Arithmetic statistics of modular symbols

Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve $L$-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture in a refined version that also allows restrictions on the location of the cusps.


Introduction
Modular symbols are fundamental tools in number theory. By the work of Birch, Manin, Cremona and others they can be used to compute modular forms, the homology of modular curves, and to gain information about elliptic curves and special values of L-functions. In this paper we study the arithmetical properties of the modular symbol map (1.1) Here f ∈ S 2 ( ) is a holomorphic cusp form of weight 2 for the group = 0 (q), and {∞, a} denotes the homology class of curves between the cusps ∞ and a. For our purposes it is convenient to work with the real-valued, cuspidal one-form α = ( f (z)dz). The finite cusps a are parametrized by Q, so for r ∈ Q we write r = 2πi r i∞ α. (1.2) Note that the path can be taken to be the vertical line connecting r ∈ Q to ∞. If a is equivalent to ∞ under the -action such that {∞, a} = {∞, γ (∞)} for some γ ∈ we write the map (1.1) as γ, α := γ (∞) = 2πi where in the last expression we have replaced ∞ by any z 0 ∈ H * .
Mazur, Rubin, and Stein [31,46] have recently formulated a series of conjectures about the value distribution of r . We now describe these conjectures. Let E be an elliptic curve over Q of conductor q with associated holomorphic weight 2 cusp form f (z). We write the Fourier expansion of f at ∞ as It is a fundamental question in number theory to understand how often the central value of L(E, χ, s), vanishes, when χ runs over the characters of Gal(Q/Q).
Mazur, Rubin, and Stein used raw modular symbols. For r ∈ Q these are defined by This corresponds roughly to taking α in (1.2) to be the real or imaginary part of the 1-form f (z)dz. See Remark 2.1 for the precise statement. Modular symbols and the central value of twists of the corresponding L-function are related by the Birch-Stevens formula e.g. [35,Eq. 2.2], [32,Eq 8.6] that is τ (χ)L(E,χ, 1) = a∈(Z/mZ) * χ(a) a/m ± for a primitive character of conductor m (here the choice of ± corresponds to the sign of χ ). To understand the vanishing of L(E,χ, s) at s = 1 Mazur, Rubin, and Stein were led to investigate the distribution of modular symbols and theta constants. In this paper we investigate modular symbols but not theta constants.
Mazur, Rubin, and Stein studied computationally the statistics of (raw) modular symbols. Since f has period 1, the same is true for the modular symbols: r + 1 = r . They observed the behavior of contiguous sums of modular symbols, defined for each x ∈ [0, 1] by Based on their computations they defined g(x) = 1 2πi n≥1 (a(n)(e(nx) − 1)) n 2 , and arrived at the following conjecture.

Conjecture 1.1 (Mazur-Rubin-Stein) As c → ∞ we have
They added credence to this conjecture with the following heuristics. If we cut off the paths in (1.2) for modular symbols at height δ > 0, then because the left-hand side is a Riemann sum for the integral. The heuristics involves interchanging this limit with the limit as δ → 0.
In another direction Mazur and Rubin investigated the distribution of a/c for (a, c) = 1 as c → ∞. Define the usual mean and variance by where φ is Euler's totient function. They conjectured the following asymptotic behavior of the variance. Moreover, The constant C f is called the variance slope and the constant D f,d the variance shift. In Conjecture 1.2 the symmetric square L-function L(sym 2 f, s) is normalized such that 1 is at the edge of the critical strip. Moreover, the numerics suggest that the normalized raw modular symbols obey a Gaussian distribution law. In this paper we prove average versions of Conjectures 1.1 and 1.2 when we average over c. We only work with q squarefree. The restriction to q squarefree may not be necessary. On the other hand we can only work with averages over c and not individual c.
Moreover, we prove a refined version of Conjecture 1.3. We can restrict the rational number a/c to lie in any prescribed interval and we can restrict to rational numbers a/c with (c, q) a fixed number.
To prove these results we specialize to 0 (q) more general results on modular symbols for cofinite Fuchsian groups with cusps. Here is a statement of our results for 0 (q), when q is squarefree. Here A d,q , B q are explicitly computable constants given by (8.12). Theorems 1.4, 1.6, and 1.7 will follow rather easily from the following general theorems for finite covolume Fuchsian groups with cusps. Let a, b be cusps of , not necessarily distinct. Let α = ( f (z)dz), where f ∈ S 2 ( ) is a holomorphic cusp form of weight 2. We do not assume that f has real coefficients. Fix scaling matrices σ a and σ b for the two cusps. Define We denote the denominator of r by c(r ). We order the elements of T ab according to the size of c(r ) and define We define general modular symbols as The constant C f is the variance slope.
The formula for D f,ab is explicit but complicated, see (7.4). It depends on f , the period b a α and data from the Kronecker limit formula for the Eisenstein series for the cusps a and b. r ab C f log c(r ) ordered according to c(r ) have asymptotically a standard normal distribution, that is, for every a, b ∈ [−∞, ∞] with a ≤ b we have Remark 1. 12 We have obtained different but related normal distribution results for modular symbols in [36][37][38]40]. One difference between these papers and the current one is in the ordering and normalization of the values of γ, α . The orderings in these papers were more geometric (action of on H) and less arithmetic, and the modular symbols used closed paths in \H. However, in Theorem 1.7 we need to combine statistics from various cusps, since not all rational cusps are equivalent to ∞ for 0 (q). Moreover, we allow to restrict γ (∞) = a/c to a general I ⊆ R/Z. This is a new feature, making our current results significantly more refined. The expression for the variance shift in Theorems 1.6 and 1.10, see also (7.4), is very explicit. This is an unexpected facet. The analogue in [36,Theorem 2.19] involves also the reduced resolvent of the Laplace operator, which is much harder to understand.

Remark 1.13
To prove Theorem 1.11 we study the asymptotic kth moments of the modular symbols r ab for all k, see Theorem 7.5. We make no effort to optimize the error bounds for the moments in Theorems 1.9, 1.10, and 7.5, but they can all be made explicit in terms of spectral gaps. Remark 1.14 An important tool in this paper and in [36] is non-holomorphic Eisenstein series twisted with modular symbols E m,n (z, s). These were introduced by Goldfeld [15,16] and studied extensively by many authors, see e.g. [3,9,25,26,34] and the references therein. For their definition see (2.11). In [36] we used the Eisenstein series twisted with the kth power of modular symbols as a generating series itself to study the kth moment of modular symbols. In this paper we need to understand the nth Fourier coefficient of E (k) a (σ b z, s), which involves twists by the kth power of r ab . The reason why the results here are more arithmetic is because the Fourier coefficients of Eisenstein series and of twisted Eisenstein series encode arithmetic data and modular symbols. To isolate the nth coefficient we use inner products with Poincaré series.

Remark 1.15
The structure of the paper is as follows. In Sect. 2 we introduce the generating functions (Dirichlet series) L (k) ab (s, m, n) for the powers of modular symbols. We also introduce Poincaré series and Eisenstein series twisted by powers of modular symbols.
In Sect. 3 we analyze L ab (s, 0, n) for (s) > 1/2, and conclude with the statement that T ab is equidistributed modulo 1.
In Sect. 4 we study Eisenstein series twisted by modular symbols, or rather a related series D (k) a (z, s). We prove the analytic continuation for (s) > 1/2 and study the order of the poles and leading singularity at s = 1. The crucial identity is Eq. (4.4) that allows us to understand D (k) a (z, s) inductively using the resolvent of the Laplace operator R(s).
In Sect. 5 we find explicit expressions for the functional equations of D  ab (s, 0, n) for k ≥ 1. When k = 1 we find the residue at s = 1, and the whole singular part when k = 2. Finally, we identify the order of the pole and the leading singularity for all k.
In Sect. 7 we prove Theorems 1.9, 1.10, and 1.11 for general finite covolume Fuchsian groups with cusps. We use the method of contour integration.
In Sect. 8 we specialize the general results to 0 (q) for squarefree q. This leads to the proofs of Theorems 1.4, 1.6, and 1.7.

Remark 1.16
In a recent preprint [27] the authors prove that, when c → ∞ through primes, the limiting behavior in Conjecture 1.1 holds. Their method is different from ours.

Generating series for powers of modular symbols
From now on we allow to be any cofinite Fuchsian group with cusps. All implied constants in our estimates depend on and f . In this section we define a generating series for modular symbols, and explain how it can be understood in terms of derivatives of Eisenstein series with characters.

Modular symbols
The modular symbols defined by are independent of z 0 ∈ H * , and independent of the path between z 0 and γ z 0 . Fix a set of inequivalent cusps for . For such a cusp a we fix a scaling matrix σ a , i.e. a real matrix of determinant 1 mapping ∞ to a, and satisfying a = σ a ∞ σ −1 a . Here a is the stabilizer of a in , and ∞ is the standard parabolic subgroup. We have γ, α = 0 for γ parabolic since α is cuspidal.
For any real number ε and every cuspidal real differential 1-form α we have a family of unitary characters χ : → S 1 defined by Note that this character is the conjugate of the one we considered in [36]. We also need the antiderivative of α. We define We expand α at a cusp b. Let us assume that where j (γ , z) = cz + d. Then We consider the line integral for z ∈ H. Consequently, we have the estimate where α g = (g(z)dz). Consequently, taking α = ( f (z)dz) covers both cases of raw modular symbols.

The generating series
Let a, b be cusps and σ a , σ b fixed scaling matrices. Then we define It is easy to see that a/c mod 1 is well-defined for the double coset containing γ . Note also that γ ∞ = a/c mod 1. and r = γ ∞, r = γ ∞. We may assume that 0 ≤ a < c and 0 ≤ a < c . The matrix γ = γ −1 γ ∈ σ −1 b σ b has lower left entry −ac + a c. If this is zero then γ ∈ ∞ and r = r mod 1. If not, then by [24, Eq. (2.30)] we have −ac + a c ≥ c b , which implies that Therefore r = r and since 0 ≤ r, r < 1 this implies that r = r mod 1.
From Proposition 2.2 we conclude the following result: Corollary 2.3 Any r ∈ T ab determines a unique number c(r ) > 0 and unique cosets a(r ) mod c(r ), and d(r ) mod c(r ) satisfying a(r )d(r ) = 1 mod c(r ) and a(r )/c(r ) = r mod 1.
From Corollary 2.3 it follows that there exists a unique pair (c(r ), a(r )) such that 0 ≤ a(r ) < c(r ) and a(r )/c(r ) = r mod 1. We can therefore put an ordering on the elements of T ab by putting the lexicographical ordering on the (c(r ), a(r )) i.e.
For r ∈ T ab we definer mod 1 byr = d(r )/c(r ). For r ∈ T ab we define We suppress α from the notation. We notice that if γ ∈ ∞ \σ −1 a σ b / ∞ corresponds to r as in Proposition 2.2 we have which we will also refer to as a modular symbol. The map r → r ab does not grow too fast in terms of c(r ):

Proposition 2.4
The following estimate holds Proof We use (2.3) with σ a γ σ −1 b and a fixed z. Writing the lower row of γ as (c(r ), d(r )) we may assume that |d(r )| ≤ |c(r )|. Writing σ −1 b z = w, we use the elementary inequalities from which the result follows.
We can now define the main generating series: ab (s, m, n) is absolutely convergent for (s) > 1, and uniformly convergent on compacta of (s) > 1. It is the analytic properties of this series that will eventually allow us to prove our main results.

Relation to Eisenstein series
To explain how L (k) ab (s, m, n) relates to Eisenstein series we recall that the generalized Kloosterman sums are defined by For m, n ∈ Z we define where the sum is over c for * * c * ∈ σ −1 a σ b . This is a version of the Selberg-Linnik zeta function. When ε = 0, that is when the character is trivial, we omit it from the notation. Using [24,Prop. 2.8] we see that for (s) > 1 this is an absolutely converging Dirichlet series. Note that We now recover L When m = 0, i.e. in the case of the usual Eisenstein series we will often omit the subscript m. When we omit to specify ε in the notation, we have set ε = 0. For m > 0 it is known that E a,m (z, s, ε) ∈ L 2 ( \H) and that it admits meromorphic continuation to s ∈ C, see [19, p. 247]. The usual Eisenstein series has Fourier expansion at a cusp b given by (see e.g [44, p. 640-641]) As usual δ ab = 1 if a = b and is 0 otherwise. The derivatives of φ ab (s, ε) and φ ab (n, s, ε) in ε are given by ab (s, 0, n). (2.9) It follows that the series L (k) ab (s, 0, n) can be understood by understanding derivatives of Eisenstein series.
We consider the kth derivative in ε of E a (z, s, ε) at ε = 0, which we denote by E (k) a (z, s). It is easily seen that, when (s) > 1, we have This series is absolutely and uniformly convergent on compact subsets of (s) > 1, as can be seen by (2.3) and using the standard region of absolute convergence of the Eisenstein series E a (z, s). We note that for k ∈ N (2.10) Here the Fourier expansion is computed by termwise differentiation of the Fourier expansion of E a (z, s, ε) at the cusp b, which is allowed. Hence, the generating series L

Automorphic Poincaré series with modular symbols
While the generating series for the modular symbols appear as Fourier coef- a (z, s) are not automorphic when k > 0. They are higher order modular forms. Properties of such forms has been studied extensively in many papers such as [2,4,6,7,10,11,22,45].
Since 2 γ, α = γ, f (z)dz + γ, f (z)dz , we see that our E (k) (z, s) is indeed a linear combination of the Eisenstein series twisted by powers of γ, f (z)dz and γ, f (z)dz with m + n = k: For background on E m,n a (z, s) see Remark 1.14. For our purpose it is convenient to consider a related function, which has the advantage of also being automorphic: Using (2.2), and by comparison with the standard Eisenstein series, we see that the function D a,m (z, s) is absolutely and uniformly convergent on compact subsets of (s) > 1. It follows immediately that D (k) a,m (z, s) is -automorphic, and holomorphic for s in this half-plane.
We consider also As usual when m = 0 we omit it from the notation.
We now explain how E a,m (z, s) are related. We arrange our Eisenstein and Poincaré series in column vectors indexed by the cusps as follows: We define the diagonal matrix U (z, ε) with diagonal entries Let A(z) be the diagonal matrix with diagonal entries the antiderivatives 2πi z a α = A a (z). It follows from (2.13) by differentiation at ε = 0 that for (s) > 1 we have the vector equations Hence, we can freely translate between D

The generating series L ab (s, 0, n)
In this section we discuss the analytic properties of L ab (s, 0, n) for (s) > 1/2. In order to do so we first need some general bounds on Eisenstein series and Poincaré series with modular symbols.

Bounds on Eisenstein series
We first discuss a construction of Eisenstein series of weight k for σ = (s) > 1/2, generalizing the construction for weight 0 in [5,Section 2]. This approach is useful to estimate Eisenstein series in the cusps for be the Laplace operator of weight k and˜ k the closure of k acting on smooth, weight k functions such that f, k f are square integrable. We fix a fundamental domain F of , and notice that σ −1 a F is a fundamental domain for σ −1 a σ a . However, the spectral analysis of the k-Laplacian remains the same, since the manifold \H is isometric to

Lemma 3.1 Let k ∈ Z. Let h(y) be a smooth function that is identically 1 for y ≥ Y + 1 and identically 0 for y
Proof If such a solution exists we denote the left-hand side of (3.1) by g(z, s, k). We apply k + s(1 − s) to deduce is a compactly supported function, is independent of k, and is holomorphic in s.
We now define H (z, s) by (3.2), and apply to it the resolvent R(s, k) The standard inequality for the operator norm of the resolvent In particular (3.4) and the estimate [24, Corollary 3.5], which can be made uniform for (s) ∈ [1 + ε, A], y ≥ Y 0 . We estimate the right-hand side of (3.4) on the compact part F(Y ) of F and on the cuspidal zones see [24, p. 40]. For different cusps a and b, the matrix σ −1 a σ b has nonzero c bounded from below by c(a, b), see [24, Section 2.6]. We have (σ −1 a σ b z) σ ≤ y σ /(cy) 2σ y −σ . For a = b the terms y σ cancel.
As usual we define the Eisenstein series of weight k by Using Lemma 3.2 we see that This equation and Lemma 3.1 show that E a (z, s, k) agrees with the construction of Lemma 3.1. Therefore, the conclusions of Lemma 3.1 hold for the Eisenstein series of weight k in the region We can also estimate D a,m (z, s) defined in (2.12) using (2.2) and Lemma 3.2. We write If m > 0, and 1 + ε ≤ (s) ≤ A we see that the contribution of the identity term A a (z) k e(mσ −1 a z) (σ −1 a z) s decays exponentially at the cusp a i.e. exp(−(2π − ε)y) and is O(y −σ +ε ) at the other cusps. When m = 0, k > 0 we notice that in the cuspidal zone for a, the expansion (2.1) shows that A a (σ a z) decays exponentially. We can deduce that for all cusps b we have D where the implied constant depends only on k, , ε, and A.
Proof We use the result from Lemma 3.1 and the notation in its proof. We have We need to analyze also It suffices to concentrate on the cuspidal sector for a, since h( (σ −1 a z)) vanishes elsewhere. We have We use (3.5). When b = a, we see that

Meromorphic continuation and bounds on L ab (s, 0, n)
It is well known that the inner product of an automorphic function G and E b,n (z, s, ε) is directly related to the nth Fourier coefficient of G at the cusp b. We first use this to find an integral expression for L ab (s, 0, n) for n > 0.
Proof We unfold and, using (2.5), we find where we have used [18, 6.621.3, p. 700]. The result follows from the above computations and (2.7).
We can now use the integral expression in Lemma 3.4 to find the analytic properties of L ab (s, 0, n). admits meromorphic continuation to s ∈ C. For n = 0 the continuation is holomorphic at s = 1, while for n = 0 the continuation has a simple pole with residue res s=1 L ab (s, 0, 0) = 1 π vol( \H) .
For 1/2 + ε < (s) < 1 + ε, and s(1 − s) bounded away from spec(−˜ ), the following estimate holds: Proof It is well known that the Fourier coefficients of E a (z, s) admit meromorphic continuation to s ∈ C and the meromorphic continuation of the functions L ab (s, 0, n) follows from (2.6) and (2.7).
To analyze further when (s) > 1/2, we use Lemma 3.4 when n = 0. Using Proposition 2.6 we see that it suffices to consider n > 0. We let w = 1+2ε+it, where t = (s). With this choice of w Stirling's formula gives that the quotient of Gamma factors is O(|s| 1/2 ). The claim about growth on vertical lines follows from Lemma 3.3. For n > 0 the residue at s = 1 has 1, E b,n (z,w, 0) as a factor. But this vanishes by unfolding.
For n = 0 we examine (2.6). Using the well-known fact that φ ab (s) has a pole of order 1 at s = 1 with residue 1/vol( \H), the claim for the residue of L ab (s, 0, 0) follows. For the growth on vertical lines, we observe that Remark 3.6 The analytic properties of L ab (s, m, n, ε) for mn = 0 have been studied in [19].

Remark 3.7
If is the full modular group and a = b = ∞ with trivial scaling matrices then T ab = Q/Z so that in this case Here S(0, n, c) is the standard Ramanujan sum. We notice that in this case the bound in Lemma 3.5 on L ab (s, 0, n) for n ∈ Z is far from optimal.

Equidistribution of T ab
We can use Lemma 3.5 to observe that T ab is equidistributed modulo 1. The generating series of e(nr) for r ∈ T ab is L ab (s, n, 0). Proposition 2.6 allows us to understand its behavior through L ba (s, 0, − n). We define Using contour integration and the polynomial estimates on vertical lines for L ba (s, 0, − n) in Lemma 3.5 we deduce that, for some δ > 0 depending on the spectral gap for , These are the Weyl sums for the sequence T ab . As usual δ m (n) = 1 if n = m and is 0 otherwise and similarly for a set A, δ A (n) = 1 if n ∈ A and is 0 otherwise. Good studied the asymptotics of such Weyl sums in [17].
To state our results we need to introduce norms on L 1 -functions on R/Z: whereĥ(n) denotes the nth Fourier coefficient of h. We then have the following result: The sequence T ab is equidistributed modulo 1, i.e. for any con- Proof The first claim follows from Weyl's equidistribution criterion. For the second we write the Fourier series of h, which is absolutely convergent, and interchange the summation over n and over r ∈ T ab (M). We have Remark 3.9 In Theorem 3.8 the Sobolev norm h H 1/2 can be replaced by h H ε for any 0 < ε < 1/2, at the expense of a worse exponent δ = δ(ε). This can be done by making the appropriate changes in the contour integration argument leading to (3.6). Note that if we do so the exponent depends both on the spectral gap and on ε.
In Theorem 3.8 we are only looking at a subsequence of the left-hand side, namely only N equal to #T ab (M) for some M. It is possible to consider the whole sequence by noticing that it follows from (3.6) that This improves on the trivial bound in [24, (2.37)].

Eisenstein series with modular symbols
In this section we analyze further the series D (k) a,m (z, s) defined in (2.12), when m = 0. We will omit m from the notation and simply write D (k) a (z, s). We will show that this function admits meromorphic continuation and prove L 2bounds for it. Let δ( pdx + qdy) = −y 2 ( p x + q y ). Define Remark 4.1 These definitions are motivated by perturbation theory. Even if we are not using perturbation theory directly, as in our previous work [36,39], it is useful to have the following in mind. Let α be a closed smooth 1-form rapidly decaying at the cusps but not necessarily harmonic. We define unitary operators We also define where the automorphic LaplacianL(ε) is the closure of the operator acting on smooth functions f with f, f ∈ L 2 ( \H, χ(·, )). This ensures that L(ε) acts on the fixed space L 2 ( \H). It is then straightforward to verify that We notice that L(ε) does not depend on the cusp a. We observe that δ(α) = 0, if α is harmonic. From now on we assume α to be harmonic. We remark that L(ε) as a function of ε is a polynomial of degree 2, and that L (1) and L (2) defined in (4.1) and (4.
when k ≥ 1, where the last term should be omitted for k = 1.
so that d A a (z) = 2πiα. Using the product rule we have that We recognize the first term as The other two terms give This proves that Now we automorphize this equation over γ ∈ a \ , and use that T γ = T γ . We notice that for any two differential forms ω 1 , ω 2 we have for the action of on functions T γ ω 1 , ω 2 = γ * ω 1 , γ * ω 2 .
Using that α is invariant under , we arrive at the result.
is a bounded operator on L 2 ( \H, dμ(z)) satisfying (3.3) and where P 0 is the projection to the eigenspace of˜ for the eigenvalue 0, and R 0 (s) is holomorphic in a neighborhood of s = 1.
Before we can prove the meromorphic continuation of D (k) a (z, s) we need a small technical lemma about Eisenstein series: are the raising and lowering operators as in [13,Eq. (3)]. It follows that Applying this with h = E a (z, s) we see that it suffices to study y f (z)K 0 E a (z, s) and y f (z)L 0 E a (z, s). We analyze the first and notice that the analysis of the latter is similar. We have K 0 E a (z, s) = s E a (z, s, 2), where E a (z, s, 2) is the Eisenstein series of weight 2, see [42,Eq. (10.8), (10.9)]. Using that f decays exponentially at all cusps the claim now follows easily from Lemma 3.1.
We are now ready to prove the meromorphic continuation of D

Recall that for a twice differentiable function
, we use Lemma 4.2 for k = 1 and elliptic regularity to conclude that D (1) a (z, s) ∈ C ∞ ( \H). By (4.7) the function L (1) D (1) a (z, s) is smooth as well. Using Lemma 4.2 for k = 1 again we deduce that D (1) a (z, s) ∈ L 2 ( \H). We find from (4.7) and the estimate y f (z) ∞ 1 that it suffices to estimate K 0 D (1) a (z, s) L 2 and L 0 D (1) a (z, s) L 2 . We use (4.8) and Lemma 4.2 to conclude where we have used Cauchy-Schwarz and Lemma 4.3. This proves the case k = 1. Assume now that the claim has been proved for every l < k. Then for s) is meromorphic, smooth, and square integrable. Hence, by (4.4), the mapping properties of the resolvent, and (3.3), we find that D

By elliptic regularity and Lemma 4.2 it follows that D (k)
a (z, s) ∈ C ∞ ( \H). Therefore, the function L (1) D (k) a (z, s) is also smooth. We now use (4.7), (4.8), and Lemma 4.2 to see that This completes the inductive step.
when k ≥ 1, where the last term should be omitted for k = 1. Proof Applying Stokes' theorem, as in e.g. [36, p. 1026], we find that L (1) is a self-adjoint operator. Alternatively, we notice that L (1) is the infinitesimal variation of the family of self-adjoint operators L(ε) given in Remark 4.1. Therefore, for k ≥ 1 we have is a differentiation operator, see (4.1). The case k = 0 involves E a (z, s), which is not square integrable. This is easily compensated by the fact that α is cuspidal.

From Corollary 4.5 it is evident that D (k)
a (z, s) has singularities at the spectrum of . We now describe the nature of the pole at s = 1.

Proposition 4.7
The functions L (1) E a (z, s) and D (1) a (z, s) are regular at s = 1.
a (z, s) = E a (z, s) has a constant residue and L (1) is a differentiation operator. It then follows from (4.5) that D (1) a (z, s) can have at most a simple pole at s = 1. By (4.5) and Proposition 4.6 we see It follows that D  To run an inductive argument we assume that the claim has been proved up to some k odd. Then k +1 is even and we see from (z, s) has a pole or order 1 + (k − 1)/2 + 1 = 1 + (k + 1)/2 with leading coefficient

If k is odd, then D
Here we have again used (4.5). It follows from Corollary 4.5 that D (k+1) a (z, s) has a pole of order 1 + (k + 1)/2 with leading coefficient . (4.10) We observe that 1, L (2) 1 L 2 = −8π 2 f 2 . The order of the pole and leading singularity of D (k+1) a (z, s) agrees with the claim of the theorem. We now prove that D (k+2) a (z, s) has at most a pole of order (k +1)/2. We use Corollary 4.5 for k +2. Since L (1) annihilates the leading term in D a (z, s) has at most a pole of order (k + 1)/2. This concludes the inductive step.
In this section we show that D (k) (z, s) and E (k) (z, s) have analogous properties.
Recall the weighted L 2 -spaces in [33, p. 573]. We choose a smooth function can be defined as a bounded operator on weighted L 2 -spaces for s away from its poles. This is achieved by first continuing meromorphically the resolvent kernel (automorphic Green's function) r (z, z , s) to C. The analytic continuation of the resolvent kernel r (z, z , s) to C satisfies the limiting absorption principle: see [28, p. 352]. We choose 0 < δ < 2π so that e δρ(z) y| f (z)| is decaying exponentially at the cusps.
a (z, s) admits meromorphic continuation to C. Moreover, we have  s) is a linear combination of E m,n a (z, s) for m + n = k, its derivatives satisfy the same upper bounds. By (2.14) the functions D (k) a (z, s) are also smooth. This proves claim (i).
For the claims (ii), (iii) we need also to control the derivatives ∂ a+b /∂z a ∂z b of A a (σ b z) j . For a + b = 0, we use (2.2). For a + b ≥ 1 we have exponential decay by (4.3). We conclude that The claim (ii) follows by taking a = b = 0, and the claim (iii) by noticing that K 0 and L 0 in (4.6) are expressed in terms of ∂/∂ x and ∂/∂y. Finally, claim (iv) follows from (4.7) using the cuspidality of f (z) at all cusps.

s) meromorphic matrices given by
Proof For notational purposes we define D (−1) (z, s) to be 0. Using Corollary 4.5 we get for k ≥ 1, and (s) > 1/2. We can extend the validity of this equation to C, since the resolvent is applied to a function belonging to L 2 δ ( \H), which follows from Lemma 5.1. Furthermore we see that D Assume the result for m < k. We define the matrix (k) (s) indexed by the cusps by (5.4). Then where we have used (5.2) and (5.5). For the first term above we now use the inductive hypothesis and finally (5.5) to see that it equals This completes the inductive step.  Proof Considering (5.4) it suffices to show that for every k we have For (5.6) we use Stokes' theorem, as in the proof of Proposition 4.6, and bound By Theorem 4.4, Lemma 4.3, and Cauchy-Schwarz this is bounded by a constant times |s|.
For (5.7) we note that we can move L (2) in front of E b (z, s), since L (2) is a multiplication operator. Since L (2)

Functional equation for E (k) a (z, s)
Let k be a natural number. Using (2.15) and (2.14) we can find the functional equation for E (k) (z, s) as follows: Setting r = j + h + l we see that

If we set
We can rewrite this to see that We emphasize that (r ) ab (s) does not depend on z. We also note that if there is only one cusp we have (r ) (s) = (r ) (s).
Looking at the a-entry of (5.8) and its Fourier expansion (2.10) at the cusp b we get for the zero Fourier coefficients: ab (s). We summarize the results for E (k) (z, s).

Theorem 5.4 The vector of Eisenstein series twisted by modular symbols E (k) (z, s) satisfies the functional equation
ab (s) are given by (5.4). Moreover, (σ b z, s), see (2.10).   The matrices (k) (s) satisfy the functional equation cf. [25,Th. 2.2]. For k = 0 this is due to Selberg and for k ≥ 1 it follows by comparing the coefficient of y s in (5.9). We notice also that for 1/2 + ε ≤ (s) ≤ A with s bounded away from the spectrum we have

Meromorphic continuation
If k = 0 we obtained the meromorphic continuation of L (k) ab (s, 0, n) in Lemma 3.5. For k ≥ 1 we consider first the case n = 0. From (2.8), and Theorem 5.4 we find that The right-hand side is meromorphic by Theorem 5.2. If n ≥ 1 we deduce from Lemma 3.4 and (2.13) that, for (s) > 1, (w) > 1, We differentiate to get The differentiation is allowed and the right-hand side is meromorphic for (s) > 1/2 + ε by Theorem 4.4, Lemma 3.3 and the fact that D b,n (z, w) is bounded for (w) > 1. Using Proposition 2.6 we can deal also with n ≤ − 1. To summarize we have proved the following result: Theorem 6.1 For any cusps a, b and any integers k ≥ 0, n ∈ Z the function L (k) ab (s, 0, n) admits meromorphic continuation to (s) > 1/2 + ε.

The first derivative
We now study in more detail the analytic properties of L (1) ab (s, 0, n).
Proof Using Proposition 2.6 the claim for n ≤ − 1 follows from the case n ≥ 1. So we can assume that n ≥ 1. Consider (6.2) when k = 1. For (w) ≥ 1 + 2ε fixed, the functions F(s, w, n) is holomorphic as long as (s) > 0, so we must analyze the three expressions To analyze (6.3) we get by Lemma 3.3 There is a pole of the Eisenstein series D a (z, s) at s = 1, which gives rise to a residue for (6.3) To see that this vanishes we unfold the integral as in the Rankin method. The integrand contains the factor e(nσ −1 b z), with n = 0, and we notice that, as γ varies over the cosets b \ , the sets σ −1 b γ F cover the strip {z ∈ H : (z) ∈ [0, 1]}.
To analyze (6.4) we note that by Theorem 4.4, Proposition 4.7 and (3.5) the term is holomorphic at s = 1 and, for s bounded away from the spectrum, satisfies \H D (1) a (z, s)D b,n (z,w)dμ(z) ε 1.
Finally, we analyze (6.5). Since the Eisenstein series D a (z, s) has a pole at s = 1 we find that the integral has a simple pole at s = 1 with residue where we have unfolded using (2.12) and (2.1). We notice also that by Lemma 3.3 we have \H D a (z, s)D (1) b,n (z,w) dμ(z) ε 1.
Using the above analysis of the three integrals we can finish the proof for n > 0 as follows. Observing that F(1, w, n) = (4π n) w−1 /( (w − 1)π) we get the residue at s = 1 for n > 0. For the growth on vertical lines we choose (w) = (s) and use Stirling's asymptotics to get The bound on vertical lines now is obvious when we choose (w) = 1 + 2ε. As far as L (1) (s, 0, 0) is concerned we use (6.1) with k = 1, which leads to analyze ab has a pole, it must be a simple pole with residue a constant times L (1) D a (z, 1)dμ(z). This vanishes by Proposition 4.6 so (1) ab is regular at s = 1. The conclusion follows.

The second derivative
We will now describe the full singular part of L (2) ab (s, 0, 0) at s = 1. We denote the constant term in the Laurent expansion of E a (z, s) by B a (z), i.e.
as s → 1. For = PSL 2 (Z), the function B ∞ (z) can be described in terms of the Dedekind eta function (Kronecker's limit formula). For general groups the function B a (z) is given in terms of generalized Dedekind sums, see e.g. [14].
Proof Using (6.1) we see that L (2) ab (s, 0, 0) equals ab (s) . (6.8) We consider each of the three terms separately: We start by noting that since (0) (s) = (s), the first term has singular part To analyze the second term we note that by Theorem 5.2 we have which is regular by Proposition 4.7 and Proposition 4.6.
To analyze the third term we note that by Theorem 5.2 (2) and analyze the contribution of the two summands. Since D (1) a (z, s) is regular at s = 1 (Proposition 4.7), the first summand has at most a first order pole.
The corresponding residue is zero by Proposition 4.6, so the first summand is regular.
It follows that the singular part of (2) ab (s) equals the singular part of (2s − 1) −1 \H E b (z, s)L (2) E a (z, s)dμ(z). It follows that the singular part of the third term of (6.8) equals the singular part of The result follows using (6.7), (4.2), and standard values of (z)/ (z) [18, 8.366].
The bound on vertical lines follow from Theorem 5.3 and Stirling's asymptotics.

Remark 6.4
We remark that Theorem 6.3 allows us to write the singular expansion of L (2) ab (s, 0, 0) exclusively in terms of data of Rankin-Selberg integrals and periods. Indeed, writing so that the singular expansion of L (2) ab (s, 0, 0) equals

Higher derivatives Theorem 6.5 If k is even the function L (k)
ab (s, 0, 0) has a pole at s = 1 of order k/2 + 1. The leading term in the singular expansion around s = 1 equals b,n (z,w, ε) dμ(z), (6.9) where g(z) is one of the coefficients in the singular expansion of D

If k is odd the function L
a (z, s). For k 2 = 0 the function g(z) is constant so, in particular, is square integrable. For k 2 > 0 we note that for some j ≥ 0 and sufficiently small r . Here C(r ) is the circle centered at 1 with radius r . The radius r is chosen so that there are no other singularities of D (k 2 ) (z, s) inside C(r ) apart from s = 1. It follows from Theorem 4.4 that g(z) is square integrable. By using Cauchy-Schwarz we see that (6.9) is bounded independently of n. See again the discussion after (3.5).

Distribution results
We are now ready to prove Theorems 1.9, 1.10 and 1.11. Since we have identified the behavior of the generating functions L (k) ab (s, 0, n) at s = 1 and on vertical lines in Sect. 6, we can use the well-known method of contour integration to deduce the asymptotics of r ab .

First moment with restrictions
In this subsection we study sums of the form where F ab is the formal series given by (a a (n)e(nt)) n .
Proof The generating series of r ab e(nr) is L (1) ab (s, n, 0). Writing h(t) = n∈Zĥ (n)e(nt), we have by Proposition 2.6, Theorem 6.2, and a complex integration argument that r ab e(nr) We deduce from Theorems 7.4 and 3.8 that

Normal distribution
In this subsection we show that the value distribution of modular symbols (appropriately normalized) obeys a standard normal distribution, even if we restrict r to any interval.
Proof We use Proposition 2.6, Theorem 6.5, and Theorem 6.6. We apply a complex integration argument in a strip of width ε around (s) = 1 to deduce that ). (7.5) We insert this in and useĥ(0) = R/Z h(t)dt to get the result. The degree of P k,m is strictly less than k/2 if either m = 0 or k is odd, and exactly k/2 for k even and m = 0. In Theorem 7.4 we identify this polynomial when k = 2 and m = 0.
Using a standard approximation argument based in Theorem 7.5 we arrive at the following corollary: Corollary 7.7 Let I ⊆ R/Z be an interval and let k ∈ N. Then The above corollary allows us to renormalize the modular symbol map and determine the distribution of the renormalized map using the method of moments:

Results for Hecke congruence groups
In this section we translate the distribution results of Sect. 7 to the case of Hecke congruence groups = 0 (q), where q is a squarefree integer. In this case the cusps of and their scaling matrices can be described as follows, see [8,Section 2.2]): a complete set of inequivalent cusps of 0 (q) are given by a d = 1/d with d|q. Notice that if d = q then 1/d is equivalent to the cusp at infinity. Write q = dv. We may take for the corresponding scaling matrix. It follows that Using this and definition (1.5) we easily see that Therefore, where r is as in (1.2).

First moment with restrictions
From Theorems 7.2 and 3.8 we now deduce the following corollary. We can also remove the condition (a, c) = 1, by summing according to (a, c) = k. Using that ζ(2) = π 2 /6 we find that

The variance
Similarly to the analysis above we can use Theorem 7.4, Corollary 7.3, and Theorem 3.8 to conclude the following corollary.

Relation with the symmetric square L-function
We explain how to relate C f , and D f,d to the symmetric square L-function. We recall the definitions but refer to [21, for additional details. Assume that f is a Hecke eigenform normalized with first Fourier coefficient equal to 1, and let λ f (n) be its nth Hecke eigenvalues. Let Using these definitions, the formula vol( 0 (q)\H) = (π/3)q p|q (1 + p −1 ), and that ζ(s) has a simple pole at s = 1 with residue 1, we find that . (8.6) This verifies the variance slope of Conjecture 1.2.
To express in more arithmetic terms the constant D f,d we notice that is the constant term in Laurent expansion at s = 1 of by definition of B ∞ (z). By unfolding we see that the last integral equals To understand the part of D d involving B 1/d we recall the Atkin-Lehner involutions [1]. We also recall that we assume that q is squarefree. For every d|q there exists an integer matrix of determinant d of the form with y, w ∈ Z. It is straightforward to verify that W d normalizes 0 (q), that is, W −1 d 0 (q)W d = 0 (q). Since f is assumed to be a Hecke eigenform it follows from Atkin-Lehner theory that d · j (W d , z) −   , . (8.12) This completes the proof of Theorem 1.6.

Numerical investigations
As an example we consider the elliptic curve 15.a1. We computed L(sym 2 f, 1) = 0.9364885435 and L (sym 2 f, 1) = 0.03534541 using lcalc in Sage [43] and the data from [29]. 1 These numbers should be accurate to at least 6 decimal places. This allows to estimate the value of the variance shift D f,d and compare with the experimental values in Table 1. We would like to thank Karl Rubin for providing us with the experimental values of the variance shift. Notice that these are the opposite of what appears in [31], because our modular symbols are purely imaginary.

Normal distribution
With the description of the modular symbols and T ∞ 1 d from (8.2), (8.3), and (8.4), we can apply Corollary 7.7 to compute the moments of r . Since r = a √ q/d c √ q/d ∈ T ∞ 1 d has c(r ) = c √ q/d, we may use log c(r ) = log c+2 −1 log(q/d) and Remark 7.9 to conclude the following corollary. This completes the proof of Theorem 1.7.