 1.2 implies that the Cohen-Eisenstein series are p -adic modular forms in the senseof Serre. This fact was proven by Koblitz in [11].Not much is known about harmonic Maass Hecke eigenforms except for the forms constructedhere. The fact that Hecke operators increase the order of singularities at cusps poses a majorroadblock in the study of harmonic Maass Hecke eigenforms. The forms constructed here standout because this issue doesn’t arise. It is an open question of Mazur to describe what the generalstructure of a “mock eigencurve” could be. For example, are there other branches of the mockeigencurve that connect together other harmonic Maass eigenforms? WAGNER In Section 2 we will give background knowledge on harmonic Maass forms and state some resultsof Zagier vital to the construction of the half-integral weight forms. Section 3 will be dedicated tothe construction of the forms in Theorem 1.1 and proving that they are Hecke eigenforms. Section4 will be used to discuss the p -adic properties of the forms in Theorem 1.2.2. Background on harmonic Maass forms and results of Zagier Basics of harmonic Maass forms. In this section we let z = x + iy ∈ H , with x, y ∈ R .We denote the space of weight k harmonic Maass forms on Γ by H k (Γ). The following details onharmonic Maass forms can be found in Chapter 4 of [1]. If the growth condition mentioned in thedefinition of harmonic Maass forms given above is given by f ( z ) = O ( e εy )as y −→ ∞ for some ε > 0, then we say that f is a weight k harmonic Maass form of manageablegrowth on Γ and we denote this space by H mg k (Γ).If f ( z ) is a weight k harmonic Maass form on a congruence subgroup, Γ ⊂ SL ( Z ), it can benaturally decomposed into its holomorphic part , f + ( z ), and its non-holomorphic part , f − ( z ). Theholomorphic part of a harmonic Maass form is often called a mock modular form . The Fourierexpansion of f also naturally splits as(2.2) f ( z ) = X n ≫−∞ c + f ( n ) q n + c − f (0) y − k + X n ≪∞ n =0 c − f ( n )Γ(1 − k, − πny ) q n . We have the following propostion about the action of the Hecke operators on f ( z ). Proposition 2.1 (Proposition 7 . . Suppose that f ( z ) ∈ H mg κ (Γ ( N ) , χ ) with κ ∈ Z . Thenthe following are true.(1) For m ∈ N , we have that f | T ( m ) ∈ H mg κ (Γ ( N ) , χ ) .(2) if κ ∈ Z , ǫ ∈ {±} , then, unless n = 0 and ǫ = − , c ǫf | T ( p ) ( n ) = c ǫf ( pn ) + χ ( p ) p κ − c ǫf (cid:18) np (cid:19) . Moreover, c − f | T ( p ) (0) = ( p κ − + χ ( p )) c − f (0) . (3) if κ ∈ Z \ Z , then, with ǫ ∈ {±} ( n = 0 for ǫ = − ), we have that c ǫf | T ( p ) ( n ) = c ǫf ( p n ) + χ ∗ ( p ) (cid:18) np (cid:19) p κ − c ǫf ( n ) + χ ∗ ( p ) p κ − c ǫf (cid:18) np (cid:19) , where χ ∗ ( n ) := (cid:18) ( − κ − n (cid:19) χ ( n ) . If n = 0 and ǫ = − , then we have that c − f | T ( p ) (0) = ( p − κ + χ ∗ ( p )) c − f (0) . Differential operators are an important tool for studying harmonic Maass forms. We will focuson the ξ -operator. Let M ! k (Γ ( N )) be the space of weakly holomorphic modular forms on Γ ( N )(see [14]). Proposition 2.2 (Theorem 5 . 10 of [1]) . For any k ≥ , we have that ξ − k : H mg2 − k (Γ ( N )) ։ M ! k (Γ ( N )) . In particular, for f ∈ H mg2 − k (Γ ( N )) , we have that ξ − k ( f ( z )) = ξ − k ( f − ( z )) = ( k − c − f (0) − (4 π ) k − X n ≫−∞ c − f ( − n ) n k − q n . ARMONIC MAASS FORM EIGENCURVES 7 The ξ -operator allows for a connection between the Hecke operators for harmonic Maass formsand modular forms (see [1]). In particular, we have that(2.3) p d (1 − κ ) ξ κ ( f | T ( p d , κ, χ )) = ξ κ ( f ) | T ( p d , − κ, χ ) , where d := ( κ ∈ Z , κ ∈ + Z . Results of Zagier. Several results of Zagier will be applicable to the construction of our forms.We will state them here. Proposition 2.3 (Zagier, [20]) . For positive integers a and c , let (2.4) λ ( a, c ) = i − c (cid:0) ac (cid:1) if c is odd, a even i a (cid:0) ca (cid:1) if a is odd, c even0 otherwise . Define the Gauss sum γ c ( n ) by (2.5) γ c ( n ) := 1 √ c c X a =1 λ ( a, c ) e − πin ac . Let n be a nonzero integer and define a Dirichlet series E n ( s ) by (2.6) E n ( s ) := 12 ∞ X c =1 c odd γ c ( n ) c s + 12 ∞ X c =2 c even γ c ( n )( c/ s , (i.e. E n ( s ) = P a m m − s where a m = ( γ m ( n ) + γ m ( n )) when m is odd, and a m = γ m ( n ) when m is even). Let K = Q ( √ n ) , D be the discriminant of K , χ D = (cid:0) D · (cid:1) be the character of K , and L ( s, χ D ) = P χ D ( n ) n s be the L -series of K (if n is a perfect square, then χ ( m ) = 1 for any m and L ( s, χ ) = ζ ( s ) ). Then if n ≡ , , we have E n ( s ) = 0 . If n ≡ , , we have (2.7) E n ( s ) = L ( s, χ D ) ζ (2 s ) X a,c ≥ ac | v µ ( a ) χ D ( a ) c s − a s = L ( s, χ D ) ζ (2 s ) T χ D s ( v ) v s − , where n = v D and (2.8) T χs ( v ) = X t | v t s − X a | t µ ( a ) χ ( a ) a s = X a | v µ ( a ) χ ( a ) a s − σ s − ( v/a ) . Furthermore, we have (2.9) E ( s ) = ζ (2 s − ζ (2 s ) . Remark. It is clear from Zagier’s proof in [20] that E n ( s ) can be continued to a meromorphic functionon the whole s -plane. It will also be beneficial to note that T χs ( v ) = v s − T χ − s ( v ).It will be useful for us to define(2.10) E oddn ( s ) = ∞ X c =1 c odd γ c ( n ) c − s , WAGNER and(2.11) E evenn ( s ) = ∞ X c =1 c even γ c ( n )( c/ − s , so that(2.12) E n ( s ) = 12 (cid:16) E oddn ( s ) + E evenn ( s ) (cid:17) . Proof of Theorem 1.1 Here we prove Theorem 1.1. There are two cases to consider, the integer weight and half-integralweight cases. In the next subsection we consider the integer weight case.3.1. Proof of Theorem 1.1 Part . We will construct the forms from Theorem 1.1 part 1 first.Let z ∈ H and k ∈ Z . Define(3.1) G ( z, − k, s ) := 12 X n,m ′ ( mz + n ) k | mz + n | s , where the primed sum means the sum runs over all ( n, m ) except (0 , G ( z, − k, s ) has a mero-morphic continuation to the whole s -plane. Let(3.2) f ( z, − k, s ) := ∞ X n = −∞ ( z + n ) k | z + n | − s . Then we have f ( z, − k, s ) = ∞ X n = −∞ h n, k ( s, y ) e πinz = ∞ X n = −∞ h n ( y, − k, s ) e πinx e − πny , where h n ( y, − k, s ) = Z iy + ∞ iy −∞ z k | z | − s e − πinz dz. After making the substitution z = yt + iy we have(3.3) h n ( y, − k, s ) = y k − s e πny Z ∞−∞ ( t + i ) k ( t + 1) − s e − πinyt dt. For n = 0, we have h ( y, − k, s ) = y k − s R ∞−∞ ( t + i ) k ( t + 1) − s dt . Following Zagier, we chooseour branch cut along the negative imaginary axis. Then using contour integration, we find that(3.4) h ( y, − k, s ) = 2 iy k − s e kπi sin( π ( s − k )) Z − i ∞− i | t + i | k − s | t − i | − s dt. We substitute t for − i (2 u + 1) to arrive at h ( y, − k, s ) = 2 k − s y k − s e kπi sin( π ( s − k )) Z ∞ u k − s ( u + 1) − s du. We make one more substitution, u = − vv . Then, we have h ( y, − k, s ) = 2 k − s y k − s e kπi sin( π ( s − k )) Z (1 − v ) k − s v s − k − dv = 2 k − s y k − s e kπi π Γ(2 s − k − s − k )Γ( s ) . (3.5)For n > 0, we define a path c as a clockwise path around − i from − i ∞ to − i ∞ . Then we have h n ( y, − k, s ) = y k − s Z c ( v + i ) k ( v + 1) − s e − πinyv dv. ARMONIC MAASS FORM EIGENCURVES 9 Substitute v for t − i and define the path c = c + i , then we have h n ( y, − k, s ) = y k − s e − πny Z c t k − s ( t − i ) − s e − πinyt dt. For n < 0, define the path c as before to circle i clockwise from i ∞ to i ∞ . Making the substitutions v = t + i and c = c − i , we arrive at h n ( y, − k, s ) = y k − s e πny Z c t − s ( t + 2 i ) k − s e − πinyt dt, for n < 0. Notice that h n ( my, − k, s ) = m k − s h mn ( y, − k, s ), so we have G ( z, − k, s ) = 12 ∞ X m = −∞ f ( mz, − k, s )= ζ (2 s − k ) + ∞ X m =1 f ( mz, − k, s )= ζ (2 s − k ) + ∞ X m =1 ∞ X n = −∞ m k − s h mn ( y, − k, s ) e πinmx . (3.6)We want to now look at the limit as s goes to zero in order to obtain a negative weight Eisensteinseries. However, it is clear that for any n ∈ Z , h n ( y, − k, 0) = 0 . Thus, our G ( z, − k, 0) functionswill also go to zero. In order to work around this we will look at the derivative of our Eisensteinseries with respect to s . Define(3.7) G ( z, − k ) := lim s −→ dds G ( z, − k, s ) . We will now calculate the q -expansion of G ( z, − k ). For n = 0, we have dds h ( y, − k, s ) | s =0 = y k k e kπi π Γ( − k − − k )= ( − k +1 y k k π k + 1 . (3.8)For n > 0, we have dds h n ( y, − k, s ) | s =0 = − y k e − πny Z c t k log( t ( t − i )) e − πinyt dt = − y k e − πny (2 πi ) Z − i ∞ t k e − πinyt dt = ( − k y k e − πny (2 π ) Z ∞ t k e − πnyt dt = ( − k (2 π ) − k n − k − Γ(2 k + 1) e − πny . (3.9)The log term jumps by 2 πi across the branch cut, while everything else is continuous. Similarly, for n < dds h n ( y, − k, s ) | s =0 = ( − k +1 (2 π ) − k n − k − e − πny Γ(1 + 2 k, − πny ) , Let h ′ n ( y, − k, 0) := dds h n ( y, − k, s ) | s =0 , then, from equation 2 . 6, we have(3.11) G ( z, − k ) = 2 ζ ′ ( − k ) + ∞ X n = −∞ h ′ n ( y, − k, σ k +1 ( n ) e πinx . Recall that σ k +1 (0) = ζ ( − k − show that G ( z, − k ) is a Hecke eigenform, notice that its image under the ξ -operator is a nonzeromultiple of the weight 2 k + 2 Eisenstein series, E k +2 ( z ). E k +2 is known to be an eigenform witheigenvalue σ k +1 ( p ) = 1 + p k +1 under the Hecke operator T ( p ). By equation (2 . 3) and inspectionit is clear that G ( z, − k ) is then an eigenform with eigenvalue 1 + p k +1 . (cid:3) Proof of Theorem 1.1 part . Let k = 2 r − r ≥ 1. We define the two Eisensteinseries F (cid:0) z, − k , s (cid:1) and E (cid:0) z, − k , s (cid:1) by(3.12) F (cid:18) z, − k , s (cid:19) = X n,m ∈ Z n> | m (cid:16) mn (cid:17) ε − kn ( mz + n ) k/ | mz + n | s , and E (cid:18) z, − k , s (cid:19) = (2 z ) k/ | z | s F (cid:18) − z , − k , s (cid:19) , where (cid:0) mn (cid:1) is the Kronecker symbol and ε n := ( if n ≡ i if n ≡ . A linear combination of these forms will have a meromorphic continuation to the whole s -plane andevaluating at s = 0 will give our weight − k form. We will abuse this fact by letting s = 0 in theassembly of the forms. We have(3.13) E (cid:18) z, − k , s (cid:19) = 2 k − s X n,m ∈ Z n> ,odd (cid:16) mn (cid:17) ε − kn ( nz − m ) k/ | nz − m | s . From this we have E (cid:18) z, − k , s (cid:19) = 2 k − s X n> ,odd ε − kn n k − s X m (mod n ) (cid:16) mn (cid:17) ∞ X h = −∞ (cid:0) z − mn + h (cid:1) k | z − mn + h | s = 2 k − s X n> ,odd ε − kn n k − s ∞ X N = −∞ X m (mod n ) (cid:16) mn (cid:17) α N (cid:18) y, − k , s (cid:19) e − πiNmn q N = ∞ X N = −∞ a ( N ) q N , where(3.14) a ( N ) = 2 k − s α N (cid:18) y, − k , s (cid:19) X n> ,odd ε − kn n k − s X m (mod n ) (cid:16) mn (cid:17) e − πiNmn , and by the Poisson summation formula α N (cid:18) y, − k , s (cid:19) = Z iy + ∞ iy −∞ z k | z | − s e − πiNz dz. Making the substitution z = yt + iy gives us α N (cid:18) y, − k , s (cid:19) = y k +1 − s e πNy Z ∞−∞ ( t + i ) k ( t + 1) − s e − πiNyt dt. ARMONIC MAASS FORM EIGENCURVES 11 Following Zagier, we choose the branch cut along the negative imaginary axis. Using contourintegration we have(3.16) α N (cid:18) y, − k , s (cid:19) = 2 e kπi sin (cid:18) π (cid:18) k − s (cid:19)(cid:19) y k +1 − s Z ∞ t k − s ( t + 2) − s e − πNyt dt. Letting s = 0 we arrive at α N (cid:18) y, − k , s (cid:19) = 2 e kπi sin (cid:18) πk (cid:19) y k +1 Z ∞ t k e − πNyt dt = 2 e kπi sin (cid:18) πk (cid:19) y k +1 (2 πN y ) − k − Z ∞ t k e − t dt = 2 e kπi sin (cid:18) πk (cid:19) (2 πN ) − k − Γ (cid:18) k (cid:19) , (3.17)for N > 0. If we evaluate the similar integral for N ≤ 0, because we do not cross a branch cut theintegral is zero. It will be useful to evaluate the derivative. We have that dds α N (cid:18) y, − k , s (cid:19) | s =0 = − y k +1 e πNy (2 πi ) Z i ∞ ( t + 2 i ) k e − πiNyt dt = − y k +1 πi k Z ∞ t k e πNyt dt = i − k (2 π ) − k N − k − Γ (cid:18) k , − πN y (cid:19) , (3.18)for N < 0, while dds α (cid:18) y, − k , s (cid:19) | s =0 = − − r i k y k +1 π r − . Similarly, for F (cid:0) z, − k , s (cid:1) we have F (cid:18) z, − k , s (cid:19) = 1 + X m> | m m k − s X n (mod m ) (cid:16) mn (cid:17) ε − kn ∞ X h = −∞ (cid:0) z + nm + h (cid:1) k | z + nm + h | s = 1 + X m> | m m k − s X n (mod m ) (cid:16) mn (cid:17) ε − kn ∞ X N = −∞ α N (cid:18) y, − k , s (cid:19) e πiNnm q N = 1 + ∞ X N = −∞ b ( N ) q N , where(3.19) b ( N ) = α N (cid:18) y, − k , s (cid:19) X m> | m m k − s X n (mod m ) (cid:16) mn (cid:17) ε − kn e πiNnm . Using Proposition 2.3 and by manipulating the inner sums of a ( N ) and b ( N ), it is not hard to showthat a ( N ) = 2 k − s α N (cid:18) y, − k , s (cid:19) X n> ,odd n k − s n X m =1 m even λ ( m, n ) e − πi ( − r N mn = 2 k − s α N (cid:18) y, − k , s (cid:19) X n> ,odd n k + − s γ n (( − r N ) = 2 k +1 − s α N (cid:18) y, − k , s (cid:19) E odd ( − r N (cid:18) − k − 12 + 2 s (cid:19) , (3.20)and b ( N ) = (1 + i r +1 )4 k + − s α N (cid:18) y, − k , s (cid:19) X m> m even γ m (( − r N )( m/ − k − +2 s = (1 + i r +1 )4 k + − s α N (cid:18) y, − k , s (cid:19) E even ( − r N (cid:18) − k − 12 + 2 s (cid:19) . (3.21)We are now able to define our forms. Define H (cid:18) z, − r + 12 (cid:19) = ∞ X N = −∞ c r ( N ) q N := lim s −→ ζ (1 + 2 r − s ) (cid:18) i r − F (cid:18) z, − r + 12 , s (cid:19) + 2 r − (1 + i r − ) E (cid:18) z, − r + 12 , s (cid:19)(cid:19) . (3.22)The rest of the construction is using Proposition 2.3. Similar calculations can be found in [3] or [21].Note that the functional equations for the zeta function and the L -function are used and that thereis pole when evaluating the non-holomorphic coefficients. The image of H (cid:0) z, − r + (cid:1) under the ξ -operator is a nonzero multiple of the weight r + Cohen-Eisenstein series. The weight r + Cohen-Eisenstein series is a Hecke eigenform with eigenvalue 1 + p r +1 under the Hecke operator T ( p ). Therefore, using equation (2 . H (cid:18) z, − r + 12 (cid:19) (cid:12)(cid:12)(cid:12) T ( p ) − (cid:18) p r +1 (cid:19) H (cid:18) z, − r + 12 (cid:19) is a weight − r + holomorphic modular form in the Kohnen plus space (see [14]). This space isempty and so H (cid:0) z, − r + (cid:1) must be a Hecke eigenform with eigenvalue 1 + p r +1 . (cid:3) Proof of Theorem 1.2 In order to discuss p -adic harmonic Maass forms we will first need to recall some facts aboutBernoulli numbers. Values of the Reimann zeta function at negative integers are tied to Bernoullinumbers. In fact we have ζ (1 − k ) = − B k k , and ζ ( − k ) = 0. In a similar way, there is a connectionbetween generalized Bernoulli numbers and the values of L -functions at negative integers. The generalized Bernoulli numbers B ( n, χ ) are defined by the generating function(4.1) ∞ X n =0 B ( n, χ ) t n n ! = m − X a =1 χ ( a ) te at e mt − , Where χ is a Dirichlet character modulo m . Generalized Bernoulli numbers are known to give thevalues of Dirichlet L-functions at non-positive integers. In fact, from [14] we know that if k is apositive integer and χ is a nontrivial Dirichlet character, then(4.2) L (1 − k, χ ) = − B ( k, χ ) k . This connection helps one define a p -adic L -function, L p ( s, χ ). The p -adic L -function is analyticexcept for a pole at s = 1 with residue (cid:16) − p (cid:17) . For n ≥ L p (1 − n, χ ) = − (1 − χ · ω − n ( p ) p n − ) B ( n, χ · ω − n ) n , where ω is the Teichm¨uller character. The Teichm¨uller character is a p -adic Dirichlet character ofconductor p if p is odd and conductor 4 if p = 2. It is best to view it as a p -adic object. For more ARMONIC MAASS FORM EIGENCURVES 13 information see Chapter 5 of [19]. Kummer famously showed that if n ≡ m (mod ( p − p a ) and( p − ∤ n, m for an odd prime p , then(4.4) (1 − p n − ) B n n ≡ (1 − p m − ) B m m (mod p a +1 ) , where a is a nonnegative integer. Similar congruences hold for generalized Bernoulli numbers aswell. For example, if we let χ = 1 be a primitive Dirichlet character with conductor not divisible by p , then if n ≡ m (mod p a ) we have(4.5) (1 − χ · ω − n ( p ) p n − ) B ( n, χ · ω − n ) n ≡ (1 − χ · ω − m ( p ) p m − ) B ( m, χ · ω − m ) m (mod p a +1 ) . Notice that twisting by the appropriate power of the Teichm¨uller character removes the dependenceon the residue class of n and m modulo p − p -adic harmonic Maass formscoming from the integer weight forms in Theorem 1.1 are constructed in the exact same way as the p -adic Eisenstein series in [18]. Equation (4 . 4) shows that the constant term, the p -adic zeta functionat a negative integer, will satisfy congruences. The other terms satisfy congruences due to Euler’stheorem which generalizes Fermat’s Little Theorem. The algebraic parts of these p -adic harmonicMaass forms enjoy similar congruences as their modular counterparts. In fact, the non-holomorphicparts are nearly identical to the p -adic Eisenstein series. The holomorphic parts behave not quiteas nicely only because the p -adic zeta function at positive integers does not behave as nicely as atnegative integers. However, it is still expected that it satisfies similar congruences modulo some p -adic regulator. For example, we have G + , (5) ( z, − 2) = − π (cid:18) ζ (5) (3) + q + 98 q + 2827 q + 7364 q + 175 q + · · · (cid:19) , while G + , (5) ( z, − 6) = − π (cid:18) ζ (5) (7) + q + 129128 q + 21882187 q + 1651316384 q + 178125 q + · · · (cid:19) . The family of p -adic harmonic Maass forms coming from the half-integral weight forms from Theorem1.1 are defined using p -adic L -functions and the fact that T χ, ( p ) r ( v ) is the p -adic limit of T χr ( v ). Asin the previous case, the non-holomorphic parts satisfy nice congruences due to equation (4 . p -adic regulator. Acknowledgements The author would like to thank Barry Mazur, J-P. Serre, Larry Rolen, Michael Griffin, KathrinBringmann, and ¨Ozlem Imamo¯glu for their comments on an earlier version of this paper. Theauthor would also like to thank Ken Ono for his numerous suggestions and the two referees for theircomments which improved the quality of this paper. References [1] K. Bringmann, A. Folsom, K. Ono, and L. Rolen. Harmonic Maass forms and mock modular forms: theory andapplications. Published by the American Mathematical Society, Providence, RI, 2017.[2] K. Bringmann, B. Kane, and R. Rhoades. Duality and differential operators for harmonic Maass forms. Develop-ments in Mathematics 28, 85-106, 2012.[3] H. Cohen. Sums involving the values at negative integers of L -functions of quadratic characters. MathematischeAnnalen 217, 271-285, 1975.[4] R. Coleman. p -adic Banach spaces and families of modular forms. Ivent. Math. 124, 215-241, 1996.[5] R. Coleman and B. Mazur. The eigencurve London Math. Soc. Lecture Note 254, 1-114, 1998.[6] W. Duke and ¨O. Imamo¯glu. A converse theorem and the Saito-Kurokawa lift. International Mathematics ResearchNotices, 7, 347-355, 1996.[7] W. Duke. ¨O. Imamo¯glu, and ´A. T´oth. Cycle integrals of the modular j -function and mock modular forms. Annalsof Mathematics, 173, 947-981, 2011.[8] H. Hida. Iwasawa modules attached to congruences of cusp forms. Annales scientifiques de l’´Ecole NormaleSup´erieure, 19, 231-273, 1986. [9] F. Hirzebruch and D. Zagier. Intersection numbers of curves on Hilbert modular surfaces and modular forms ofnebentypus. Invent. Math. 36, 57-113, 1976.[10] K. Iwasawa. On p -adic l -functions. Ann. of Math. 89, 198-205, 1969.[11] N. Koblitz. p -Adic congruences and modular forms of half integer weight. Mathematische Annalen, 274, 199-220,1986.[12] T. Kubota and H.W. Leopoldt. Eine p -adische Theorie der Zetawerte. J.Crelle, 214-215, 328-339, 1964.[13] Y. Morita. A p -adic analogue to the Γ-function. J. Fac. Sci. Tokyo, 22, 255-266, 1975.[14] K. Ono. The web of modularity: arithmetic of the coefficients of modular forms and q-series, volume 102 of CBMSRegional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences,Washington, DC; by the American Mathematical Society, Providence, RI, 2004.[15] R. Rankin. Contributions to the theory of Ramanujan’s function τ ( n ) and similar arithmetical functions. I. Thezeros of the function P ∞ n =1 τ ( n ) /n s on the line R s = 13 / 2. II. The order of the Fourier coefficients of integralmodular forms. Proc. Cambridge Philos. Soc. 35, 351-372, 1939.[16] R. Rhoades and M. Waldherr. A Maass lifting of θ and class numbers of real and imaginary quadratic fields.Mathematical Research Letters, 18, 1001-1012, 2011.[17] A. Selberg. Bemerkungen ¨uber eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbundenist. Arch. Math. Naturvid. 43, 47-50, 1940.[18] J.P. Serre. Formes modulaires et fonctions zˆeta pp