Exponential sums in prime fields for modular forms

The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on Shparlinski’s bound for exponential sums attached to certain linear recurrence sequences over finite fields.


Introduction
Let f be a modular form of weight k ∈ 2Z and level N such that it has a Fourier expansion with a(n) be the nth Fourier coefficient. In this article, we shall restrict to the family of modular forms with rational coefficients, that is, f (z) with a(n) ∈ Q for every n. We first consider Hecke eigenforms or simply eigenforms in the space of cusp forms of weight k for the congruence subgroup 1 (N ) with trivial nebentypus. When f is an eigenform with integer Fourier coefficients, it follows from Deligne-Serre that for any prime , there exists a corresponding Galois representation such that tr(ρ ( ) f (Frob p )) = a(p), for any prime p N . For a quick reference about this correspondence, we refer the interested reader to [8,Chapter 3].
In particular, a(p) (mod ) is determined by the trace of the corresponding Frobenius element in GL 2 (Z / Z ) = GL 2 (F ). In certain cases, Chebotarev's density theorem implies that given any λ ∈ F , there exists a prime p such that a(p) ≡ λ (mod ). However, the set of such primes p come with density strictly less than 1. So what about the other primes p? In this context, we address the following Waring-type question.
Question Does there exist an absolute constant s such that for any given primes p and , any element of F can be written as a sum of at most s elements of the set {a(p n )} n≥1 ?
In [24], it is proved that the set {τ (n)} n≥1 is an additive basis modulo any prime , that is, there exists an absolute constant s such that the Waring-type congruence τ (n 1 ) + · · · + τ (n s ) ≡ λ (mod ) is solvable for any residue class λ (mod ).
Shparlinski's work was later generalized by Garaev, García and Konyagin over the global field Q. More precisely, in [10], the authors proved that for any λ ∈ Z, the equation always has a solution for s =74,000.
Later García and Nicolae [12] extended this result for coefficients a(n) of normalized Hecke eigenforms of weight k in S new k ( 0 (N )). More precisely, they proved that for any λ ∈ Z, the equation We are studying the finite field version of this additivity problem by obtaining nontrivial exponential sums associated with coefficients of modular forms, in the sense of [24]. We are working with the class of forms that García and Nicolae [12] considered, but with Fourier coefficients evaluated only at prime powers. Our results are recorded in Corollaries 15 and 16. Our main tool is Theorem 1 which provides a nontrivial bound for exponential sums with coefficients of modular forms. To study this problem, we shall primarily focus on the exponential sums of type max ξ ∈F * n≤τ e (ξ a(p n )) where p, are primes, and τ is a suitable parameter which we shall specify later. This is done in Theorems 2 and 3. When f is a normalized eigenform, it is well known that a(n) is a multiplicative function and for any prime p N satisfies the relation a(p n+2 ) = a(p)a(p n+1 ) − p k−1 a(p n ), n ≥ 0. (1) Moreover, we have a(p n ) = a(p) n for any prime p | N . These facts come from the properties of Hecke operators, see [5,Proposition 5.8.5]. If a(p) ∈ Q, then one can consider a(p) (mod ) ∈ F naturally for any large enough prime . For instance, can be taken to be any prime not dividing the denominators of the Fourier coefficients. On the other hand, any cuspform can be uniquely written as a C-linear combination of pairwise orthogonal eigenforms with Fourier coefficients coming from C. See [5,Chapter 5] for a brief review of the Hecke theory of modular forms. However, here we are concerned with all such cuspforms which can be uniquely written as a Q-linear combination of pairwise orthogonal eigenforms with Fourier coefficients coming from Q. Note that, in this case, the sequence {a(p n )} is a linear recurrence sequence of possibly higher degree. We now turn to discuss the basic theory of linear recurrence sequences. We will also discuss the bounds of their associated exponential sums.

Linear recurrence sequences and exponential sums
Let r ≥ 1 be an integer and p be an arbitrary prime number. A linear recurrence sequence {s n } of order r in F p consists of a recursive relation s n+r ≡ a r−1 s n+r−1 + · · · + a 0 s n (mod p), with n = 0, 1, 2, . . . , and initial values s 0 , . . . , s r−1 ∈ F p . Here a 0 , . . . , a r−1 ∈ F p are fixed. The characteristic polynomial ω(x) associated to {s n } is ω(x) = x r − a r−1 x r−1 − · · · − a 1 x − a 0 .
We see from Eq. (1) that {a(p n )} is a linear recurrence sequence of order 2 when f is an eigenform. We shall prove the results in Sect. 2, by studying exponential sums associated to a much more general class of linear recurrence sequences.
Let p be a prime number and ω(x) be the characteristic polynomial of a linear recurrence sequence {s n } defined by Eq. (2). If (a 0 , p) = 1 and at least one of the s 0 , . . . , s r−1 are not divisible by p, then the sequence {s n } is periodic modulo p, that is for some T ≥ 1, The least positive period is denoted by τ . Moreover, τ ≤ p r − 1 and τ divides T for any period T ≥ 1 of the sequence {s n }.
In 1953, Korobov [16] obtained bounds for rational exponential sums involving linear recurrence sequences in residue classes. In particular, for the fields of order p, if {s n } is a linear recurrence sequence of order r with (a 0 , p) = 1 and period τ , it follows that n≤τ e p (s n ) ≤ p r/2 .
Note that such a bound is nontrivial if p r/2 < τ and asymptotically effective only if p r/2 /τ → 0 as p → ∞. Estimate (3) is optimal in general terms, indeed Korobov [15] showed that there is a linear recurrence sequence {s n } with length r satisfying In turn, for any given ε > 0, it has been proved that there exists a class of linear recurrence sequences with a better upper bound n≤τ e p (s n ) ≤ τ 1/2+ε .
However, the proof of the existence is ineffective in the sense that we do not know any explicit characteristics of such family, see [7, In particular, if r is fixed then the upper bound is non trivial for τ ≥ p r/2−1/6+ε . We already pointed out that the inequality (3) is nontrivial for τ > p r/2+ε , so the most important case occurs when τ ≤ p r/2+ε . If τ ≤ p r/2+ε , then condition (4) is needed to obtain a non trivial bound suggested by an example given in [23,Section 1]. In this particular example, the exponential sums of type are considered for certain a in F * p m with g a generator of F * p m and m be any even integer. It is worth noting that {Tr(ag 2n )} is indeed a linear recurrence sequence of order m in F p .
Moreover, we consider the general case when the associated polynomial ω(x) is not necessarily irreducible, and deduce the following key result.
Theorem 1 Let p be a large prime number and ε > ε > 0. Suppose that {s n } is a nonzero linear recurrence sequence with positive order and period τ in F p such that its characteristic polynomial ω(x) has distinct roots in its splitting field, and (ω(0), p) = 1. Set , and for each i, α i denotes a root of ω i (x). If all polynomials ω i (x) have the same degree, i.e. deg ω i (x) = r > 1, and the system τ i = ord α i , satisfies then there exists a δ = δ(ε, ε ) > 0 such that It turns out that, this extends [2,Corollary] due to Bourgain, where all of the irreducible factors have degree r = 1, while Theorem 1 deals with the case r ≥ 2.
This will be of immense use in what follows, roughly because the characteristic polynomial associated to {a(p n )} have degree two. Theorem 1 will be essential to establish Theorem 2 and Corollaries 12 and 17. Our approach, which relies on the sum-product phenomenon, provides an improvement over Theorem 3.1 of [23] for the same class of linear recurrence sequences, obtaining non trivial exponential sums in a larger range. To be more precise, if p(r) denotes the least prime divisor of r then any τ > p r/p(r)+ε satisfies In particular, our result works for any τ > p r/p(r)+ε , while bound in [23] is nontrivial if τ > p r/2−1/6+ε . This is an improvement if p(r) > 2, more precisely when r is odd.

Main results for exponential sum with modular forms
We now quickly discuss the main results obtained in this article. In the list, our first result is the following: Theorem 2 Let f (z) be an eigenform with rational coefficients a(n). Let P be the set of primes p such that a(p u ) = 0 for any u ∈ N. Then the following is true.
(i) The set of primes P satisfies that given p ∈ P, for any 0 < ε < 1/2 there exists a δ = δ(ε) > 0 such that the following estimate holds for π(y) + O(y 2ε ) many primes ≤ y, where the least period τ of the linear recurrence sequence {a(p n )} (mod ) depends on both p and , and π(y) denotes the number of primes up to y which is asymptotically equivalent to y log y . (ii) For the exceptional set of primes p / ∈ P, let u be the least natural number such that a(p u ) = 0. Then for any 0 < ε < 1/2, there exists a δ = δ(ε) > 0 such that the following estimate holds for π(y) + O(y 2ε ) many primes ≤ y.
Roughly speaking, a newform of level N is a normalized eigenform which is not a cuspform of level N for any proper divisor N of N. For details and basics on modular forms, we refer the reader to [5]. A newform is said to have complex multiplication (CM) by a quadratic Dirichlet character φ if f = f ⊗ φ, where we define the twist as In part (i) of Theorem 2, the condition a(p u ) = 0 holds for almost all prime p provided that f is a newform without CM. This is a consequence of Sato-Tate conjecture and we shall discuss this again in the proof of Lemma 11. In particular, we have a non trivial estimate for the following exponential sum max ξ ∈F * n≤τ e (ξ a(p n )) .
Let us recall that any general cusp form f can be uniquely written as C-linear combination of eigenforms. These eigenforms will be called as components of f . We then have the following result.
Theorem 3 Let f (z) be a cusp form which is not necessarily an eigenform, and can be written as a Q-linear combination of newforms with rational coefficients. Suppose that there are r 2 many components with CM, then under the assumption of GST hypothesis 1 there exists a set of primes p with density at least 2 −r 2 such that for any 0 < ε < 1/2 there exists a δ = δ(ε) > 0 for which the following estimate holds for c f π(y) + O(y 2ε ) many primes ≤ y, where c f > 0 is a constant.
In both of the above theorems, we took a fixed prime p and looked for primes for which a non trivial estimate to (9) holds. However, these results are valid for almost all primes , and we do not know explicitly which of the primes are being excluded in this process. Thus, one may naturally ask, what if we now fix a prime and find out for how many primes p the sum at (9) is non trivial. In this regard, we have the following results.
Theorem 4 Let f (z) be a newform of weight k, without CM, and with integer Fourier coefficients. Consider the set P = prime | (k − 1, − 1) = 1 . Then, for any fixed ε > 0 and any large enough ∈ P, the set of primes p satisfying max ξ ∈F * n≤τ e (ξ a(p n )) ≤ τ −δ (11) have density at least 1 Intuitively, this theorem can be regarded as the inverse of Theorem 2, and in this analogy, the following result as the inverse of Theorem 3. Just for the sake of simplicity we are assuming (k − 1, − 1) = 1, which can be easily avoided and will be evident from the proof of the following theorem.
Theorem 5 If f (z) is a cuspform, and can be written as Q linear combination of r many newforms without CM and with integer coefficients, such that all of these components satisfies GST hypothesis. Then, for any fixed ε > 0 and large enough , the set of primes p satisfying max ξ ∈F * n≤τ e (ξ a(p n )) ≤ τ −δ (12) have density at least 2 −r + O ε , where δ = δ(ε) is same as in Theorem 1.

Waring type problems over finite fields
Given a sequence {x n }, one of the classical questions are to decide whether {x n } is an additive basis. More precisely, is there an absolute constant k ≥ 1 such that any residue class λ modulo p can be represented as for infinitely many primes p?
In this article, we are concerned about the case when {x n } is a linear recurrence sequence in F p . For a simple sequence 2 n (mod p), it follows combining a result of Erdős and Murty [6] and a result of Glibichuk [13] that for almost all primes p, every residue class modulo p can be represented in the following form 2 n 1 + · · · + 2 n 8 (mod p), for certain positive integers n 1 , · · · , n 8 .
Note that 2 n (mod p) is a linear recurrence sequence of order 1. For higher order cases, one can ask about the classical case of Fibonacci sequences. The third author proved in [11,Theorem 2.2], that given a parameter N → ∞, for π(N )(1 + o(1)) primes p ≤ N, every residue class modulo p can be written as F n 1 + · · · + F n 16 ≡ λ (mod p), provided that n 1 , . . . , n 16 ≤ N 1/2+o (1) . The method is based on the distribution properties of sparse sequences for almost all primes and particular identities of Lucas sequence. It does not seem easy to extend such ideas for general linear recurrence sequences. In this article, we prove that if {s n } is a linear recurrence sequence in Z, whose characteristic polynomial ω(x) ∈ Z[x] is monic, irreducible, and having prime degree, then there exists an absolute constant k such that every residue class λ(mod p) can be represented as for a set of primes p with positive density. We record this in Theorem 13 in Sect. 6.

Exponential sums with linear recurrence sequences
In this section, our main goal is to prove Theorem 1, which is one of our key tool in establishing several important results of this article. Recalling the example of Shparlinski in [23, Section 1], we already noticed in Sect. 1 . We shall discuss more about this condition later in Remark 1. Now, we illustrate with an example that all of the gcd(τ i , τ j ) s cannot be too large. In other words, we need condition (b) (or some other condition) to obtain a non trivial bound in Theorem 1. For example, let r = 2 and g be a generator of F * 2 . Then, consider the sequence It is easy to see that gcd( 2 − 1, ( 2 + 1)/2) = 1, so gcd(τ 1 , τ 2 ) = 2 − 1. On another hand we note that gcd(τ 1 , − 1) = − 1. Then, one can show that Let p be any prime and q be any power of p. Then, the classical theorem about additive sums for one-variable polynomial, due to A. Weil (see [17,Theorem 3.2]), states that, for a given polynomial f (x) ∈ F q [x] with degree d, d < q, gcd(d, q) = 1 and a nontrivial additive character ψ in F q , we have Therefore, the linear recurrence sequence {s n } satisfies We now need to discuss some necessary background. Let K be a finite field of characteristic p and F be an extension of K with [F : K ] = r. The trace function Tr F /K : F → K is defined by The following properties of Tr F /K (z) are well known.
Tr F /K (a) = ra, for any a ∈ K.
Throughout this section, F = F q , K = F p with q = p r and we will simply write Tr (z) instead Tr F /K (z). Let {s n } be a linear recurrence sequence of order r ≥ 1 in F p with characteristic polyno- . It is well known that nth-term can be written in terms of the roots of the characteristic polynomial, see Theorem 6.21 in [18]. Therefore, if the roots α 0 , . . . , α r−1 of ω(x) are all distinct in its splitting field, then where β 0 , . . . , β r−1 are uniquely determined by initial values s 0 , . . . , s r−1 , and belong to the splitting field of ω(x) over F p . If the characteristic polynomial ω(x) is irreducible and α is a root, then its r distinct conjugates are Hence, the coefficients s n are given by One of our main tools is the bound for Gauss sum in finite fields given by Bourgain and Chang [3,Theorem 2]. This will be required to prove Theorem 1. Assume that for a given α ∈ F q and ε > 0, such that ord α = t satisfies t > p ε and max 1≤d<r d|r Then, there exists a δ = δ(ε) > 0 such that for any nontrivial additive character ψ of F q , we have Note that the second assumption in (18) implies the first one whenever r ≥ 2.

Proof of Theorem 1
We proceed by induction over ν. Before that, following properties (14) and (15) of trace function we write In other words, any such α i is in F p r . We then have, r = [F p (α 1 , . . . , α ν ) : F p ] and z p r = z for any z ∈ F p (α 1 , . . . , α ν ). In addition, from (16) it follows that, Tr (z p ) = Tr (z) for any z ∈ F p (α 1 , . . . , α ν ). Then, for each pair (i, j), raising each argument β i,j α p j n i to the power p r−j This implies that where The case ν = 1 follows from Bourgain and Chang [3, Theorem 2]. We shall now proceed inductively, and ν = 2 will be the base case. We start by denoting h = gcd(τ 1 , τ 2 ). It is clear that lcm(τ 1 , Hence, it is enough to prove that Let (n 1 , n 2 ) be a tuple with Moreover, the pair (m 1 , m 2 ) has the following property: given (n 1 , n 2 ), with 1 ≤ n i ≤ τ i /h, then n = n 1 m 1 and n is unique modulo Combining (21) and (22), we have with Each e p Tr ξγ i z corresponds to a nontrivial additive character, say ψ i (z), in F p (α i ) = F p r . In order to satisfy condition (18), we first recall assumptions h < p ε , ε > ε > 0 and max d<r d|r gcd(τ i , p d − 1) < τ i p −ε for some i ∈ {1, 2}. Without loss of generality, let us assume that i = 1. Then, for any d|r with 1 ≤ d < r, we have Therefore, by Bourgain and Chang [3, Theorem 2] it follows that On the other hand, bounding trivially we have Thus, combining above equations with (20) and (23) we get Now, let us assume that one of the λ i = 0, say for i = 2. Arguing exactly as few lines above, it follows from assumption (a) that Hence, the desired bound follows. This conclude the case ν = 2. Now, we proceed by induction over ν, and assume Theorem 1 to be true up to ν − 1. We follow the idea due to Garaev [9,Section 4.4]. Considering (19) and periodicity, for any t ≥ 1 we get Raising to the power 2t, and applying Cauchy-Schwarz, we have Note that writing J ν (λ 1 · · · , λ ν ) in terms of character sums, it follows that In particular, we note that J t,ν ≤ J t,ν−1 . From (24), it follows that Tr Note that aγ λ, with aγ = 0, runs over λ ∈ F q , then e p ( Tr (aθλz)) runs through all additive characters ψ in F q , evaluated at z. Then, the above expression can be written as We now require an estimate for J t,ν−1 , and write Finally, we note that s m = Tr λ 1 α m 1 + · · · + λ ν−1 α m ν−1 defines a linear recurrence sequence with period τ dividing τ , which in particular satisfies induction hypothesis. Therefore m≤τ e p Tr λ 1 α m 1 + · · · + λ ν−1 α m and combining with (26), we get We conclude the proof combining the above estimate with (25) to get 2 max ξ ∈F * p n≤τ The following is an immediate corollary of this theorem which will be quite handy in establishing several results in Sects. 3 and 6.

Corollary 6
Suppose that {s n } is a nonzero linear recurrence sequence of order r ≥ 2 such that its characteristic polynomial Remark 1 It is possible to relax the condition (a) by assuming that max d<r d|r Since {s n } is a nonzero linear recurrence sequence, there exists some 1 ≤ i ≤ ν for which λ i = 0. We discussed in Sect. 1.1 that why (a) (or some other condition) is needed to prove the irreducible case of Theorem 2. Now, for the reducible case, some of the λ i could be 0. For the worst case scenario, let us assume that only one of them is nonzero, say for i = 1. Then, it follows from (19) that, we are back to considering the irreducible case and then we need the condition (a) for i = 1. In particular, we need (a) (or some other condition) for each irreducible component of the underlying ω(x).

Exponential sums for modular forms
In this section, we study the effect of linear recurrence sequence and Theorem 1 in the behaviour of the exponential sums associated with certain Fourier coefficients of modular forms. As a consequence, we obtain interesting results which have been summarized earlier in the form of Theorems 2 and 3.

Order of the roots of the characteristic polynomial
In the case of normalized eigenforms, the sequence {a(p n )} defines a linear recurrence sequence of order two when p N, otherwise it is of order one. This is one of the tools for Theorem 2. However, we do not need to assume that the form is normalized because the normalizing factor is in Q, and we can realize that to be an element of F * for any large enough prime . Before going into the proof of this theorem, we develop a tool which will be quite useful throughout. We state it in the form of following lemma.
Proof Given a large positive parameter T, we begin by considering the polynomial It is clear that ω(x) (mod ) has distinct roots for all but finitely many primes , since a 2 − 4b = 0. For any such prime , let α and β be the distinct roots in its splitting field. We now consider the resultant Res(ω(x), G T (x)), and note that where each μ i is a root of G T (x) in its splitting field over F .
In particular, Res(ω(x), G T (x)) ≡ 0 (mod ) if and only if ω(x) (mod ) and G T (x)(mod ) have common roots in some finite extension of F . Additionally, since α β = b, it follows that ord (α β −1 ) ≤ T if and only if α 2t − b t = 0 (or β 2t − b t = 0), for some t ≤ T. Therefore, α (or β ) is a common root of ω(x) (mod ) and G T (x) (mod ) if ord α or ord (α β −1 ) (or ord β or ord (α β −1 )) is less than T. Now, the Sylvester matrix of ω(x) and G T (x) is a square matrix of order 2 + deg(G T (x)) T 2 , and entries bounded by an absolute constant M (which depends on a, b and not on or the parameter T ). Then, by Hadamard's inequality, the determinant Note that Res(ω(x), G T (x)) is zero if and only if α t = 1, β t = 1 or (αβ −1 ) t = 1 for some t ≤ T, which, following our assumption, can not happen. In particular, the resultant has at most O T 2 many distinct prime divisors. This shows that Choosing T = y ε , the number of primes ≤ y such that Let us now proceed to prove the main result of this section.

Proof of Theorem 2
If p | N, then a(p n ) = a(p) n for any n. We only need to consider max ξ ∈F * n≤τ e (ξ a(p) n ) .
If p / ∈ P, then there exists u such that a(p u ) = 0. Since p | N, we have a(p) = 0. In this case, the sum is O(1) because we have τ = 1.
On the other hand, if p ∈ P, then for any prime large enough τ is simply the order of a(p) (mod ) in F * . Due to Lemma 7, we may assume that τ > p ε holds for π(y) + O(y 2ε ) many primes < y. Hence, this case is settled down by [4,Theorem 6].
Let us now consider the case p N . The characteristic polynomial of (1) is and has discriminant a 2 (p) − 4p k−1 . We note that in our case the discriminant does not vanish, otherwise |a(p)| = 2p (k−1)/2 is absurd, with a(p) being integer and p (k−1)/2 irrational. Let P be the set of all primes. We divide the proof for primes p ∈ P and p ∈ P \ P. Since a 2 (p) − 4p k−1 = 0, for any p ∈ P, we write a 2 (p) − 4p k−1 = u 2 D p , with D p < 0 square-free and u = 0. Let us split the cases according to D p (mod ) is quadratic residue, zero or non quadratic residue modulo . Set For ν = 0, 1, −1, we also define It is clear that π ν (x) = π(x)(κ ν + o(1)), and κ 0 + κ 1 + κ −1 = 1. Note that for a given prime p, the associated polynomial ω(x)(mod ) has a single root in F if and only if u 2 D p ≡ 0(mod ). Since such equation has finitely many solutions for , we get κ 0 = 0. On the other hand, Chebotarev's density theorem implies that the uniform distribution of primes such that ω(x) (mod ) is irreducible or has distinct roots in F . Equivalently, the primes satisfying D p = ±1 are distributed in the same proportion, therefore κ −1 = κ 1 = 1/2. We now turn to establish nontrivial exponential sums for {a(p n )}(mod ) with ∈ P ν , for ν = ±1.
Applying Corollary 6 we complete the proof of this case.

Case 2. ∈ P 1 :
let α, β be the roots of ω(x)(mod ) inside F * . From (17) it follows that for n ≥ 0, a(p n ) ≡ cα n + dβ n (mod ), for some constants c, d in F , with (α, β) = (0, 0). It is clear that − 1 is a period of the sequence a(p n )(mod ), and hence τ divides − 1. We have From Lemma 7, there is a subset of P 1 with π (y) 2 + O(y 2ε ) many primes ≤ y such that ord α, ord β and ord (αβ −1 ) are bigger than ε . It follows from [2, Corollary, page 479] that there exists a δ = δ(ε) > 0 such that Hence, (i) of Theorem 2 holds. Now, assume that p belongs to the exceptional set P \ P, that is a(p u ) = 0 for some u ≥ 1. We consider u = u(p) to be the least such integer. Since the discriminant is nonzero (the roots α and β of (28) are distinct), we get 3 First of all observe that u is odd. As otherwise, if u is even then we would get which is absurd as α u+1 +β u+1 is a rational, but p (u+1)(k−1) 2 is not. Now, for any 0 < e < u+1 we have where the sign on the right hand side above depends on the sign of α u+1 . Without loss of generality, we are assuming that this sign is negative. It is easy to see that our next argument applies to the positive sign case as well. Since u is fixed, so are all the e's up to u − 1. In particular, we may consider large primes for which all of the a(p e ) ≡ 0(mod ) for any 1 ≤ e ≤ u − 1. Then, we have Due to Lemma 7, we may assume that t u = ord (−p (k−1)(u+1)/2 ) > ε holds for π(y) + O(y 2ε ) many primes ≤ y. Now, by [4, Corollary 1] it follows that and for any t u ≥ t > ε . Writing [τ /(u + 1)] = qt u + r, with 0 ≤ r < t u it follows that The estimate n≤t u e ξα (u+1)n a(p e−1 ) ≤ t u −δ follows from (31). If r ≤ ε/2 , then we get trivially n≤r e ξα (u+1)n a(p e−1 ) ≤ ε/2 . If ε/2 ≤ r < t u , then from (31) it follows that n≤r e ξα (u+1)n a(p e−1 ) ≤ t u −δ .
Recalling that t u ≥ ε , we can also assume that t u −δ ≥ ε/2 by taking small enough δ. Thus, Finally, combining the above inequality with (30) we obtain max ξ ∈F * n≤τ e (ξ a(p n )) = τ u + 1 This conclude the proof for all exceptional set of primes p ∈ P \ P.

Consequences of Theorem 2
Let us consider an exponential sum of type S(p, x, α) = p n ≤x e(αa(p n )), for α ∈ [0, 1]. As one of the consequences of Theorem 2, we want to study this exponential sum when α is a rational whose denominator is a prime. In this regard, we have the following result.
Corollary 8 Let f be an eigenform of weight k and level N with rational coefficient. Then for a given 0 < ε < 1/2, there exists a δ(ε) > 0 such that for at least log log x many primes , we have the following estimates: Proof Consider the same δ := δ(ε) as in Theorem 2 and any prime Following Theorem 2, we have holds, for at least primes . For these primes, we also have τ ≤ 2 < log x log p . In particular, max ξ ∈F * p n ≤x e (ξ a(p n )) ≤ log x δ log p On the other hand, let p ∈ P be a prime, then by Theorem 2 we have max ξ ∈F * n≤τ e (ξ a(p n )) = τ u + 1 holds, for some u depending on p, and for at least primes . Due to Lemma 7, we can assume that τ > δ holds by choosing small enough δ, for at least log log x primes . Arguing similarly as in the previous case, we get the desired main term, and the error term that we get where the last equality holds because τ > δ .
We can bound the error term by Theorem 2 and without loss of generality assuming δ < 1, we get the sum above is simply τ 1 + O(τ 1 −δ ). This is further bounded by 2τ 1 −δ , because the explicit constant in Theorem 2 is exactly 1. Therefore, for some δ = δ(ε) > 0. This shows that the inequality max ξ ∈F * n 1 ≤τ 1 · · · n ν ≤τ ν e ξ a(p n 1 1 · · · p n ν ν ) ≤ 3τ 1 · · · τ ν −δ holds for almost all prime and this completes the proof because we can remove the extra factor 3 by taking primes large enough.

Exponential sums for modular forms: beyond eigenforms
We shall now prove Theorem 3. Write where a i ∈ Q, and f i is a newform with rational coefficients for every 1 ≤ i ≤ r. Let ω (i,p) be the characteristic polynomial of a f i (p n ) and D i (p) be its discriminant. Consider It is clear that S 1 has positive density. One can verify this by considering primes congruent to 1 modulo 8 r i=1 D i (p). This works well because, we then have where odd(.) denotes odd part of the corresponding number. These conditions altogether imply ∈ S 1 . Let α (i,p) and β (i,p) be the roots of ω (i,p) . So for any ∈ S 1 , we can write where for every i, j, α (i,p) , β (j,p) are in F . Now, we consider the set of primes Lemma 10 For any prime p ∈ S 2 , the following inequalities are true for π(y) + O(y 2ε ) many primes ≤ y.
For any prime in the set above, and for any prime L in O K lying over , ω 1 (x) (mod L) and g T (x) (mod L) have a common root, Therefore, R(T )(mod L) = 0. Since both ω 1 (x) and g T (x) are in Z[x], it is clear that R(T ) ∈ Z, and so R(T ) (mod ) = 0 as well. Now, one can estimate the number of prime divisors of R(T ) similar to as in Lemma 7. This shows that ord α (i,p) (β (j,p) ) −1 > ε , and ord α (i,p) (α (j,p) ) −1 > ε holds for all i = j ∈ {1, 2}, and π(y) + O(y 2ε ) many primes ≤ y. Rest of the cases can be dealt with Lemma 7.

GST: Beyond Sato-Tate
We shall now give a short overview of Sato-Tate distribution. When f is a newform without CM, then Sato-Tate conjecture says that the normalized coefficients a(p) On the other hand, if f is with CM, then the corresponding Sato-Tate distribution is Moreover at θ p = π 2 , a(p) becomes zero and it is known that the set of such primes p has density exactly 1 2 . Now, consider the L-function defined by where α p , β p are normalized roots of (28). In other words, if α p , β p are the roots of (28), then we define α p = . Serre in [22] showed that if for all integer m ≥ 0, L(s, Sym m (f )) extends analytically to Re(s) ≥ 1 and does not vanish there, then the Sato-Tate conjecture holds true for f. Note that Barnet-Lamb et al. have proved the conjecture in [1] working with this L-function. However, to estimate the size of S 2 we will have more than one newform to play with, and it will be helpful to have their distributions independent. This independency property is stated as Generalized Sato-Tate (GST) hypothesis. In this article, we shall always work with the newforms that obey this hypothesis. For example, in Theorem 3, it is assumed that all the associated newforms satisfy the GST hypothesis.

A consequence of GST
To prove Theorem 3, we need to study the set S 2 . We have that luxury when the associated newforms satisfy GST.

Lemma 11
Suppose that there are r 1 many components without CM and r 2 many components with CM in f. Then under the GST hypothesis, density of S 2 is 2 −r 2 .
Proof We start by writing So, the problem reduced to study the set of primes It follows from the discussion above that the density of this set is bounded by 2 π r 1 1 2π r 2 · · · S sin 2 (θ 1 ) sin 2 (θ 2 ) · · · sin 2 (θ r 1 ) dθ 1 dθ 2 · · · dθ r , where S = (θ 1 , θ 2 , · · · , θ r ) ∈ [0, π] r | θ i ± θ j ∈ Q × π for some 1 ≤ i, j ≤ r . Just for the sake of simplicity and to have a feel of what is going on, let us first do the case when there is only one component. Case 1 r = 1: suppose that the given component is without is a root of unity then this implies that θ 1,p ∈ π × Q. By Sato-Tate, density of such primes is bounded by 2 π θ ∈π ×Q sin 2 (θ ) dθ.
Since the integral above runs over a set of measure zero, the integral is zero, and for this particular case density of S 2 is indeed 1. Now, suppose that the given component is with CM. In this case, the density of S 2 is 1 2π Case 2 r ≥ 2: for this general case, it is enough to show that the integral over S in (36) is zero. This is because, due to GST, we are now working with the measure 2 π r 1 1 2π r 2 · · · sin 2 (θ 1 ) sin 2 (θ 2 ) · · · sin 2 (θ r 1 ) dθ 1 dθ 2 · · · dθ r , and with respect to this measure, [0, π] r has measure 1 2 r 2 . We can write S = 1≤i,j≤r S i,j , where the set S i,j is defined to be the tuples for which θ i ± θ j ∈ Q × π. It is now enough to show that each of these sets S i,j has a zero measure. Note that the integral over S i,j is crudely bounded by as Q × Q has zero measure. We now note that for any b > a. In particular, for any ε > 0, The last implication above follows from the standard argument to show a countable set always has a zero measure. In particular, the second integral of (38) is zero. On the other hand, just by replacing θ j with π − θ j , we get This just shows that the integral over S i,j at (38) is zero, which completes the proof.

Proof of Theorem 3
Let p ∈ S 2 be a prime, then we can write where a ( ) i , c (i, ) and d (i, ) are all in F . On the other hand, all the roots α (i, ) and β (i, ) are in F , as ∈ S 1 . The proof now follows by [2,Corollary,p. 479] combining with Lemmas 10 and 11.
Remark 2 It is known, due to Thorner, that GST holds for r = 2 when both f 1 and f 2 are without CM and not twist-equivalent. We say that f 1 and f 2 are twist-equivalent if there exists a primitive Dirichlet character χ such that f 1 = f 2 ⊗ χ. For more details, we refer the reader to Theorem 1.3 in [27].

Exponential sums for modular forms: the inverse case
One may now ask that for a given prime and small enough ε, how many primes p are there for which an estimate like (7) holds. Our attempt to answer this question is summarized in the form of Theorems 4 and 5. Let us begin with the proof of Theorem 4.

Proof of Theorem 4
For any prime p, let us denote the roots of x 2 − a(p)x + p k−1 (mod ) by α From the proof of Theorem 2, we know that an estimate of type (7) holds provided that, Let us define, where λ 1,A , λ 2,A are the eigenvalues of A in F * 2 . Now the problem is about computing the density of primes p for which the corresponding ρ f, Frob p is in C. Note that C is a subset of GL 2 (F ) stable under conjugation. Hence, by Chebotarev's density theorem, the required density is at least |C| |GL 2( F )| . For each a = b ∈ F * , let C a,b be the conjugacy class of a 0 0 b . It is known that |C a,b | = ( + 1) . For any element λ in F 2 \ F , we denote c λ to be the conjugacy class of matrices in GL 2 (F ) having eigenvalue λ. It is known that |C λ | = ( − 1). Now, we consider the following sets: and realize that |C| = 1 2 (( + 1) |S 1 | + ( − 1)|S 2 |). This reduced to the problem of estimating S 1 and S 2 . Let us first estimate S 1 . Take σ to be a generator of F * . For any divisor d of − 1, the set of all elements of F * having order exactly d is of the form σ −1 d i with (i, d) = 1. In particular, the number of elements of F * with order greater than ε is given by where d(·) is the divisor function, and here we are using the well known upper bound on divisor function (see [20]) for any prime large enough. Now note that ord ab −1 < ε implies that ab −1 belongs to a set with only k| −1,k< ε φ(k) many elements. By the argument above, this set has only O ε 2ε many elements. This observation implies that a, b ∈ F * | ord (a), ord (b), or ord (ab −1 ) < ε = O ε ( 2ε+1 ).
In particular, we then have |S 1 | = 2 + O ε ( 2ε+1 ). Let us now estimate |S 2 |. Take τ to be a generator of F * 2 , then any λ ∈ S 2 , of order d, is of the form τ 2 −1 d i , with (i, d) = 1. We also have an order restriction on λ −1 , which implies that d (d, −1) > ε . Hence, Note that, the condition d (d, −1) < ε implies that d < ε+1 . Therefore, Therefore, the required density is at least is O ε ( 2ε ). Moreover, ord (a) < ε or ord (b) < ε holds for only O ε ( 2ε ) many elements a or b. In particular, the number of tuples that does not come into our consideration is In particular, we then have The extra factor ( +1) 2 r is coming because each conjugacy class C a i,i+1 has ( + 1) many elements and taking into consideration that C a i ,a i+1 = C a i+1 ,a i , ∀i odd, the extra factor 1 2 is coming for each component. The proof is now complete because |

Impact on Waring-type problems
In the present section we combine Theorem 1 with classical analytical tools to prove that a linear recurrence sequence {s n } is an additive basis over prime fields, under some assumptions. Moreover, we discuss about the advantages of nontrivial exponential sums obtained in Theorem 1 to prove it.

Waring-type problems with linear recurrence sequences
Let {s n } be a nonzero linear recurrence sequence modulo as in (2) with order r, period τ and (a 0 , ) = 1. Given an integer k ≥ 2, for any residue class λ(mod ), we denote by T k (λ) the number of solutions of the congruence s n 1 + · · · + s n k ≡ λ (mod ), with 1 ≤ n 1 , . . . , n k ≤ τ .
Taking away the term ξ = 0 and using triangle inequality, it is clear that n k ≤τ e ξ (s n 1 + · · · + s n k − λ) n k ≤τ e ξ (s n 1 + · · · + s n k ) Assume that we have an exponential sum bound of the type max ξ ∈F * n≤τ e (ξ s n ) ≤ R .

Corollary 12
Let be a prime number, k > 0 be any integer, ε > 0, and {s n } be a linear recurrence sequence of order r ≥ 2 in F . If the characteristic polynomial ω(x) in F [x] is irreducible with (ω(0), ) = 1, the least period τ satisfies max d<r d|r and for every integer λ, let T k (λ) denote the number of solutions of the congruence s n 1 + · · · + s n k ≡ λ (mod ), with 1 ≤ n 1 , . . . , n k ≤ τ , then there exists an integer k 0 > 0 such that for any k ≥ k 0 , T k (λ) = τ k (1 + o(1)).
We are now ready to prove the main result of this section.
is solvable for any residue class λ (mod ).
Proof Let Q ω denote the splitting field of ω and G ω be Gal (Q ω /Q). Note that deg(ω) divides |G ω | and G ω is contained in the symmetric group S deg(ω) . By the Cauchy's theorem, there exists an element in G ω of order deg(ω). In particular, there is a deg(ω)-cycle in G ω because deg(ω) is prime. By Chebotarev's density theorem, the set of such primes for which ω(x) (mod ) is irreducible, have positive density, see Theorem of Frobenius in [26,Page 11]. We are now interested to work with these primes.
For any integer t, we have the following In particular, α is a root of both ω(x)(mod ) and t≤T x 2rt − ω(0) 2t (mod ). Now, given a large positive parameter T , we consider the resultant Counting the number of distinct prime factors of the resultant as in the proof of Lemma 7, we see that |{ prime | ord (α −1 ) ≤ T }| = O(T 2 ). For any large y > 0, taking T = y ε , we see that there exists a δ such that max ξ ∈F * n≤τ e (ξ s n ) ≤ τ −δ holds, for at least c ω π(y) + O(y 2ε ) many primes ≤ y, for some constant (which depends only on ω) c ω > 0. Now, the proof follows immediately from Corollary 12.
For further explanation, one can consider the following example.

Example 14
Consider the classical case of Fibonacci sequence {F n }. In the beginning of this section, the result of the third author is discussed for this special case. We can however get a slightly weaker result from Corollary 12. In this case, the characteristic polynomial is x 2 − x − 1. It is of course a monic, irreducible and of a prime degree. This polynomial is irreducible modulo prime , iff we have the Legendre symbol 5 = −1. The set of such primes have density 1/2. Corollary 12 says, for almost all of these primes, {F n } is an additive basis modulo . For the other set of primes, we use Lemma 7. Given any 0 < ε < 1/2, for π(y) + O(y 2ε ) many primes ≤ y, we have ord α > ε , ord β > ε and ord (α β −1 ) > ε , where α and β are the roots of x 2 − x − 1(mod ). It then follows from [2, Corollary, page 479] that there exists a δ = δ(ε) > 0 such that max (c,d)∈F ×F (c,d) =(0,0) n≤ −1 e (cα n + dβ n ) ≤ 1−δ .
In particular, we then have max ξ ∈F * n≤τ e (ξ F n ) ≤ τ −δ , which guarantees the existence of an absolute constant, as we saw in the proof of Theorem 13. With this, we have an inexplicit result for the Fibonacci sequences compared to what the third author had in [11]. However, Theorem 13 provides a general result for a large class of linear recurrence sequences.

Waring-type problems for modular forms
Let us recall our discussion from the introduction about Waring problem for modular forms. In this section, we are assuming that the modular form is a newform without CM. Fix any 0 < ε < 1 2 , say ε = 1 3 . Then taking δ := δ(ε) as in Theorem 2, the following estimate max ξ ∈F * n≤τ e (ξ a(p n )) ≤ τ −δ , holds for almost all primes p and . The discussion in Sect. 6.1 shows that T s (λ) > 0 for any λ ∈ F , and s > 1/δ, where T s (λ) is the number of solutions of the congruence a(p n 1 ) + · · · + a(p n s ) ≡ λ (mod ), with 1 ≤ n 1 , . . . , n s ≤ τ .
Moreover, this s does not depend on the choice of the eigenform because δ does not. More precisely, we have the following result. As an immediate consequence of Theorem 4, we obtain the following result.

Corollary 16 Suppose the newform is without CM and with integer Fourier coefficients.
Then there exists an absolute constant s 0 such that, for any large prime satisfying the coprimality condition ( − 1, k − 1) = 1, the proposition Q f (p, , s 0 ) is true for a set of primes p with density at least 1 + O 1 √ . Moreover, s 0 does not depend on the choice of f.
Bounds for W (b) have cryptographic significance, see [25] and references therein. Shparlinski and Winterhof [25,Theorem 1] proved that max 0≤b≤ r −1 |W (b)| τ 3/4 r 1/4 r/8 , whenever the characteristic polynomial of {s n } is irreducible. Such bound is asymptotically effective if r r/2 /τ → 0. Combining Corollary 6 and the ideas of [25], we are able to improve such bound for a large class of linear recurrence sequences in the range τ > ε . For example, assuming hypothesis of Corollary 6, if r is fixed then |W (b)| τ −δ as → ∞ for some δ > 0. In general, we get |W (b)| = o(τ ) if r log / δ → 0 as → ∞. log . Clearly there are many such cases, for instance, one can consider any element in F * of order smaller than 1 2 .