Bounded gaps between primes in Chebotarev sets

A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes p1, p2 with |p1-p2| ≤ 600 as a consequence of the Bombieri-Vinogradov Theorem. In this paper, we apply his general method to the setting of Chebotarev sets of primes. We use recent developments in sieve theory due to Maynard and Tao in conjunction with standard results in algebraic number theory. Given a Galois extension K/Q, we prove the existence of bounded gaps between primes p having the same Artin symbol K/Qp . We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over , congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms. Primary 11N05; 11N36; Secondary 11G05


Introduction
The long-standing twin prime conjecture states that there are infinitely many primes p such that p + 2 is also prime.The fact that there is a large amount of numerical evidence supporting the twin prime conjecture is fascinating, considering that the Prime Number Theorem tells us that on average, the gap between consecutive primes p 1 , p 2 is about log(p 1 ).A resolution to the twin prime conjecture seems beyond the reach of current methods.The next best result for which one could hope is that there are bounded gaps between primes; that is, there exist a constant C > 0 and infinitely many pairs of distinct primes p 1 , p 2 satisfying |p 1 − p 2 | ≤ C.
In [8], Maynard proved that there are infinitely many pairs of distinct primes p 1 , p 2 satisfying |p 1 − p 2 | ≤ 600.(Tao developed the underlying sieve theory independently, but arrived at slightly different conclusions.)This result follows from a dramatic improvement to the GPY method arising from the use of more general sieve weights.Once we have this improvement, all that one must know in order to obtain bounded gaps between primes is the distribution of primes within the integers (which is given by the Prime Number Theorem) and the fact that the level of distribution θ is positive (which is given by the Bombieri-Vinogradov Theorem).
In this paper, we exploit the flexibility in the methods presented in [8] to obtain analogous results on bounded gaps between primes in Chebotarev sets P.These sets are characterized as follows.Let K/Q be a Galois extension of number fields with Galois group G and discriminant ∆.For a prime p ∤ ∆, there corresponds a certain conjugacy class C ⊂ G consisting of the set of Frobenius automorphisms attached to the prime ideals of K which lie over p. Denote this conjugacy class by the Artin symbol [ K/Q p ].We say that a subset P of the primes is a Chebotarev set, or that P satisfies a Chebotarev condition, if there exists an extension K/Q and a collection of conjugacy classes C ⊂ G such that P is a union of sets of the form {p prime : p ∤ ∆, [ K/Q p ] = C}.The Chebotarev Density Theorem asserts that P has relative density within the primes that is both positive and rational, and a result of Murty and Murty [10] tells us that P has a positive level of distribution if we omit certain arithmetic progressions.These two ingredients in conjunction with the sieve developed in [8] enable us to prove bounded gaps between primes any Chebotarev set.
By the Kronecker-Weber Theorem, if K/Q is abelian, then P is determined by congruence conditions.Thus finding bounded gaps between primes in Chebotarev sets determined by abelian extensions is equivalent to finding bounded gaps between primes in arithmetic progressions, which is proven in [7].In this paper, we handle the nonabelian extensions, proving a complete characterization of bounded gaps between primes in Chebotarev sets.
Theorem 1.1.Let K/Q be a Galois extension of number fields with Galois group G and discriminant ∆, and let C be a conjugacy class of G. Let P be the set of primes p ∤ ∆ for which is an abelian extension, let q be the smallest positive integer so that K ⊂ Q(e 2πi/q ).There exist infinitely many pairs of distinct primes We use Theorem 1.1 to prove several results in algebraic number theory.The first two of these are immediate.
Corollary 1.2.Let K/Q be a Galois extension of number fields with ring of integers O K .There exist a constant c(K) > 1 and infinitely many pairs of non conjugate prime ideals a, b Let f ∈ Z[x] be monic polynomial of degree d and discriminant ∆ that is irreducible over Q, and let G be the permutation representation of the Galois group of f .Let p ∤ ∆ be a prime, let 1 ≤ r ≤ d, and suppose that f ≡ r i=1 f i (mod p) with the f i distinct irreducible polynomials in (Z/pZ)[x] of degree n i .Then G contains a permutation σ p that is a product of disjoint cycles of length n i ; we call the cycle type of σ p the factorization type of f mod p.
Corollary 1.3.Assume the above notation.Let f ∈ Z[x] be an irreducible monic polynomial.There exists a constant c(f ) > 1 and infinitely pairs of distinct primes p 1 , p 2 such that (1) (2) f mod p 1 and f mod p 2 have the same factorization type.
Theorem 1.1 has many interesting applications to the theory of elliptic curves.Let E/Q be an elliptic curve, and let E d /Q denote the quadratic twist of E by d.Our applications are related to the following conjecture due to Silverman regarding the rank of E ±p (Q) when p is prime.
Conjecture.There are infinitely many primes p for which E ±p (Q) has rank zero, and there are infinitely many primes ℓ for which E ±ℓ (Q) has positive rank.
In light of Silverman's conjecture, we prove the following for certain "good" elliptic curves, which is related to the rank zero component of Silverman's conjecture.
Theorem 1.4.Let E/Q be a "good" elliptic curve, where "good" is defined in Section 4.There exist a constant c(E) > 1 and infinitely many pairs of distinct primes p 1 , p 2 such that (1) In light of recent results by Coates, Li, Yian, and Zhai [2], we use our results to study ranks of twists of the elliptic curve E = X 0 (49), whose minimal Weierstrass equation is given by E : y 2 + xy = x 3 − x 2 − 2x − 1.Let p > 7 be a prime such that p ≡ 3 (mod 4) and p is inert in the field Q( √ −7).For k ≥ 0, let q = k i=1 q i be a product of distinct primes q i = p, each of which splits completely in Q(E [4]).Suppose further that the ideal class group of Q( √ −pq) has no element of order 4.Under these hypotheses, Coates, Li, Yian, and Zhai prove that the Hasse-Weil L-function L(E −pq , s) has a simple zero at s = 1, E −pq (Q) has rank 1, and the Shafarevich-Tate group of E −pq is finite of odd cardinality.They predict that every elliptic curve should satisfy a property similar to this.We prove the following.Theorem 1.5.Let E = X 0 (49).There exist infinitely many pairs of distinct primes p 1 , p 2 such that (1) (2) The Hasse-Weil L-functions L(E −p1 , s) and L(E −p2 , s) have a simple zero at s = 1, the ranks of E −p1 (Q) and E −p2 (Q) are 1, and the Shafarevich-Tate groups of E −p1 and E −p2 are finite of odd cardinality.
A specific elliptic curve for which the entirety of Conjecture 1 is true is the congruent number elliptic curve E ′ : y 2 = x 3 − x.We call a positive squarefree integer d congruent if d is the area of a right triangle with sides of rational length.It is well-known that d is a congruent number if and only if E ′ d (Q) has positive rank.If p is prime, it is also well-known that For such primes, the existence of bounded gaps follows immediately from the second part of Theorem 1.1.We obtain the following result for twists E ′ p (Q), but one can easily adapt the statement to suit the twists E ′ 2p (Q).
Theorem 1.6.There exist infinitely many pairs of distinct primes p 1 , p 2 such that (1) the ranks of E ′ p1 (Q) and E ′ p2 (Q) are either both zero or both positive.In particular, we have bounded gaps between congruent primes and between noncongruent primes.
We also have applications to congruence conditions satisfied by the Fourier coefficients of normalized Hecke eigenforms, i.e. newforms, on congruence subgroups of SL 2 (Z).
] be a newform of even weight k ≥ 2, let ℓ be prime, and let p 0 ∤ N ℓ be prime.There exist a constant c(f, ℓ) > 1 and infinitely many pairs of distinct primes p 1 , p 2 such that (1) In particular, there are bounded gaps between primes p for which a f (p) ≡ 0 (mod ℓ).
As an application of Theorem 1.7, let where τ is the Ramanujan tau function.In this case, we have bounded gaps between primes p for which τ (p) ≡ 0 (mod ℓ) for any prime ℓ.If k = 2, then f is the newform associated to an elliptic curve E/Q with conductor N .In this case, a f (p) = p + 1 − #E(F p ), and we have bounded gaps between primes p for which #E(F p ) ≡ p + 1 (mod ℓ) for any prime ℓ.Finally, we consider primes represented by binary, positive-definite, integervalued quadratic forms.
(2) Both p 1 and p 2 are represented by Q.
In particular, if −4n < 0 is a fundamental discriminant, then there are bounded gaps between primes of the form x 2 + ny 2 .
Acknowledgements.I would like to thank Tristan Freiberg and Ethan Smith for their useful comments and suggestions on this paper.I would particularly like to thank James Maynard for spending time discussing his exciting recent work and answering my many questions, as well as Ken Ono for his support and for suggesting this project.Numerical computations were performed using Mathematica 9.

Notation
Given functions f, g defined on a subset of R, we say that f Additionally, we use the notation f (x) ≪ g(x) to mean the same as f (x) = O(g(x)).
We will let k be a fixed positive integer.The set H = {h 1 , . . ., h k } ⊂ Z will always be admissible; that is, for every prime p, the set {h 1 mod p, . . ., h k mod p} does not contain all residue classes of Z/pZ.Any constants implied by the asymptotic notation o, O, or ≪ may depend on k, H, and the Galois extension K/Q.We let N denote a large positive integer, and all asymptotic notation refers to behavior as N tends to infinity.All sums, products, and suprema are taken over variables in either the primes, which we denote as P, or the positive integers unless otherwise noted.The functions ϕ, τ r (n), and µ refer to the Euler totient function, the number of representations of n as a product of r positive integers, and the Möbius function, respectively.We let ǫ > 0 be a sufficiently small real number, where sufficiency will be obvious by context.We let p be a rational prime; given a set P ⊂ P, p n will denote the n-th prime of P. We let #S or |S| denote the cardinality of a finite set S.

Bounded Gaps Between Primes
The variant of the Selberg sieve developed in [8] eliminates the θ ≥ 1 2 barrier to achieving bounded gaps between primes that the original GPY method encountered.By studying the proof of the following theorem, it is clear that we obtain bounded gaps between primes as long as θ > 0, a condition which is guaranteed by the Bombieri-Vinogradov Theorem.
We provide a brief outline the important components of the proof given in [8].For a fixed admissible set H = {h 1 , . . ., h k }, we consider the sums where w n are nonnegative weights, ρ > 0, χ P is the indicator function of the primes, and The goal is to show that S(N, ρ) > 0 for all sufficiently large N .This would imply that for infinitely many N , there exists n ∈ [N, 2N ) for which at least ⌊ρ + 1⌋ of the n + h i are prime, establishing an infinitude of intervals of bounded length containing ⌊ρ + 1⌋ primes.First, estimates on the sums S 1 (N ) and S 2 (N ) are established.
Proposition 3.2.Let P have level of distribution θ > 0. We have provided that I k (F ) = 0 and J (i) k (F ) = 0 for each i, where Proof.This is proven in Sections 5 and 6 of [8].
Following the GPY method, we want S 2 − ρS 1 to be positive for all sufficiently large N , ensuring that for infinitely many n, several of the n + h i are prime.The following proposition states this formally.Proposition 3.3.Let P have level of distribution θ > 0, and let H = {h 1 , . . ., h k } be an admissible set.Let S k denote the set of piecewise differentiable functions There are infinitely many n such that at least r k of the n + h i are prime.Furthermore, if p n is the n-th prime, then Proof.This is proven in Section 4 of [8].
All that remains is to find a suitable lower bound for M k .
The exact manner in which these propositions are put together is outlined in Section 4 of [8].We emulate those arguments in the next section.

Proof of Theorem 1.1
One fascinating aspect of the proof of Theorem 3.1 is how adaptable it is to exploring bounded gaps between primes in special subsets of the primes.In this section, we will modify the proof to obtain a version applicable to sets of primes satisfying a Chebotarev condition.
Let K/Q be a Galois extension of number fields with Galois group G and discriminant ∆, and let C be a conjugacy class of G. Let (4.1) where [ K/Q • ] is the Artin symbol, and define π P (N ; q, a) = #{N ≤ p < 2N : p ∈ P : p ≡ a (mod q)}.(1) We have π P (N ) = δx/ log(x) + O(x/ log(x) 2 ).
(2) Let θ = 2/|G|.For any fixed A > 0 and any small ǫ > 0, Proof.The first part is the Chebotarev Density Theorem with error term.The second part follows from the main result in [10].
In order to use the second part of Lemma 4.1, we must slightly modify the work in the previous section.Let W be defined as in (3.4), and let H = {h 1 , . . ., h k } be admissible.For a positive integer n, let rad(n) = p|n p. Define By the Chinese Remainder Theorem and the admissibility of H, there exists an integer u 0 satisfying ( Instead of the restriction n ≡ v 0 (mod W ), we use n ≡ u 0 (mod U ).We note that when N is sufficiently large, rad(∆ det(H)) divides W .As in the previous section, λ d1,...,d k will be supported when Therefore, if N is sufficiently large, then (3.4), (4.4), and (4.where ρ > 0. With θ = 2/|G| as in Lemma 4.1, let R = N θ/2−ǫ .We have the following estimate S 1 (N, P).Proposition 4.2.Assume the above notation.If the primes in P have level of distribution θ > 0, then where I k (F ) is defined in Proposition 3.2.
Proof.The only difference between S 1 from Proposition 3.2 and S 1 (P) is that instead of the condition n ≡ v 0 (mod W ), we have n ≡ u 0 (mod U ).Following the proof of Lemma 5.1 in [8], we will alleviate S 1 (P) of any conditions in the sums that depend on U .Then the Selberg sieve manipulations and analysis from [8] will give us the desired estimates.
Expanding the square gives us Using the Chinese Remainder Theorem, we conclude that the inner sum can be written as a sum over a single residue class modulo q = U k i=1 [d i , e i ] when U and each [d i , e i ] are pairwise coprime, in which case the inner sum is N/q + O(1).
Otherwise, λ d1...,d k λ e1,...,e k = 0. Using Lemma 5.1 of [8] which is a condition that is independent of the arithmetic progression containing n.Therefore, the condition ′ is independent of our modulus U , as desired.We now see that S 1 (N, P) is a multiple (depending only on ∆) of S 1 (N ) in one of the intermediate steps in Lemma 5.1 of [8].Therefore, the proposition follows from Lemmata 5.1 and 6.2 of [8].
We will use the reasoning from the above proof to estimate S 2 (N, P). where Proof.The desired result follows from estimating each S (m) 2 (N, P).Expanding the square gives us As with S 1 (N, P), the inner sum can be written as a sum over a single residue class We can conclude from the support of λ d1,...,d k and our choices of u 0 and a m that Therefore, (q, a m + h m ) = 1 if and only if d m = e m = 1.In this case, the inner sum will have size π P (N )/ϕ(q) + O(E(N, q)), where If (q, a m +h m ) = 1, then the inner sum equals either 0 or 1.The inner sum equals 1 if and only if there exists a prime p satisfying n + h m = p for some n ∈ [N, 2N ) with p | q.Since N is large, we have Thus n + h m = p for some n ∈ [N, 2N ) implies that p > R, so if p | q, then λ d1,...,d k λ e1,...,e k = 0. Thus the inner sum only contributes to S (m) 2 (P) when (q, a m + h m ) = 1.We conclude that where q = U k i=1 [d i , e i ] and ′ denotes the restriction that U and each [d i , e i ] be pairwise coprime.
We first analyze the error term.From the support of λ d1,...,d k , we only need to consider squarefree q < R 2 U ≤ N θ−ǫ satisfying (q, ∆) = 1, where ǫ > 0 is sufficiently small.Given a squarefree integer r, there are at most τ 3k (r) choices of Using the Cauchy-Schwarz inequality and the trivial bound E(N, q) ≪ N/ϕ(q), the error term is It follows from Lemma 4.1 that the error is ≪ λ 2 max N/ log(N ) A for any fixed A > 0, which is also true of the error term in S (m) 2 (N ) in Lemma 5.2 of [8].Using (4.4), for any fixed A > 0, we have where ′ denotes the restriction that U and each [d i , e i ] be pairwise coprime.As in the proof of Proposition 4.2, we can take ′ to denote the restriction that i =j ([d i , e i ], [d j , e j ]) = 1.We now see that up to the choice of prime counting function (which results in the factor of δ in the statement of the proposition), S Then there are infinitely many n such that at least r k of the n + h i are in P. Furthermore, if p n is the n-th prime in P, then Proof.We want to show that S(N, ρ, P) > 0 for all sufficiently large N .Recall that R = N θ/2−ǫ for some small ǫ > 0. By the definition of M k , we can choose Using Propositions 4.2 and 4.3 and the identity ϕ(rad(∆)) rad(∆) = ϕ(∆) ∆ , we have If ǫ is sufficiently small, then S(N, ρ, P) > 0 for all sufficiently large N .Thus there are infinitely many n for which at least ⌊ρ + 1⌋ of the n + h i are in P. If ǫ is sufficiently small, then ⌊ρ + 1⌋ = δθϕ(∆)M k

2∆
, and we obtain the claimed result.
It remains to find a suitable lower bound for M k .Proposition 3.4 gives us a lower bound on M k when k is sufficiently large.We will now establish the full range of k for which this lower bound holds.Although this lower bound is far from optimal for k low in the range, the following suffices for the purposes of this paper since k will typically be very large.Proposition 4.5.Let k ≥ 213 be a positive integer.We have Proof.By the analysis in Section 8 of [8], for some positive constant A, we have , provided that the right hand side is positive.Let A = log(k) − 2 log(log(k)) as in [8].With this choice of A, the left hand side of the above inequality is bounded below by log(k) − 2 log(log(k)) − 2 for k ≥ 16.Since log(k) − 2 log(log(k)) − 2 > 0 when k ≥ 213, we have the desired result.
Proof of Theorem 1.1.The second part is proven in [7]; it remains to prove the first part.Suppose that K/Q is nonabelian.Since |G| ≥ 4, Lemma 4.1 tells us that P has level of distribution θ = 2/|G|.By Proposition 4.5, if k ≥ 213 is an integer, then Choosing ǫ = θ k , it follows from a numerical calculation that (4.11) is greater than .
The above expression is minimized when δ = θ = ∆ = 1, in which case k ≥ 213 as required by Proposition 4.5.Thus for any admissible set H = {h 1 , . . ., h k } with k as above, at least 2 of the n + h i are in P for infinitely many integers n.We can choose h j = q π(k)+j , where 1 ≤ j ≤ k and q j is the j-th prime in P. For n ≥ 6, we have [4] n Therefore, if p n is the n-th prime of P, then .
We obtain the desired result upon substituting 2 |G| for θ and |C| |G| for δ.
5. Proofs of Theorems 1.4, 1.5, 1.7, and 1.8 To prove Theorems 1.4, 1.5, 1.7, and 1.8, it suffices to prove that the set of primes in each theorem is a Chebotarev set.The claimed bounds then follow from Theorem 1.1.
To prove Theorem 1.4, let E/Q be an elliptic curve with Weierstrass form where the discriminant of the cubic is nonzero.We will assume that E and its points are Q-rational.If d is a squarefree integer, we define E d to be the d-quadratic twist of E given by E d : dy 2 = x 3 + ax 2 + bx + c.Let E be an elliptic curve without Q-rational 2-torsion.Following [1], we call E good if E satisfies the following criteria: (1) The 2-Selmer rank of E is zero.
(2) The discriminant ∆ of E is negative.
(3) If p is any prime for which E has bad reduction, then E has multiplicative reduction at p, and v p (∆) is odd.(4) E has good reduction at 2 and the reduction of E modulo 2 has j-invariant zero.A prototypical example of a good elliptic curve is E = X 0 (11), which has Weierstrass form E : y We define a squarefree integer d to be 2-trivial for E if E has no rational 2-torsion modulo p for every odd prime p | d.For good elliptic curves, the following is proven in [1].
In particular, for such odd d, we have that the rank of E d (Q) is zero.
We now prove Theorem 1.4.
Proof of Theorem 1.4.We write E in Weierstrass form E : y 2 = f (x), where f (x) = x 3 + ax 2 + bx + c ∈ Z[x] has Galois group G and discriminant ∆.Since E is good, f is irreducible over Z and G ∼ = S 3 .By the above discussion, the primes p satisfying the hypotheses of Theorem 5.1 are exactly the primes p ∤ ∆ such that f mod p is irreducible, that is, the factorization type of f mod p corresponds to 3-cycles in S 3 .
The desired result now follows from Corollary 1.3.
The following result is proven in [2], which we will use to prove Theorem 1.5.
(3) The ideal class group H N of the field Q( √ −N ) has no element of order 4.
Then the Hasse-Weil L-function L(E −N , s) has a simple zero at s = 1, E −N (Q) has rank 1, and the Shafarevich-Tate group of E −N is finite of odd order.
To prove Theorem 1.5, we consider the case of Theorem 5.2 where k = 0.
Proof of Theorem 1.5.Using the theory of quadratic forms, Gauss proved that if p ≡ 3 (mod 4), then |H p | is odd.Thus Theorem 5.2 holds when N is a prime such that N = 7 such that N ≡ 3 (mod 4) and N is a quadratic non-residue modulo 7.
Every prime p congruent to 3, 19, or 27 modulo 28 satisfies this condition, and the desired result follows from the second part of Theorem 1.1.
Following Murty and Murty [9], let q = e 2πiz , and let be a newform of even weight k ≥ 2 and character χ. (This forces χ to be real, and χ is nontrivial if and only if f has complex multiplication.)Let G = Gal( Q/Q), and let ℓ be prime.By the work of Deligne, there exists a representation ρ ℓ : G → GL 2 (Z/ℓZ) with the property that if p ∤ N ℓ is prime and σ p is a Frobenius element at p in G, then ρ ℓ is unramified at p and trρ ℓ (σ p ) = a f (p), det ρ ℓ (σ p ) = χ(p)p k−1 .
Let H ℓ be the kernel of ρ ℓ , let K ℓ be the subfield of Q fixed by H ℓ , and let G ℓ = G/H ℓ = Gal(K ℓ /Q).Let p 0 ∤ N ℓ be prime, and let C p0,ℓ be the union of conjugacy classes of ρ ℓ (G) consisting of elements of trace a f (p 0 ).
Proof of Theorem 1.7.For a prime p ∤ N ℓ, the condition a f (p) ≡ a f (p 0 ) (mod ℓ) is equivalent to the condition that for any Frobenius element σ p of p, we have ρ ℓ (σ p ) ∈ C p0,ℓ .Therefore, the primes p satisfying p ∤ N ℓ and a f (p) ≡ a f (p 0 ) (mod ℓ) form a Chebotarev set of the primes p unramified in K ℓ with [ K ℓ /Q p ] ⊂ C p0,ℓ .The desired result now follows from Theorem 1.1.We obtain the final claim by choosing p 0 so that a f (p 0 ) ≡ 0 (mod ℓ).
We now prove Theorem 1.8.
, P) is a multiple (depending only on δ and ∆) of S (m) 2 (N ) in one of the intermediate steps in Lemma 5.2 of[8].Therefore, the proposition follows from Lemmata 5.2 and 6.3 of[8] and Lemma 4.1.We now modify Proposition 3.3 accordingly.

Proposition 4 . 4 .
Let H = {h 1 , . . ., h k } be an admissible set, and let and (4.4), we have max = sup d1,...,d k |λ d1,...,d k | and ′ denotes the restriction that U and each [d i , e i ] are pairwise coprime and each d i , e i is squarefree.If a prime p divides ([d i , e i ], U ) for some i, then we have already shown that λ d1,...,d k λ e1,...,e k = 0. Therefore, we may take ′ to denote the condition that i =j ([d i , e i ], [d j , e j