 Research
 Open Access
Bounded gaps between primes in Chebotarev sets
 Jesse Thorner^{1}Email author
https://doi.org/10.1186/2197984714
© Thorner; licensee Springer. 2014
 Received: 20 February 2014
 Accepted: 14 March 2014
 Published: 17 June 2014
Abstract
Purpose
A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes p_{1}, p_{2} with p_{1}p_{2} ≤ 600 as a consequence of the BombieriVinogradov Theorem. In this paper, we apply his general method to the setting of Chebotarev sets of primes.
Methods
We use recent developments in sieve theory due to Maynard and Tao in conjunction with standard results in algebraic number theory.
Results
Given a Galois extension $K/\mathbb{Q}$, we prove the existence of bounded gaps between primes p having the same Artin symbol $\left[\frac{K/\mathbb{Q}}{p}\right]$
.
Conclusions
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.
AMS subject classification
Primary 11N05; 11N36; Secondary 11G05
Keywords
 Twin primes
 Chebotarev Density Theorem
 Elliptic curves
Background
Introduction and statement of results
The longstanding 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.
As an application of the large sieve, Bombieri and Vinogradov established that (1) holds when $0<\theta <\frac{1}{2}$. It was conjectured by Elliott and Halberstam [2] that (1) holds when 0<θ < 1. The GPY method produces bounded gaps between primes assuming that $\theta >\frac{1}{2}$. In [3], Zhang proved that there are infinitely many pairs of distinct primes p_{1},p_{2} satisfying p_{1}p_{2} ≤ 7 × 10^{7} by finding a suitable modification for (1) which is valid for $\theta >\frac{1}{2}$. Zhang’s work is inspiring but seems difficult to adapt to other settings.
In [4], 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 of the primes is positive (which is given by the BombieriVinogradov Theorem).
In this paper, we exploit the flexibility in the methods presented in [4] to obtain analogous results on bounded gaps between primes in Chebotarev sets . These sets are characterized as follows: Let $K/\mathbb{Q}$ be a Galois extension of number fields with Galois group G and discriminant Δ. For a prime p ∤ Δ, 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. We denote this conjugacy class by the Artin symbol $\left[\frac{K/\mathbb{Q}}{p}\right]$. We say that a subset of the primes is a Chebotarev set, or that satisfies a Chebotarev condition, if there exists an extension $K/\mathbb{Q}$ and a union of conjugacy classes C ⊂ G such that is a union of sets of the form $\{p\phantom{\rule{2.77626pt}{0ex}}\text{prime}\phantom{\rule{0.3em}{0ex}}:p\nmid \mathrm{\Delta},\left[\frac{K/\mathbb{Q}}{p}\right]=C\}$. The Chebotarev Density Theorem asserts that has relative density within the primes that is both positive and rational, and a result of Murty and Murty [5] tells us that we can extend the notion of a positive level of distribution to if we omit certain arithmetic progressions. These two ingredients in conjunction with the sieve developed in [4] enable us to prove the existence of bounded gaps between primes in any Chebotarev set.
By the KroneckerWeber Theorem, if $K/\mathbb{Q}$ is an abelian extension, then 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 [6] using a combinatorial argument. In this paper, we handle the nonabelian extensions, proving a complete characterization of bounded gaps between primes in Chebotarev sets.
Theorem 1
Let$K/\mathbb{Q}$be a Galois extension of number fields with Galois group G and discriminant Δ, and let C be a conjugacy class of G. Let be the set of primes p ∤ Δ for which$\left[\frac{K/\mathbb{Q}}{p}\right]=C$.
1. If$K/\mathbb{Q}$is a nonabelian extension, then there exist infinitely many pairs of distinct primes${p}_{1},{p}_{2}\in \mathcal{P}$such that${p}_{1}{p}_{2}\le 825{\left(\frac{\leftG{}^{2}\right\mathrm{\Delta}}{\leftC\right\phi \left(\right\mathrm{\Delta}\left\right)}\right)}^{3}exp\left(\frac{\leftG{}^{2}\right\mathrm{\Delta}}{\leftC\right\phi \left(\right\mathrm{\Delta}\left\right)}\right)$.
2. If$K/\mathbb{Q}$is an abelian extension, let q be the smallest positive integer so that$K\subset \mathbb{Q}\left({e}^{2\pi i/q}\right)$. There exist infinitely many pairs of distinct primes${p}_{1},{p}_{2}\in \mathcal{P}$such that p_{1}p_{2} ≤ 600q.
We use Theorem 1 to prove several results in the algebraic number theory. The first two of these are immediate.
Corollary 1
Let$K/\mathbb{Q}$be a Galois extension of number fields with ring of integers${\mathcal{O}}_{K}$. There exist a constant c(K) > 1 and infinitely many pairs of non conjugate prime ideals such that .
Let $f\in \mathbb{Z}\left[x\right]$ be monic polynomial of degree d and discriminant Δ that is irreducible over , and let G be the permutation representation of the Galois group of f. Let p ∤ Δ be a prime, let 1 ≤ r ≤ d, and suppose that $f\equiv \prod _{i=1}^{r}{f}_{i}\phantom{\rule{2.77626pt}{0ex}}\left(\text{mod}\phantom{\rule{2.77626pt}{0ex}}p\right)$ with the f_{ i } distinct irreducible polynomials in $(\mathbb{Z}/p\mathbb{Z})\left[x\right]$ 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 2
 1.
p _{1}p _{2} ≤ c(f).
 2.
f mod p _{1} and f mod p _{2} have the same factorization type.
Theorem 1 has many interesting applications to the theory of elliptic curves. Let $E/\mathbb{Q}$ be an elliptic curve, and let ${E}_{d}/\mathbb{Q}$ denote the quadratic twist of E by d. We denote the rank of the group of rational points $E\left(\mathbb{Q}\right)$ by rk(E). Our applications are related to the following conjecture due to Silverman regarding rk(E_{±p}) when p is prime.
Conjecture 1
There are infinitely many primes p for which rk(E_{ p }) = 0 or rk(E_{p}) = 0, and there are infinitely many primes ℓ for which rk(E_{ ℓ }) > 0 or rk(E_{ℓ}) > 0.
In light of Silverman’s conjecture, we prove the following result for certain ‘good’ elliptic curves, which is related to the rank zero component of Silverman’s conjecture.
Theorem 2
 1.
p _{1}p _{2} ≤ c(E).
 2.
$\text{rk}\left({E}_{{p}_{1}}\right)=\text{rk}\left({E}_{{p}_{2}}\right)=0.$
In light of recent results by Coates, Li, Tian, and Zhai, as described in [7], 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} + x y = 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 $\mathbb{Q}\left(\sqrt{7}\right)$. For k ≥ 0, let $q=\prod _{i=1}^{k}{q}_{i}$ be a product of distinct primes q_{ i } ≠ p, each of which splits completely in $\mathbb{Q}\left(E\left[4\right]\right)$. Suppose further that the ideal class group of $\mathbb{Q}\left(\sqrt{\mathit{\text{pq}}}\right)$ has no element of order 4. Under these hypotheses, Coates, Li, Yian, and Zhai prove that the HasseWeil Lfunction L(E_{p q},s) has a simple zero at s = 1, rk(E_{p q}) = 1, and the ShafarevichTate group $\u0428\left({E}_{\mathit{\text{pq}}}/\mathbb{Q}\right)$ is finite of odd cardinality. They predict that every elliptic curve should satisfy a property similar to this. We prove the following:
Theorem 3
 1.
p _{1}p _{2} ≤ 16,800.
 2.
Both $L\left({E}_{{p}_{1}},s\right)$ and $L\left({E}_{{p}_{2}},s\right)$ have a simple zero at s = 1, $\text{rk}\left({E}_{{p}_{1}}\right)=\text{rk}\left({E}_{{p}_{2}}\right)=1$, and $\u0428\left({E}_{{p}_{1}}/\mathbb{Q}\right)$ and $\u0428\left({E}_{{p}_{2}}/\mathbb{Q}\right)$ are both finite of odd cardinality.
A specific elliptic curve for which the entirety of Silverman’s conjecture 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 wellknown that d is a congruent number if and only if ${E}_{d}^{\prime}\left(\mathbb{Q}\right)$ has positive rank. If p is prime, it is also known [8] that

If p ≡ 3 (mod 8), then $\text{rk}\left({E}_{p}^{\prime}\right)=0$.

If p ≡ 5 (mod 8), then $\text{rk}\left({E}_{2p}^{\prime}\right)=0$.

If p ≡ 5 or 7 (mod 8), then $\text{rk}\left({E}_{p}^{\prime}\right)=1$.

If p ≡ 3 (mod 4), then $\text{rk}\left({E}_{2p}^{\prime}\right)=1$.
For such primes, the existence of bounded gaps follows immediately from the second part of Theorem 1. We obtain the following result for twists ${E}_{p}^{\prime}\left(\mathbb{Q}\right)$, but one can easily adapt the statement to suit the twists ${E}_{2p}^{\prime}\left(\mathbb{Q}\right)$.
Theorem 4
There exist infinitely many pairs of distinct primes p_{1},p_{2}such that
1. p_{1}p_{2} ≤ 4800.
2. Either$\text{rk}\left({E}_{{p}_{1}}^{\prime}\right)=\text{rk}\left({E}_{{p}_{2}}\right)=0$ or $\text{rk}\left({E}_{{p}_{1}}^{\prime}\right)=\text{rk}\left({E}_{{p}_{2}}^{\prime}\right)=1$.
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 ${\text{SL}}_{2}\left(\mathbb{Z}\right)$.
Theorem 5
 1.
p _{1}p _{2} ≤ c(d, f).
 2.
a _{ f }(p _{0}) ≡ a _{ f }(p _{1}) ≡ a _{ f }(p _{2}) (mod d).
In particular, we have bounded gaps between primes p satisfying a_{ f }(p) ≡ 0 (mod d).
where τ is the Ramanujan tau function. In this case, we have bounded gaps between primes p for which τ(p) ≡ 0 (mod d) for any positive integer d. If k = 2, then f is the newform associated to an elliptic curve $E/\mathbb{Q}$ with conductor N. In this case, ${a}_{f}\left(p\right)=p+1\mathrm{\#E}\left({\mathbb{F}}_{p}\right)$, and we have bounded gaps between primes p for which $\mathrm{\#E}\left({\mathbb{F}}_{p}\right)\equiv p+1\phantom{\rule{2.77626pt}{0ex}}\left(\text{mod}\phantom{\rule{2.77626pt}{0ex}}d\right)$ for any positive integer d.
Finally, we consider primes represented by binary quadratic forms. Let $Q(x,y)=a{x}^{2}+\mathit{\text{bxy}}+c{y}^{2}\in \mathbb{Z}\left[x,y\right]$ be a primitive, positivedefinite quadratic form with discriminant D = b^{2}4a c < 0. It is known that the primes represented by Q form a Chebotarev set. The proportion of primes that are represented by Q is either $\frac{1}{h\left(D\right)}$ or $\frac{1}{2h\left(D\right)}$, where h(D) is the class number for quadratic forms Q of discriminant D. (See [9] for further discussion.) The Chebotarev condition satisfied by these primes is in the extension ${H}_{D}/\mathbb{Q}$, where H_{ D } is the Hilbert class field of $\mathbb{Q}\left(\sqrt{D}\right)$.
Theorem 6
Let$Q(x,y)=a{x}^{2}+\mathit{\text{bxy}}+c{y}^{2}\in \mathbb{Z}\left[x,y\right]$be a primitive, positivedefinite quadratic form with b^{2}4a c < 0. There exist a constant c(Q) > 1 and infinitely many pairs of distinct primes p_{1}, p_{2}such that
1. p_{1}p_{2} ≤ c(Q).
2. Both p_{1}and p_{2}are represented by Q.
In particular, if n is a positive integer, then there are bounded gaps between primes of the form x^{2} + n y^{2}.
Remark 1
The abelian case and all of the above applications have similar results. The proof is similar to the case of pairs of primes presented here.
Notation
Given functions f, g defined on a subset of , we say that f(x) = O(g(x)) if ${limsup}_{x\to \infty}\left\phantom{\rule{0.3em}{0ex}}f\right(x)/g(x\left)\right<\infty $, and f(x) = o(g(x)) if $\underset{x\to \infty}{lim}f\left(x\right)/g\left(x\right)=0$. Additionally, we use the notation f(x) ≪ g(x) to mean the same as f(x) = O(g(x)). We let δ, ε > 0 be sufficiently small real numbers, where sufficiency will be obvious by context. The values of δ,ε may vary at each occurrence. 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 , or the positive integers unless otherwise noted. Any constants implied by the asymptotic notation o,O, or ≪ will not depend on N, but may depend on k, , δ, ε, or other quantities specified in the paper.
We will let k be a fixed positive integer. The set $\mathcal{H}=\{{h}_{1},\dots ,{h}_{k}\}\subset \mathbb{Z}$ 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 $\mathbb{Z}/p\mathbb{Z}$. 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 # S or S denote the cardinality of a finite set S. For any $x\in \mathbb{R}$, we let $\lfloor x\rfloor =\text{max}\{a\in \mathbb{Z}:a\le x\}$ and $\lceil x\rceil =min\{a\in \mathbb{Z}:a\ge x\}$. We let $(a,b)=gcd(a,b)$ and [a, b] = lcm(a, b).
Bounded gaps between primes
The variant of the Selberg sieve developed in [4] eliminates the $\theta >\frac{1}{2}$ 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 BombieriVinogradov Theorem.
Theorem 7 (Maynard)
There are infinitely many pairs of distinct primes p_{1}, p_{2}with p_{1}p_{2} ≤ 600.
By the Prime Number Theorem, W ≪ (log logN)^{2}.
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.
Setting $d=\prod _{i=1}^{k}{d}_{i}$, we choose ${\lambda}_{{d}_{1},\dots ,{d}_{k}}$ to be supported when d < R, (d, W) = 1, and μ(d)^{2} = 1.
First, estimates on the sums S_{1}(N) and S_{2}(N) are established.
Proposition 1
Proof
This is proven in sections 5 and 6 of [4].
Following the GPY method, we want S_{2}(N)ρ S_{1}(N) 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 2
Proof
This is proven in section 4 of [4].
All that remains is to find a suitable lower bound for M_{ k }.
Proposition 3
Proof
This is proven in sections 7 and 8 of [4].
The exact manner in which these propositions are put together is outlined in section 4 of [4]. We emulate those arguments in the next section.
Methods
Proof of Theorem 1
One fascinating aspect of the proof of Theorem 7 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.
Lemma 1
Assume the above notation. Let δ = C / G. Suppose that G ≥ 4.
1. We have ${\pi}_{\mathcal{P}}\left(N\right)=\mathrm{\delta N}/log\left(N\right)+O(N/{(logN)}^{2}).$
2. Equation 12 holds when M = Δ and 0 < θ < 2 / G.
Proof
The first part is the Chebotarev Density Theorem with error term. The second part follows from the main result in [5].
where ρ > 0. For a fixed θ > 0 satisfying (12), let R = N^{θ/2δ}. We have the following estimate ${S}_{1}(N,\mathcal{P})$.
Proposition 4
where I_{ k }(F) is defined in Proposition 1.
Proof
The only difference between S_{1} from Proposition 1 and ${S}_{1}\left(\mathcal{P}\right)$ 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 [4], we will alleviate ${S}_{1}(N,\mathcal{P})$ of any conditions in the sums that depend on U. Then the Selberg sieve manipulations and analysis from [4] will give us the desired estimates.
 1.
If a prime p divides (U, [d _{ i }, e _{ i }]) for some i, then p ∣ (U, d _{ i }) or p ∣ (U, e _{ i }). Thus, p ∣ (d, U) or p ∣ (e, U), and ${\lambda}_{{d}_{1},\dots ,{d}_{k}}{\lambda}_{{e}_{1},\dots ,{e}_{k}}=0$ by (15).
 2.If a prime p divides ([d _{ i }, e _{ i }], [d _{ j }, e _{ j }]) for some i ≠ j, then$p\mid d\text{or}p\mid \text{e,}\phantom{\rule{2em}{0ex}}p\mid n+{h}_{i},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}p\mid n+{h}_{j}.$
Thus, p ∣ (d, h_{ i }h_{ j }) or p ∣ (e, h_{ i }h_{ j }). Therefore, $p\mid (d,det(\mathcal{H}\left)\right)$ or $p\mid (e,det(\mathcal{H}\left)\right)$, and ${\lambda}_{{d}_{1},\dots ,{d}_{k}}{\lambda}_{{e}_{1},\dots ,{e}_{k}}=0$ by (15).
where ${\lambda}_{\text{max}}=\underset{{d}_{1},\dots ,{d}_{k}}{sup}\left{\lambda}_{{d}_{1},\dots ,{d}_{k}}\right$ and ${\sum}^{\prime}$ 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 ${\lambda}_{{d}_{1},\dots ,{d}_{k}}{\lambda}_{{e}_{1},\dots ,{e}_{k}}=0$. Therefore, we may take ${\sum}^{\prime}$ to denote the condition that $\prod _{i\ne j}\left(\left[{d}_{i},{e}_{i}\right],\left[{d}_{j},{e}_{j}\right]\right)=1,$ which is a condition that is independent of the arithmetic progression containing n. Therefore, the condition ${\sum}^{\prime}$ is independent of our modulus U, as desired.
We now see that ${S}_{1}(N,\mathcal{P})$ is a multiple (depending only on Δ) of S_{1}(N) in one of the intermediate steps in Lemma 5.1 of [4]. Therefore, the proposition follows from Lemmata 5.1 and 6.2 of [4].
We will use the reasoning from the above proof to estimate ${S}_{2}(N,\mathcal{P})$.
Proposition 5
where ${J}_{k}^{\left(i\right)}\left(F\right)$ is defined in Proposition 1.
Proof
As with ${S}_{1}(N,\mathcal{P})$, the inner sum can be written as a sum over a single residue class a_{ m } modulo $q=U\prod _{i=1}^{k}\left[{d}_{i},{e}_{i}\right]$ when U and each [d_{ i }, e_{ i }] are pairwise coprime, and ${\lambda}_{{d}_{1},\dots ,{d}_{k}}{\lambda}_{{e}_{1},\dots ,{e}_{k}}=0$ otherwise.
where $q=U\prod _{i=1}^{k}\left[{d}_{i},{e}_{i}\right]$ and ${\sum}^{\prime}$ denotes the restriction that U and each [d_{ i }, e_{ i }] be pairwise coprime.
It follows from elementary bounds on τ_{3k}(r) and Lemma 1 that the error is $\ll {\lambda}_{\text{max}}^{2}N/{(logN)}^{A}$ for any fixed A > 0, which is also true of the error term in ${S}_{2}^{\left(m\right)}\left(N\right)$ in Lemma 5.2 of [4].
where ${\sum}^{\prime}$ denotes the restriction that U and each [d_{ i }, e_{ i }] be pairwise coprime. As in the proof of Proposition 4, we can take ${\sum}^{\prime}$ to denote the restriction that $\prod _{i\ne j}\left(\left[{d}_{i},{e}_{i}\right],\left[{d}_{j},{e}_{j}\right]\right)=1$. We now see that up to the choice of prime counting function (which results in the factor of in the statement of the proposition), ${S}_{2}^{\left(m\right)}(N,\mathcal{P})$ is a multiple (depending only on and Δ) of ${S}_{2}^{\left(m\right)}\left(N\right)$ in one of the intermediate steps in Lemma 5.2 of [4]. Therefore, the proposition follows from Lemmata 5.2 and 6.3 of [4] and Lemma 1.
We now modify Proposition 2 accordingly.
Proposition 6
Proof
Let $\rho ={M}_{k}\frac{\mathfrak{d}\theta \phi \left(\right\mathrm{\Delta}\left\right)}{2\left\mathrm{\Delta}\right}\mathrm{\epsilon .}$ By choosing δ suitably small (depending on ε), we have $S(N,\rho ,\mathcal{P})>0$ for all sufficiently large N. Thus, there are infinitely many n for which at least ⌊ρ + 1⌋ of the n + h_{ i } are in . If ε is sufficiently small, then $\lfloor \rho +1\rfloor =\u2308\frac{\mathfrak{d}\theta \phi \left(\mathrm{\Delta}\right){M}_{k}}{2\mathrm{\Delta}}\u2309,$ and we obtain the claimed result.
It remains to find a suitable lower bound for M_{ k }. Proposition 3 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 7
Proof
provided that the righthand side is positive. Let A = log(k)  2 log log(k) as in [4]. With this choice of A, the lefthand 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.
We now prove Theorem 1.
Proof
Results and discussions
Proofs of Theorems 2, 3, and 5
To prove Theorems 2, 3, and 5, it suffices to prove that the set of primes in each theorem is a Chebotarev set. The claimed bounds will follow from Theorem 1.
Ranks of elliptic curves
Definition 1
Let$E/\mathbb{Q}$be an elliptic curve without rational 2torsion. Following[12], we call E good if E satisfies the following criteria:
1. The 2Selmer rank of E is zero.
2. The discriminant Δ 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 }(Δ) is odd.
4. E has good reduction at 2 and the reduction of E modulo 2 has jinvariant zero.
A prototypical example of a good elliptic curve is E = X_{0}(11), which has Weierstrass form E:y^{2} = x^{3}  4x^{2}  160x  1264.
We define a squarefree integer d to be 2trivial for E if E has no rational 2torsion modulo p for every odd prime p ∣ d. For good elliptic curves, the following is proven in [12].
Theorem 8
In particular, for such odd d, we have rk(E_{ d }) = 0.
We now prove Theorem 2.
Proof
We write E in Weierstrass form E:y^{2} = f(x), where $f\left(x\right)={x}^{3}+a{x}^{2}+\mathit{\text{bx}}+c\in \mathbb{Z}\left[\phantom{\rule{0.3em}{0ex}}x\right]$ has Galois group G and discriminant Δ. Since E is good, f is irreducible over and G ≅ S_{3}. By the above discussion, the primes p satisfying the hypotheses of Theorem 8 are exactly the primes p ∤ Δ such that f mod p is irreducible, that is, the factorization type of f mod p corresponds to 3cycles in S_{3}. The desired result now follows from Corollary 2.
We will use Theorem 91 of [7], which we now state, to prove Theorem 3.
Theorem 9
Let E = X_{0}(49). For k ≥ 0, let p, q_{1}, …, q_{ k }be prime, and let N = p q_{1}q_{2} ⋯ q_{ k }be a product of distinct primes satisfying
1. p ≡ 3 (mod 4), p ≠ 7, and p is a quadratic nonresidue modulo 7.
2. q_{1}, …, q_{ k }split completely in$\mathbb{Q}\left(E\left[4\right]\right)$.
3. The ideal class group ${\mathcal{H}}_{N}$ of the field $\mathbb{Q}\left(\sqrt{N}\right)$ has no element of order 4.
Then the HasseWeil Lfunction L(E_{  N}, s) has a simple zero at s = 1, ${E}_{N}\left(\mathbb{Q}\right)$has rank 1, and the ShafarevichTate group of E_{N}is finite of odd order.
We now prove Theorem 3.
Proof
We consider the case of Theorem 9 where k = 0. Using the theory of quadratic forms, Gauss proved that if p ≡ 3 (mod 4), then $\left{\mathcal{H}}_{p}\right$ is odd. Thus, Theorem 9 holds when N is a prime such that N ≠ 7 such that N ≡ 3 (mod 4) and N is a quadratic nonresidue 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.
Coefficients of newforms
Let ${\stackrel{~}{\rho}}_{d}:G\to {\text{GL}}_{2}(\mathbb{Z}/d\mathbb{Z})$ be the reduction modulo d of ρ_{ d }. Let H_{ d } be the kernel of ${\stackrel{~}{\rho}}_{d}$, let K_{ d } be the subfield of $\stackrel{\u0304}{\mathbb{Q}}$ fixed by H_{ d }, and let ${G}_{d}=\text{Gal}({K}_{d}/\mathbb{Q})$. If q ∤ d N is prime, then the condition a_{ f }(q) ≡ 0 (mod d) means that for any Frobenius element σ_{ q } of q, ${\stackrel{~}{\rho}}_{d}\left({\sigma}_{q}\right)\in {C}_{d}$. Since C_{ d } contains the image of complex conjugations, C_{ d } is nonempty.
We now prove Theorem 5.
Proof
By the preceding discussion, the set of primes p for which a_{ f }(p) ≡ 0 (mod d) is a Chebotarev set. By a slight variation of the preceding discussion, we find that for any fixed prime p_{0}∤d N, the set of primes p for which a_{ f }(p) ≡ a_{ f }(p_{0}) (mod d) is also Chebotarev set. The desired result now follows from Theorem 1.
Conclusion
We obtained all of the claimed results in the ‘Introduction and statement of results’ section. A discussion of the importance and relevance of the results may be found in the same section.
Declarations
Acknowledgements
I would like to thank Tristan Freiberg, Ethan Smith, and the anonymous referees 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.
Authors’ Affiliations
References
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