Convergence to the Plancherel measure of Hecke Eigenvalues

We give improved uniform estimates for the rate of convergence to Plancherel measure of Hecke eigenvalues of holomorphic forms of weight 2 and level N. These are applied to determine the sharp cutoff for the non-backtracking random walk on arithmetic Ramanujan graphs and to Serre's problem of bounding the multiplicities of modular forms whose coefficients lie in number fields of degree d.


Introduction
It is well known that the distribution of Hecke eigenvalues of modular forms at primes p 1 , . . ., p r converges to the product of the corresponding p-adic Plancherel measures as one varies over certain families ( [Sar ], [Ser ], [CDF ]).Our aim in this paper is to establish uniform rates on this convergence and to apply these to problems of sharp cuto for random walks on Ramanujan graphs ( [NS ]) and to the factorization of the Jacobian of the modular curve X 0 (N) as in [Ser ].
The Eichler-Selberg trace formula expresses the trace of the Hecke operator T n on the space S(N) := S 2 (N) of holomorphic cusp forms of weight 2 for Γ 0 (N) in terms of class numbers of binary quadratic forms.Using this, one can show ( [Ser ], Prop.4) that as f runs over a Hecke basis H(N) of such eigenforms with eigenvalues we will have where δ N (n, ) := 1 if n is a square modulo N, 0 otherwise, d(N) := d|N 1 ≪ ε N ε is the divisor function, and ψ(N) := N p|N (1 + 1/p) is the Dedekind psi function.Murty and Sinha ([MS ]) give explicit and e ective bounds in ( ).
To extend the range of n for which the left-hand side of ( ) goes to 0 with N, we introduce and remove 1/L(1, Sym 2 f ) weights into the sum.This allows us to use the Petersson trace formula, replacing class numbers with Kloosterman sums, which enjoy sharp bounds coming from the arithmetic geometry of curves ( [Wei ]).This technique applied to a similar problem is outlined in [Sar ] and is used on other problems in [Iwa ] and [ILS ].It allows us to double the exponent in the range of n, which is crucial for certain applications.
In what follows, our aim is to establish sharp estimates, and to simplify the analysis, we assume that N is prime.In much of what we do, this assumption can be removed.We address this further in the nal section .
Theorem .Let ε > 0 and let m, n be integers coprime to N. Then: With this theorem, we can formulate and prove a corresponding uniform convergence to the Plancherel measure.Let r ≥ 1, and for ℓ 1 , . . ., ℓ r ≥ 0, let P ℓ 1 ,...,ℓr denote the set of polynomials in x 1 , . . ., x r of degrees at most ℓ 1 , . . ., ℓ r , respectively, that is, (such a θ f exists because of self-adjointness of T p and thanks to the Ramanujan bound |λ f (p)| ≤ 2 due to Eichler [Eic ]).Let µ p be the p-adic Plancherel measure: We have the following uniform convergence result: Theorem .Let r ≥ 1 and η > 0. Then uniformly for p ℓ 1 1 • • • p ℓr r < N 2−η and P ∈ P ℓ 1 ,...,ℓr , This result with an exponent of N larger than 1 (which corresponds to mn going up to N 2−ε in Theorem ) is what is needed to settle the cuto window for the non-backtracking random walks on Ramanujan graphs ( [NS ]).In fact, it yields the conjectured asymptotics for the variance for these walks (see end of section ).
Another application of Theorem is to multiplicities of f 's in a Hecke basis with given For a xed y, Theorem implies that uniformly in ϕ, where r = π(y) is the number of primes up to y.If y is allowed to increase with N, then one can exploit that for f in the set de ning M N (y, ϕ), we also have λ f (m) = λ ϕ (m) for all y-smooth numbers m (which are numbers all of whose prime factors are at most y).This allows one to improve ( ) vastly.
Such an argument using the large sieve for Dirichlet characters is due to Linnik ([Lin ]).In the modular form setting, Duke and Kowalski ([DK ]) establish that the number of non-monomial newforms of square-free level up to N that have prescribed eigenvalues λ ϕ (p) at primes p ≤ y = (log N) β satis es M ≤N (y, ϕ) # ≪ β N 10/β+ε , which is non-trivial for β > 5. Lau and Wu ([LW ]) show that for y = C log N with C a large constant, there is c > 0 s.t.
We apply this to a question of Serre ([Ser ]).Assume that all f = n≥1 a(n)e(nz) ∈ H(N) are normalized so a(1) = 1.The Fourier coe cients a(n) are algebraic integers in a totally real eld of degree d(f ).For d ≥ 1, let s(N) d denote the number of f 's for which d(f ) = d.Serre shows that for d xed, s(N) d = o(s(N)) (see also [MS ], Theorem 5), and asks for stronger upper bounds.Theorem implies such a bound.

Weight Removal in the Petersson Formula
Throughout this section we assume N is prime.Let H(N) denote a simultaneous eigenbasis of Hecke operators T k , (k, N) = 1, acting on the space S(N) of dimension s(N) of weight 2 level N cusp forms for Γ 0 (N), and for f ∈ H(N), let a f (n) and λ f (n) be such that , where e(z) := e 2πiz .Assume f are normalized so a f (1) = 1.
Our starting point for this section is the Petersson trace formula estimated via the Weil bound on Kloosterman sums, as presented in [ILS ], Corollary 2.2 or [IK ], Corollary 14.24: Petersson Formula.With H(N) as above, (mn, N) = 1, and ε > 0, Here the h superscript signifies adding "harmonic" weights: We derive Theorem from the Petersson formula by removing the harmonic weights.The Petersson norm is related to the special value of the symmetric square L-function at 1 ([ILS ], Lemma 2.5) via: and to prove Theorem , we need to derive a suitable approximation for L(Sym 2 f, s).
For a parameter x > 0, let by the Mellin inversion theorem.Shifting the integral de ning A to Re(s) = −1/2 picks up the simple pole of Ψ at s = 0, so by the residue theorem, According to the Lindelöf on average result due to Iwaniec and Michel for this family of L-functions Substituting ( ) and ( ) into ( ), where where we used Cauchy-Schwartz and ( ).
To estimate I, we use Hecke relations and the formula From this, so by the Petersson formula, Combining estimates of I and II with ( ), Choosing x = N (mn) 1/4 , we nish the proof of Theorem .
sin θ be the nth Chebyshev polynomial of the second kind.U n is a degree n polynomial in cos θ with real coe cients, so we can nd a t 1 ,...,tr ∈ C such that so by Theorem , and Applying Cauchy-Schwartz and summing the geometric series, Let µ ∞ (θ) := 2 π sin 2 θdθ be the Sato-Tate measure on [0, π].From the de nition of dµ p , it follows that Hence, from ( ) and the orthonormality of U n with respect to dµ ∞ , we have We conclude that It remains to evaluate I. To interpret I as the integral against the Plancherel measure, we need the following observation: Proposition . .Let m, n ≥ 0, and let dµ p be the p-adic Plancherel measure.Then: if m ≡ n (mod 2) 0 otherwise.

( )
We leave the proof until the end of the section.From ( ), We substitute the inner product ( ) into ( ).Observe that Combining I and II, Suppose now that the conditions of Theorem are met, i.e., )) = P 2 µ 1 ,...,µr (1 + o(1)) for small enough ε.This concludes the proof of Theorem .
Proof of Proposition . .From the trigonometric identity for the product of sines, +sin 2 θ dθ.
Since sin 2 (θ) = sin 2 (π − θ) and cos(k(π − x)) = − cos kx for odd k, I(k) = 0 for odd k, so the integral is 0 when m and n have di erent parity.To prove the proposition, it su ces to show that for all integers T ≥ 0, We prove this statement by induction.Let ζ := e ix , c := (p−1) 2 4p , and let α := 2 + 4c = p + 1/p.Then: We evaluate the integral using the residue theorem.For T = 0, the poles are at ± 1/p, and both residues are equal to For T = 1, the pole at 0 has residue −1 and the poles at ±1/ √ p have residues has three poles inside the unit circle: 0, ω = 1/ √ p and −ω, and the two latter ones have the same residue.Let A(T ), B(T ) be the residues at 0 and ω respectively.Then: and Notice that both A(T ) and B(T ) satisfy the recurrence relation It remains to notice p (p − 1)p T satis es the same recurrence relation, which proves ( ).
To end this section, we apply Theorem to the question of sharp cuto of random walks on certain Ramanujan graphs.Let ℓ be a xed prime and p a large prime, p ≡ 1 (mod 12) (the notation here is made to conform with [CGL ]).The Brandt-Ihara-Pizer "super singular isogeny graphs," G(p, ℓ), are d := ℓ + 1 regular graphs on n := p−1 12 + 1 vertices (see [CGL ], page 4, for a description).The nontrivial eigenvalues of G(p, ℓ) are the numbers 2 √ ℓ cos(θ f (ℓ)) for f ∈ H(N) (N = p in our notation).The G(p, ℓ)'s are d-regular Ramanujan graphs on n vertices.The L 2 -variance, W 2 (t), for the t-step nonbacktracking random walk on G(p, ℓ) is given by (see [NS ], page 13) where R t is the t th orthogonal polynomial on [0, π] with respect to dµ ℓ , normalized so that Applying Theorem with r = 1, p 1 = ℓ, and ℓ 1 = t yields that uniformly for t < Note that N(t), the number of non-backtracking walks of length t, is (ℓ + 1)ℓ t−1 , so that This proves conjecture 1.8 in [NS ] for graphs G(p, ℓ).For the application to bounded window cuto one needs t to be as large as (1 + ε) log ℓ n, which is provided by the key doubling of the degree of P in Theorem .In order to prove Conjecture 1.8 in [NS ] for the more general Ramanujan graphs constructed using modular forms, one would need to identify the images of division algebras in H(N) under the Jacquet-Langlands correspondence and restrict the sums in Theorem 2 to those forms.

Multiplicity of Eigenvalue Tuples
Recall that for a xed prime level N and ϕ ∈ S(N) a weight 2 holomorphic cusp form for Γ 0 (N), we let M N (y, ϕ) be the multiplicity of the tuple of eigenvalues of ϕ at primes up to y in a Hecke basis H(N), i.e.
In this section we bound M N (y, ϕ) uniformly in ϕ in the range y = (log N) β for a xed β ∈ (0, 1).Speci cally, we prove Theorem via the large sieve and smooth number estimates.
From now on, we assume y = o(log N).We let p 1 , . . ., p r denote the rst r prime numbers, where r = π(y) is the number of primes up to y.
An integer m is called y-smooth if all primes p|m satisfy p ≤ y.The set of y-smooth numbers is denoted with S y , and the de Bruijn function Ψ(y, M) is the counting function for y-smooth numbers up to M: We use the large sieve inequality as in [IK ], Theorem 7.26 (the inequality is stated in [IK ] for weight k > 2 but holds for k = 2 as well -see comment after the proof): Large Sieve Inequality.Let F be an orthonormal basis of S(N), f (z) := ρ f (n)e(nz) for f ∈ F .Then for any complex numbers {c n } we have where c 2 = n≤M |c n | 2 and the implied constant is absolute.
We apply this with M = N and and the de nition of S y , we have

Number of Forms with Degree d Hecke Fields
For a prime level N, let H(N) d ⊆ H(N) denote Hecke forms whose Hecke eigenvalues span a number eld of degree exactly d.We bound the size of H(N) d using the multiplicity bound from the previous section.
Speci cally, let y > 0, r = π(y), and for f ∈ H(N) d , and let a f (p) = λ f (p) √ p be the p th Hecke operator eigenvalue of f .To prove Theorem , we combine the multiplicity bound with an upper bound on the set of possible tuples of eigenvalues of a Hecke form at the rst r primes.We get this bound by exploiting that a f (p) is a totally real algebraic integers whose conjugates are bounded by 2 √ p in size.
. Then: where κ = κ(d) is a constant depending on d.
be of degree d i ≤ d with discriminant ∆ i , and let P i (x) = (x − β j ) be the minimal polynomial of a f (p i ).Then Since K has degree at most d, it can be expressed as a composition of at most log 2 d elds K i , so the discriminant ∆ of K satis es |∆| ≪ d y k for some constant k depending only on d.This implies a bound of the same form on the number of possibilities for K by the Theorem of Schmidt ([Sch ]).
Lemma . .Let K be a totally real number field of degree ≤ d.
where the last step uses the prime number theorem.On the other hand, from Lemma ., the number of choices for K is exp(O d (log y)) = exp(o d (y)), so multiplying the two proves the proposition statement.
Combining this Proposition with Theorem , which concludes the proof of Theorem .Note that for the coe cient of y to be negative, we have to choose β small, which is why we dealt with multiplicity bounds in Theorem 3 only for 0 ≤ β ≤ 1.

Composite Level
The discussion up to this point was restricted to weight k = 2 and level N being prime.Theorems and can be extended without much change to allow varying weight and general N, as long as the relatively prime conditions (mn, N) = 1 and (p j , N) = 1 are maintained.Indeed, the starting point, which is an application of the Petersson formula ([ILS ], Corollary 2.2 or [IK ], Corollary 14.24), gives the desired uniformity in the "harmonic" weighted form.To remove the weights, one has to take care with new and old forms and the Atkin-Lehner involutions in relating f 2 2 and L(1, Sym 2 f ).On the other hand, in the proof of the multiplicity bounds in Theorem , we used y-smooth numbers and the assumption that (p, N) = 1 for p ≤ y < log N. we show in Theorem below, similar bounds can be proved for N's that do not have an abnormal number of small prime factors.For "super-smooth" numbers, such as N = p≤t p, we cannot make use of the approach to the Plancherel measure of the Hecke eigenvalues for small primes, and our bounds in Theorem and don't apply.
In what follows, we restrict ourselves to the s * (N)-dimensional space S * (N) of weight 2 level N newforms, which admits a simultaneous eigenbasis H * (N) with respect to Hecke operators T n with (n, N) = 1 (we assume these forms are normalized to have constant Fourier coe cient 1).
y = y(N) = o(log N) be a parameter going to in nity with N, and let r := π(y) ∼ y/ log y.
K , the coordinates of ι(α) are the Galois conjugates of α; their product is a non-zero integer, so the non-zero vectors in the lattice formed by the image of O K under ι have length ≥ 1.From this, sphere packing bounds imply immediately that the number of lattice points in the box [−M, M] d is bounded by O d (1)M d (this can be seen, for example, by placing (disjoint) balls of diameter 1 at each lattice point in the box and comparing volumes). of Proposition . .From Lemma ., we see that for a xed degree K number eld, the number of possible tuples (a f (p 1 ), . . ., a f Then for M > 1, the number of α ∈ O K such that all the Galois conjugates of α are bounded by M is at most C(d)M d for some constant C(d) which does not depend on K.Proof.Consider that standard embedding ι : K ֒→ R d .For α ∈ O