Volume 126, Issue 6 p. 1884-1922
RESEARCH ARTICLE
Open Access

The elliptic sieve and Brauer groups

Subham Bhakta

Subham Bhakta

Mathematisches Institut, Georg-August-Universität Göttingen, Göttingen, Germany

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Daniel Loughran

Corresponding Author

Daniel Loughran

Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, UK

Correspondence

Daniel Loughran, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.

Email: [email protected]

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Simon L. Rydin Myerson

Simon L. Rydin Myerson

Mathematics Institute, Zeeman Building, University of Warwick, Coventry, UK

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Masahiro Nakahara

Masahiro Nakahara

Department of Mathematics, University of Washington, Seattle, Washington, USA

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First published: 11 May 2023

Abstract

A theorem of Serre states that almost all plane conics over Q ${{\mathbb {Q}}}$ have no rational point. We prove an analogue of this for families of conics parametrised by elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve for elliptic curves. We also give more general results for specialisations of Brauer groups, which yields applications to norm form equations.

1 INTRODUCTION

1.1 Sums of two squares

A famous theorem of Landau and Ramanujan states that almost all integers are not sums of two squares, when ordered by absolute value. In this paper we prove a version of this result for elliptic curves.

Theorem 1.1.Let E $E$ be an elliptic curve over Q ${{\mathbb {Q}}}$ given by an integral Weierstrass equation. Let P E ( Q ) $P \in E({{\mathbb {Q}}})$ have infinite order with P E ( R ) 0 $P \in E({{\mathbb {R}}})^0$ . Then there exists ω = ω ( E , P ) > 0 $\omega =\omega (E,P) > 0$ such that

# { n Z : | n | B , y ( n P ) is a sum of two squares } E , P B / ( log B ) ω . $$\begin{equation} \#\lbrace n \in {{\mathbb {Z}}}\colon |n| \leqslant B, y(nP) \text{ is a sum of two squares} \rbrace \ll _{E,P} B/(\log B)^{\omega }. \end{equation}$$ (1.1)

Here E ( R ) 0 $E({{\mathbb {R}}})^0$ denotes the connected component of the identity of E ( R ) $E({{\mathbb {R}}})$ , and y ( n P ) $y(nP)$ denotes the y $y$ -coordinate of the point n P $nP$ ; this is a rational number and we are asking that this is the sum of two rational squares. The result shows that for almost all multiples of P $P$ , the y $y$ -coordinate is not a sum of two (rational) squares. (See Section 1.4 for a discussion on sharpness of this result.)

The reader may wonder: Why consider the y $y$ -coordinate and not the x $x$ -coordinate? Well, the corresponding result is false otherwise: Consider the elliptic curve
y 2 = x 3 k 2 . $$\begin{equation} y^2=x^3-k^2. \end{equation}$$ (1.2)
Evidently x = ( y 2 + k 2 ) / x 2 $x=(y^2+k^2)/x^2$ is always a sum of two squares. Our methods are able to handle the x $x$ -coordinate problem, but only with additional assumptions on E $E$ (see Theorem 5.3).

The assumption that P E ( R ) 0 $P \in E({{\mathbb {R}}})^0$ is slightly deeper and is intimately connected with our method; however without it we can obtain counter-examples to similar statements (see Example 1.3).

1.2 Conic bundles

To understand exactly what is happening, we put our results into the more geometric framework of conic bundles. In an influential paper [22], Serre proved that almost all plane conics over Q ${{\mathbb {Q}}}$ have no rational point, when ordered by the size of their coefficients. This was a special case of a more general result [22, Theorem 2] on conic bundles π : X P Q n $\pi :X \rightarrow {{\mathbb {P}}}^n_{{\mathbb {Q}}}$ , which says that providing π $\pi$ has no rational section we have
# { x P n ( Q ) : H ( x ) B , x π ( X ( Q ) ) } X , E , P B n + 1 / ( log B ) Δ ( π ) $$\begin{equation} \#\lbrace x \in {{\mathbb {P}}}^n({{\mathbb {Q}}}) : H(x) \leqslant B, x \in \pi (X({{\mathbb {Q}}})) \rbrace \ll _{X,E,P} B^{n+1}/(\log B)^{\Delta (\pi )} \end{equation}$$ (1.3)
for some Δ ( π ) > 0 $\Delta (\pi ) > 0$ , where H $H$ denotes the usual naive height on projective space. Here by a conic bundle, we mean a surjective morphism of varieties all of whose fibres are isomorphic to plane conics. This result was generalised in [18, Theorem 1.2] to other families of varieties over P n ${{\mathbb {P}}}^n$ .
One of the aims of the paper is to obtain a version of Serre's result for conic bundles over elliptic curves. Here the crucial concept is that of a non-split fibre: This is an irreducible fibre isomorphic to two lines over a quadratic extension (called the splitting field of the fibre). The relevant conic bundle for Theorem 1.1 is
x 1 2 + x 2 2 = y x 0 2 , $$\begin{equation} x_1^2 + x_2^2 = y x_0^2, \end{equation}$$ (1.4)
where the equation is an affine patch of the surface in P 2 × E ${{\mathbb {P}}}^2 \times E$ . Here a non-split fibre occurs over the point at infinity, but in the example (1.2) one can check that every fibre of the relevant conic bundle is split. So to get savings one requires at least a non-split fibre; this is essentially the content of our next theorem.

Theorem 1.2.Let E $E$ be an elliptic curve over Q ${{\mathbb {Q}}}$ given by an integral Weierstrass equation and π : X E $\pi : X \rightarrow E$ a non-singular conic bundle. Let P E ( Q ) $P \in E({{\mathbb {Q}}})$ have infinite order with P E ( R ) 0 $P \in E({{\mathbb {R}}})^0$ . Assume that π 1 ( m P ) $\pi ^{-1}(mP)$ is non-split with imaginary quadratic splitting field, for some m Z $m \in {{\mathbb {Z}}}$ . Then there exists ω = ω ( X , E , P ) > 0 $\omega =\omega (X,E,P) > 0$ such that

# { n Z : | n | B , n P π ( X ( Q ) ) } X , E , P B / ( log B ) ω . $$\begin{equation*} \#\lbrace n \in {{\mathbb {Z}}}\colon |n| \leqslant B, nP \in \pi (X({{\mathbb {Q}}}))\rbrace \ll _{X,E,P} B/(\log B)^{\omega }. \end{equation*}$$

The theorem says that under some technical assumptions, for almost all multiples of a given non-torsion rational point the associated conic has no rational point. The assumption P E ( R ) 0 $P \in E({{\mathbb {R}}})^0$ may look artificial, however it is necessary for the conclusion.

Example 1.3.Consider the elliptic curve with the point P $P$ of infinite order

E : y 2 = x ( x + 2 ) ( x 3 ) , P = ( 1 , 2 ) $$\begin{equation*} E: y^2 = x(x+2)(x-3), \quad P = (-1,2) \end{equation*}$$
and the conic bundle
t 0 2 + t 1 2 = ( x x 1 ) ( x x 3 ) t 2 2 , $$\begin{equation*} t_0^2 + t_1^2 = (x - x_1)(x - x_3)t_2^2, \end{equation*}$$
where x i $x_i$ denotes the x $x$ -coordinate of i P $iP$ . This has non-split fibres over ± P $ \pm P$ and ± 3 P $\pm 3P$ . Here P E ( R ) 0 $P \notin E({{\mathbb {R}}})^0$ so the assumptions of Theorem 1.2 do not hold.

We claim that the fibre over every even multiple of P $P$ contains a rational point, so the conclusion of Theorem 1.2 in fact does not hold either. This is a special case of a more general construction (Proposition 5.7), but we explain the key ideas here.

Firstly, one checks that the fibre over O $O$ has a rational point. So let 2 n P $2nP$ be a non-trivial even multiple of P $P$ . It is clear there are always p $p$ -adic points for p 1 mod 4 $p \equiv 1 \bmod 4$ . We will show that for every prime p 3 mod 4 $p \equiv 3 \bmod 4$ , we have v p ( ( x x 1 ) ( x x 3 ) ) = 0 $v_p((x-x_1)(x-x_3)) = 0$ , which implies that the fibre has a Q p ${{\mathbb {Q}}}_p$ -point. Moreover as every element of 2 Z P $2 {{\mathbb {Z}}}P$ lies in the real component of the identity it satisfies x 3 $x \geqslant 3$ . Hence ( x x 1 ) ( x x 3 ) > 0 $(x - x_1)(x - x_3) > 0$ so the conic has a real point. Hilbert's version of quadratic reciprocity now shows that every conic has a Q 2 ${{\mathbb {Q}}}_2$ -point, hence has a Q ${{\mathbb {Q}}}$ -point by Hasse–Minkowski.

So assume for a contradiction that there is some even multiple 2 n P $2nP$ and p 3 mod 4 $p \equiv 3 \mod {4}$ such that v p ( ( x x 1 ) ( x x 3 ) ) > 0 $v_p((x-x_1)(x-x_3)) > 0$ . (One can check that v p ( x 3 ) 0 $v_p(x_3) \geqslant 0$ for all p 3 mod 4 $p \equiv 3 \bmod 4$ , so the valuation cannot be negative.) Suppose for example that v p ( x x 1 ) > 0 $v_p(x - x_1) > 0$ , so that 2 n P ± P mod p $2nP \equiv \pm P \bmod p$ . Then P $P$ is divisible by 2 modulo p $p$ . However, our example was chosen so that 2-division field of P $P$ is Q ( i ) ${{\mathbb {Q}}}(i)$ (this can be shown using the criterion from [4, Section 2]), and since p $p$ is inert in Q ( i ) ${{\mathbb {Q}}}(i)$ it follows that P $P$ is not 2-divisible modulo p $p$ ; a contradiction. The case 2 n P ± 3 P mod p $2nP \equiv \pm 3P \bmod p$ is analogous.

1.3 Proof ingredients

Our key tools are sieves and elliptic divisibility sequences (EDSs).

In the classical sieve setting one usually sieves with respect to the homomorphisms Z Z / p Z ${{\mathbb {Z}}}\rightarrow {{\mathbb {Z}}}/p{{\mathbb {Z}}}$ , for primes p $p$ , or more general prime powers. We originally tried to mimic this setting by sieving with respect to the homomorphisms E ( Q ) E ( F p ) $E({{\mathbb {Q}}}) \rightarrow E({{\mathbb {F}}}_p)$ , inspired by Kowalski's elliptic sieve [14, Section 4.4] used to study prime divisors in EDSs. However our method quickly diverges from the classical setting and Kowalski's setting, as in our case information modulo p $p$ is insufficient. A significant technical step in our proof is trying to control the p $p$ -adic valuations of the rational points we are sieving, which we achieve by sieving modulo p n p $p^{n_p}$ for varying primes p $p$ and growing exponents n p $n_p$ , so our sieve has no classical analogue. This difficulty is related to the fact that elliptic curves do not satisfy weak approximation, and does not arise in the classical sieve setting where p $p$ -adic valuations are easy to control. Our exact valuation theoretic problems are closely related to p $p$ being a ‘non-Wieferich prime for base P E $P\in E$ ’ in the sense of Voloch [33], and it is not even known whether there exists a single elliptic curve with infinitely many such primes [24]. These issues greatly complicate our sieve set up, and we have to work with the filtration structure on E ( Q p ) $E({{\mathbb {Q}}}_p)$ to control valuations. (See Remark 3.20.)

To come up with a sieve criterion we use EDSs: these are defined via the denominators of the x $x$ -coordinates of the multiples of P $P$ , multiplied by a sign to obtain better recurrence properties. We recall the relevant definitions in Section 3. The key property for us is that EDSs are periodic modulo an arbitrary integer (Proposition 3.7). We achieve this using the work of Verzobio [32] and does not seem to have been proven in the literature before in this generality. In our proof we also have to be careful with signs, which requires us to use the work [27] as well as equidistibution results for multiples of irrational numbers modulo 1. Whilst these sign issues may seem a mere technical step, in fact they are crucial and related to our necessary assumptions on the real components of E ( R ) $E({{\mathbb {R}}})$ (cf. Example 1.3).

Theorem 1.2 is a quantitative strengthening of a result of the fourth author and Berg [5], which proves under suitable assumptions that for certain conic bundles π : X E $\pi :X \rightarrow E$ , the image π ( X ( Q ) ) $\pi (X({{\mathbb {Q}}}))$ does not contain a translate of a subgroup of finite index. In [5] the authors only consider elliptic curves which are Galois general in a sense captured by conditions (1)–(4) in their Theorem 3.5, and their results only apply to special conic bundles given by pulling back Châtelet surfaces from P 1 ${{\mathbb {P}}}^1$ which also have a non-split fibre over a rational point. Our results apply to an overlapping collection of elliptic curves and conic bundles, but the key point is that our conclusion is stronger: a subset of Z ${{\mathbb {Z}}}$ which contains no arithmetic progression may still have positive density (for example the set of squarefree numbers in Z ${{\mathbb {Z}}}$ ).

1.4 Lower bounds and counting by height

We believe that our paper demonstrates the usefulness of sieve techniques and EDSs to counting problems on elliptic curves. The proof of Theorem 1.2 gives an explicit value for ω $\omega$ , but we doubt that our upper bound is sharp. Proving any kind of lower bound seems very difficult in general; we are only able to do this in various trivial cases where ω = 0 $\omega = 0$ , so that π ( X ( Q ) ) $\pi (X({{\mathbb {Q}}}))$ has positive density in E ( Q ) $E({{\mathbb {Q}}})$ (see Section 5). The following question seems quite challenging.

Question 1.4.Does there exist an elliptic curve E $E$ over Q ${{\mathbb {Q}}}$ such that the set

{ ( x , y ) E ( Q ) : y is a sum of two squares } $$\begin{equation} \lbrace (x,y) \in E({{\mathbb {Q}}}) \colon y \text{ is a sum of two squares} \rbrace \end{equation}$$ (1.5)
is infinite?

Standard conjectures in arithmetic geometry seem to have nothing to say about this question, as the associated conic bundle surface is neither rationally connected nor of general type (over Q ( i ) ${{\mathbb {Q}}}(i)$ it is birational to P 1 × E ${{\mathbb {P}}}^1 \times E$ ).

However, the following heuristic suggests (1.5) should be quite sparse. Let E ( Q ) $E({{\mathbb {Q}}})$ have rank r $r$ . Fix a norm · $\Vert \,\cdot \,\Vert$ on R r ${{\mathbb {R}}}^r$ . The numerator and denominator of y ( n 1 P 1 + + n r P r ) $y(n_1P_1+\cdots + n_r P_r)$ are both integers of size exp ( O E , P ( n 2 ) ) $ \exp (O_{E,P}(\Vert \vec{n}\Vert ^2))$ , with the denominator being a perfect cube. A proportion 1 / n 2 $1/\Vert \vec{n}\Vert ^2$ of such rational numbers are sums of two squares. One might therefore speculate that the set (1.5) has size n N r 1 / n 2 $ \ll \sum _{\vec{n}\in {{\mathbb {N}}}^r} 1/\Vert \vec{n}\Vert ^2$ , so can be infinite only when r > 1 $r>1$ .

Versions of this problem were raised by Poonen [19, Questions 23, 33] and Browning [8, Problem 10, pp. 3181–3182]. Browning in particular asked about the number of points for which the denominator of y $y$ is a sum of two squares. A similar heuristic suggests the number of points { n P : n B } $\lbrace nP: n \leqslant B\rbrace$ with this property might be around n B 1 / n log B $\sum _{n\leqslant B}1/n \sim \log B$ . Our methods give upper bounds for problem without alteration.

One can rephrase our results in terms of the canonical height h ̂ $\widehat{h}$ on E $E$ , as n P $nP$ has height roughly n 2 $n^2$ . In this language the heuristic just discussed suggests the following.
# { Q E ( Q ) : h ̂ ( Q ) H , y ( Q ) is a sum of two squares } 1 , if rank E ( Q ) 1 , log H , if rank E ( Q ) = 2 , H ( r 2 ) / 2 , if rank E ( Q ) > 2 ; # { Q E ( Q ) : h ̂ ( Q ) H , the denominator of y ( Q ) is a sum of two squares } log H , if rank E ( Q ) = 1 , H ( r 1 ) / 2 , if rank E ( Q ) 2 . $$\begin{align*} &\#\lbrace Q \in E({{\mathbb {Q}}}):\widehat{h}(Q) \leqslant H, y(Q) \text{ is a sum of two squares} \rbrace \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \asymp {\begin{cases} 1,& \text{if }\operatorname{rank}E({{\mathbb {Q}}})\leqslant 1, \\ \log H, & \text{if }\operatorname{rank}E({{\mathbb {Q}}})=2, \\ H^{(r-2)/2}, & \text{if }\operatorname{rank}E({{\mathbb {Q}}})>2; \end{cases}} \\ &\#\lbrace Q \in E({{\mathbb {Q}}}):\widehat{h}(Q) \leqslant H, \text{the denominator of } y(Q) \text{ is a sum of two squares} \rbrace \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \asymp {\begin{cases} \log H, & \text{if }\operatorname{rank}E({{\mathbb {Q}}}) = 1, \\ H^{(r-1)/2}, & \text{if }\operatorname{rank}E({{\mathbb {Q}}})\geqslant 2. \end{cases}} \end{align*}$$

Our present method cannot handle the case of rank greater than 1; the key stumbling block is that we have no control over the prime p $p$ constructed in Proposition 3.19, which would be necessary to combine p $p$ -adic information at sums of points. We are however able to prove non-trivial upper bounds provided the curve has rank 1. Rather than stating the most general result in terms of conic bundles, we content ourselves with the following variant of Theorem 1.1.

Theorem 1.5.Let E $E$ be an elliptic curve over Q ${{\mathbb {Q}}}$ given by an integral Weierstrass equation. Assume that E $E$ has rank 1. Let

η = 1 if # E ( Q ) tors has two distinct prime divisors, 1 otherwise. $$\begin{equation*} \eta = {\begin{cases} 1&\text{ if }\# E({{\mathbb {Q}}})^{\operatorname{tors}}\text{ has two distinct prime divisors,} \\ -1 &\text{ otherwise.} \end{cases}} \end{equation*}$$
Then
# Q E ( Q ) E ( R ) 0 : h ̂ ( Q ) H , y ( Q ) is a sum of two squares E H 1 / 2 ( log log log H ) η / 2 ( log log H ) 1 / 2 . $$\begin{multline*} \#{\left\lbrace Q \in E({{\mathbb {Q}}}) \cap E({{\mathbb {R}}})^0 : \def\eqcellsep{&}\begin{array}{l}\widehat{h}(Q) \leqslant H, y(Q) \text{ is a }\\[3pt] \text{sum of two squares} \end{array} \right\rbrace} \ll _E \frac{H^{1/2}(\log \log \log H)^{\eta /2}}{(\log \log H)^{1/2}}. \end{multline*}$$

The total number of rational points in E ( R ) 0 $ E({{\mathbb {R}}})^0$ of height at most H $H$ is H 1 / 2 $\gg H^{1/2}$ , since 2 E ( Q ) E ( R ) 0 $2E({{\mathbb {Q}}})\subset E({{\mathbb {R}}})^0$ , so the theorem indeed shows that 0 % $0\%$ of these have y $y$ -coordinate which is a sum of two squares. Note that if E ( R ) $E({{\mathbb {R}}})$ is connected, then the upper bounds applies to all rational points on E $E$ .

1.5 Generalisation to Brauer groups

We now state our most general result, of which Theorem 1.2 is a special case. Firstly, we are able to prove results for conic bundles whose non-split fibres have real quadratic splitting field. Secondly, we make explicit the dependence of ω $\omega$ and the leading constant on P $P$ . Thirdly, our methods are sufficiently robust that they allow applications to specialisations of Brauer group elements on elliptic curves. This is also the viewpoint taken by Serre in his paper [22], as well in the more recent papers [16, 18]. Brauer groups are formally easier to work with than conic bundles, since one does not require explicit equations and one can make use of Grothendieck's residue map. Here we take a Brauer group element which is ramified at a rational point. The ramification gives rise to a cyclic extension of Q ${{\mathbb {Q}}}$ to which we associate a Dirichlet character using Kronecker–Weber (see Section 4 for details and relevant background on Brauer groups). Our result here is as follows.

Theorem 1.6.Let E $E$ be an elliptic curve over Q ${{\mathbb {Q}}}$ given by an integral Weierstrass equation. Let P E ( Q ) $P \in E({{\mathbb {Q}}})$ have infinite order. Assume there is m Z $m\in {{\mathbb {Z}}}$ such that b Br Q ( E ) $b \in \operatorname{Br}{{\mathbb {Q}}}(E)$ is ramified at m P $mP$ and let the associated Dirichlet character χ $\chi$ have modulus q ( χ ) $q(\chi )$ . Let β n $\beta _n$ be the EDS associated to P $P$ and let π $\pi$ be the period of the sequence β n mod q ( χ ) $\beta _n \bmod q(\chi )$ .

Assume that there is some index α N $\alpha \in {{\mathbb {N}}}$ with gcd ( α , π ) = 1 $\gcd (\alpha ,\pi ) = 1$ which satisfies either

  • (1) χ ( | β α | ) 0 , 1 $\chi (|\beta _\alpha |) \ne 0,1$ ; or
  • (2) χ ( | β α | ) 0 , 1 $\chi (-|\beta _\alpha |) \ne 0,1$ and P E ( R ) 0 $P \in E({{\mathbb {R}}})^0$ ; or
  • (3) χ ( | β α | ) 0 , 1 $\chi (-|\beta _\alpha |) \ne 0,1$ and 4 π $4 \nmid \pi$ .
Then we have π E , χ 1 $\pi \ll _{E,\chi } 1$ and
# { | n | B : b ( n P ) = 0 Br Q } C E , P , b B log log B ( log B ) 1 / 2 φ ( π ) , $$\begin{equation*} \#\lbrace |n| \leqslant B : b(nP) = 0 \in \operatorname{Br}{{\mathbb {Q}}}\rbrace \leqslant C_{E,P,b} \frac{B\log \log B}{ (\log B)^{1/2\varphi (\pi )}}, \end{equation*}$$
where φ $\varphi$ denotes Euler's totient function and C E , P , b $C_{E,P,b}$ is a positive constant depending on E , P $E,P$ and b $b$ only and given by (4.4).

In the statement b ( Q ) Br Q $b(Q) \in \operatorname{Br}{{\mathbb {Q}}}$ denotes the evaluation of the Brauer element b $b$ at Q $Q$ . We abuse notation slightly and implicitly ignore the finitely many points where b $b$ is not defined.

The Brauer group framework essentially allows us to replace quadratic extensions by arbitrary cyclic extensions and conic bundles by higher dimensional Brauer–Severi varieties. Moreover, we can even handle some non-abelian extensions. As an example application in the style of Theorem 1.1, we have the following.

Theorem 1.7.Let K / Q $K/{{\mathbb {Q}}}$ be a number field which contains a cyclic subfield which is not totally real. Let E $E$ be an elliptic curve over Q ${{\mathbb {Q}}}$ given by an integral Weierstrass equation and let P E ( Q ) $P\in E({{\mathbb {Q}}})$ be a point of infinite order with P E ( R ) 0 $P \in E({{\mathbb {R}}})^0$ . Then there exists ω = ω ( E , K ) > 0 $\omega =\omega (E,K)> 0$ such that

# { | n | B : y ( n P ) is a norm from K } E , P , K B / ( log B ) ω , $$\begin{equation*} \#\lbrace |n| \leqslant B \colon y(nP) \text{ is a norm from } K \rbrace \ll _{E,P,K} B/(\log B)^{\omega }, \end{equation*}$$
where the implicit constant is exp ( O E , K ( h ̂ ( P ) / log h ̂ ( P ) ) ) $\exp (O_{E,K}({\widehat{h}(P)}/{\log \widehat{h}(P)}))$ .

The fields K = Q ( a n , μ n ) $K = {{\mathbb {Q}}}(\sqrt [n]{a}, \mu _n)$ satisfy these hypotheses, for a Q × $a \in {{\mathbb {Q}}}^\times$ and n 3 $n \geqslant 3$ . The explicit dependence on the point P $P$ is included as it is needed to prove Theorem 1.5. (A version of Theorem 1.5 replaced with the condition that y ( Q ) $y(Q)$ is a norm from K $K$ , where K $K$ is of the type in Theorem 1.7, would follow by the same proof.)

Let us consider the technical assumptions on EDSs in Theorem 1.6. We suspect that Condition (1) holds for all but finitely many χ $\chi$ of given order; but it seems incredibly difficult to prove this, and even the analogous statement for the much simpler cases of Fibonacci or Mersenne numbers seems to be an open problem [15, 30]. However if | β α | $|\beta _\alpha |$ is a non-square (say), which indeed holds for all but finitely many α $\alpha$ [11, Theorem 1.1], then χ ( | β α | ) 0 , 1 $\chi (|\beta _\alpha |) \ne 0,1$ for a positive proportion of quadratic Dirichlet characters; the challenge is it show that these characters cover all but finitely many as α $\alpha$ varies. We are able to show the modest result that Condition (1) holds 100 % $100\%$ of the time under suitable assumptions; see Section 6 for details.

As for our applications, if χ $\chi$ is an odd Dirichlet character then we simply use that β 1 = 1 $\beta _1 = 1$ and χ ( 1 ) = 1 0 , 1 $\chi (-1) = -1 \ne 0,1$ to see that Condition (2) or (3) is satisfied. This is what makes stating Theorem 1.2 so simple, as for an imaginary quadratic extension Q ( D ) ${{\mathbb {Q}}}(\sqrt {D})$ with D $D$ a fundamental discriminant, the associated Dirichlet character is simply the Kronecker symbol ( D · ) $(\frac{D}{\cdot })$ , which takes the value 1 $-1$ at 1 $-1$ as D $D$ is negative.

We finish by returning to Example 1.3.

Example 1.8.Consider the elliptic curve from Example 1.3

E : y 2 = x ( x + 2 ) ( x 3 ) $$\begin{equation*} E: y^2 = x(x+2)(x-3) \end{equation*}$$
with P = ( 1 , 2 ) $P = (-1,2)$ . As we have seen, this pair does not satisfy the conclusion of Theorem 1.2 with respect to Q ( i ) ${{\mathbb {Q}}}(i)$ . Let us verify that the hypotheses of Theorem 1.6 do not hold in this case. Firstly, P E ( R ) 0 $P \notin E({{\mathbb {R}}})^0$ so Condition (2) does not hold. Here P $P$ has everywhere good reduction and the EDS β n $\beta _n$ is
0 , 1 , 4 , 65 , 504 , 242 369 , 58 888 180 , 66 048 490 369 , 60 955 459 632 144 , $$\begin{equation*} 0, 1, 4, -65, -504, 242\,369, -58\,888\,180, -66\,048\,490\,369, 60\,955\,459\,632\,144, \dots \end{equation*}$$
The relevant Dirichlet character is the unique non-principal character χ $\chi$ modulo 4. The sequence β n mod 4 $\beta _n \bmod 4$ is just
0 , 1 , 0 , 3 , 0 , 1 , 0 , 3 , 0 , 1 , 0 , 3 , 0 , 1 , 0 , 3 , 0 , 1 , 0 , 3 , 0 , $$\begin{equation*} 0,1, 0, 3, 0, 1, 0, 3, 0, 1, 0, 3, 0, 1, 0, 3, 0, 1, 0, 3, 0,\ldots \end{equation*}$$
which has period 4. Thus Condition (3) from Theorem 1.6 does not hold. Finally, the sequence | β n | mod 4 $|\beta _n| \bmod 4$ is
0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , $$\begin{equation*} 0,1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0,\ldots \end{equation*}$$
so (1) does not hold either. In particular, this demonstrates that all hypotheses in Theorem 1.6 are necessary in general for the conclusion to hold.

Outline of the paper

In Section 2 we recall various facts about elliptic curves over Q p ${{\mathbb {Q}}}_p$ . The following Section 3 contains a detailed study of elliptic divisibility sequences. Here we prove periodicity modulo an arbitrary integer, and show our main technical result (Proposition 3.19) on such sequences. In Section 4 we prove the theorems from the introduction. We give various examples illustrating our results in Section 5, as well as further examples which demonstrate that the conclusion of Theorem 1.2 does not hold for arbitrary conic bundles over elliptic curves. We finish in Section 6 by showing that the technical assumption in Theorem 1.6 holds for 100 % $100\%$ of suitable Dirichlet characters of prime moduli.

1.6 Notation and conventions

We choose an embedding Q / Z C × ${{\mathbb {Q}}}/{{\mathbb {Z}}}\subset {{\mathbb {C}}}^\times$ . This corresponds to a choice of a compatible system of n $n$ th roots of unity for all n $n$ . We denote by O = ( 0 : 1 : 0 ) P 2 $O= (0:1:0) \in {{\mathbb {P}}}^2$ . By a Weierstrass equation with coefficients a i $a_i$ we mean
y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 . $$\begin{equation} y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6. \end{equation}$$ (1.6)
We denote by h ̂ $\widehat{h}$ the canonical height on E ( Q ) $E({{\mathbb {Q}}})$ [26, Chapter VIII.9]. This extends to a positive definitive quadratic form on E ( Q ) R $E({{\mathbb {Q}}}) \otimes {{\mathbb {R}}}$ [26, Proposition VIII.9.6]. By O a , , z ( A ) $O_{a,\ldots\,,z}(A)$ we mean a quantity with absolute value at most C A $CA$ for some positive constant C $C$ depending on a , , z $a,\ldots\,,z$ only; if the subscripts are omitted the implied constant is absolute. We write A a , , z B $A\ll _{a,\ldots\,,z}B$ for A = O a , , z ( B ) $A=O_{a,\ldots\,,z}(B)$ and A = o ( B ) $A=o(B)$ for A / B 0 $A/B\rightarrow 0$ .

2 ELLIPTIC CURVES OVER Q p $\mathbb {Q}_p$

In this section let E $E$ be an elliptic curve over Q p ${{\mathbb {Q}}}_p$ given by a (not necessarily minimal) Weierstrass equation with coefficients in Z p ${{\mathbb {Z}}}_p$ . Denote by E 0 ( Q p ) $E_0({{\mathbb {Q}}}_p)$ the set of points of E ( Q p ) $E({{\mathbb {Q}}}_p)$ with non-singular reduction modulo p $p$ . We say that P E ( Q p ) $P \in E({{\mathbb {Q}}}_p)$ has bad reduction if P E 0 ( Q p ) $P \notin E_0({{\mathbb {Q}}}_p)$ . There is a subgroup filtration
E 2 ( Q p ) E 1 ( Q p ) E 0 ( Q p ) , E i ( Q p ) = { P E 0 ( Q p ) : P O mod p i } , i 1 . $$\begin{equation*} \cdots \subset E_2({{\mathbb {Q}}}_p) \subset E_1({{\mathbb {Q}}}_p) \subset E_0({{\mathbb {Q}}}_p), \quad E_i({{\mathbb {Q}}}_p) = \lbrace P \in E_0({{\mathbb {Q}}}_p) : P \equiv O \bmod p^i\rbrace , \, i \geqslant 1. \end{equation*}$$

Definition 2.1.If P E ( Q p ) E 1 ( Q p ) $P\in E({{\mathbb {Q}}}_p)\setminus E_1({{\mathbb {Q}}}_p)$ we set v p ( P ) = 0 $v_p(P)=0$ . If P E 1 ( Q p ) $P \in E_1({{\mathbb {Q}}}_p)$ we define

v p ( P ) = sup { i : P E i ( Q p ) } . $$\begin{equation*} v_p(P) = \sup \lbrace i : P \in E_i({{\mathbb {Q}}}_p) \rbrace. \end{equation*}$$

Definition 2.2.For P E 0 ( Q p ) $P \in E_0({{\mathbb {Q}}}_p)$ and k N $k \in {{\mathbb {N}}}$ we denote by P mod p k $P \bmod p^k$ the image of P $P$ in E 0 ( Q p ) / E k ( Q p ) $E_0({{\mathbb {Q}}}_p) / E_k({{\mathbb {Q}}}_p)$ . We denote by ord ( P mod p k ) $\operatorname{ord}(P \bmod p^k)$ its order.

Lemma 2.3.Let P = ( x , y ) E ( Q p ) $P =(x,y) \in E({{\mathbb {Q}}}_p)$ . Then v p ( P ) = max { 0 , v p ( x ) / 2 } $v_p(P) = \max \lbrace 0,-v_p(x)/2\rbrace$ .

Proof.If v p ( x ) 0 $v_p(x) \geqslant 0$ then v p ( P ) = 0 $v_p(P) = 0$ so the result holds. So assume v p ( x ) < 0 $v_p(x) &lt; 0$ . As the rational function x / y $x/y$ is a uniformising parameter at O $O$ , we find that v p ( P ) = v p ( x / y ) $v_p(P) = v_p(x/y)$ . However, using v p ( x ) < 0 $v_p(x) &lt; 0$ and the Weierstrass equation, one finds that 2 v p ( y ) = 3 v p ( x ) $2v_p(y) = 3v_p(x)$ , and the claim easily follows. $\Box$

Lemma 2.3 gives a more explicit definition of the filtration which is often used in texts (for example [26, Example VII.7.4]). We have the following inequality for the valuation of a multiple of a point.

Lemma 2.4.Let P E 1 ( Q p ) $P \in E_1({{\mathbb {Q}}}_p)$ . Then v p ( n P ) v p ( P ) + v p ( n ) $v_p(nP)\geqslant v_p(P) + v_p(n)$ , with equality if p n $p \nmid n$ .

Proof.Hensel's lemma [7, Lemma 2.1] shows that | E i ( Q p ) / E i + 1 ( Q p ) | = p $|E_{i}({{\mathbb {Q}}}_p)/E_{i+1}({{\mathbb {Q}}}_p)| = p$ for all i 1 $i \geqslant 1$ , thus this quotient is isomorphic to Z / p Z ${{\mathbb {Z}}}/p{{\mathbb {Z}}}$ . The result now easily follows. $\Box$

Remark 2.5.Using the formal group law on E $E$ [26, Theorem IV.6.4(b), Proposition VII.2.2], one can show that equality in fact holds except possibly if p = 2 , v p ( P ) = 1 $p=2, v_p(P) = 1$ and p n $p\mid n$ . (See also [29, Theorem 3] for a version over number fields.) The hypothesis is required for p = 2 $p=2$ . Take

E : y 2 + x y = x 3 + 4 x + 1 , P = ( 15 / 4 , 83 / 8 ) . $$\begin{equation*} E: y^2 +xy = x^3 + 4x + 1, \quad P = (15/4, -83/8). \end{equation*}$$
Then v 2 ( P ) = 1 $v_2(P) = 1$ , but one calculates that v 2 ( 2 P ) = 4 $v_2(2P) = 4$ . (The issue here is that E 1 ( Q 2 ) $E_1({{\mathbb {Q}}}_2)$ has non-trivial 2-torsion, so is not isomorphic to 2 Z 2 $2{{\mathbb {Z}}}_2$ .)

3 ELLIPTIC DIVISIBILITY SEQUENCES

3.1 Basic properties

Now let E / Q $E/{{\mathbb {Q}}}$ be an elliptic curve given by a Weierstrass equation (1.6) with coefficients a i Z $a_i\in {{\mathbb {Z}}}$ . Let P E ( Q ) $P \in E({{\mathbb {Q}}})$ be a non-torsion point. Throughout this section we consider E $E$ and P $P$ as being fixed.

For any integer n 0 $n\geqslant 0$ , define the n $n$ th division polynomial ψ n Z [ x , y ] $\psi _n\in {{\mathbb {Z}}}[x,y]$ as follows:
ψ 0 = 0 , ψ 1 = 1 , ψ 2 = 2 y + a 1 x + a 3 , ψ 3 = 3 x 4 + b 2 x 3 + 3 b 4 x 2 + 3 b 6 x + b 8 , ψ 4 = ψ 2 ( 2 x 6 + b 2 x 5 + 5 b 4 x 4 + 10 b 6 x 3 + 10 b 8 x 2 + ( b 2 b 8 b 4 b 6 ) x + b 4 b 8 b 2 6 ) , $$\begin{align*} &\psi _0=0,\quad \psi _1=1,\quad \psi _2=2y + a_1x + a_3,\\ &\psi _3=3x^4 + b_2x^3 + 3b_4x^2 + 3b_6x + b_8,\\ &\psi _4=\psi _2(2x^6 + b_2x^5 + 5b_4x^4 + 10b_6x^3 + 10b_8x^2+(b_2b_8 - b_4b_6)x + b_4b_8 - b_2^6), \end{align*}$$
where the b i $b_i$ are defined in [26, Chapter III], with subsequent polynomials given by
ψ 2 n + 1 = ψ n + 2 ψ n 3 ψ n + 1 3 ψ n 1 , n 2 , ψ 2 n ψ 2 = ψ n ( ψ n + 2 ψ n 1 2 ψ n 2 ψ n + 1 2 ) , n 3 , $$\begin{equation} \begin{split} \psi _{2n+1}=\psi _{n+2}\psi _n^3-\psi _{n+1}^3\psi _{n-1},\quad n\geqslant 2, \\ \psi _{2n}\psi _2=\psi _n(\psi _{n+2}\psi _{n-1}^2-\psi _{n-2}\psi _{n+1}^2),\quad n\geqslant 3, \end{split} \end{equation}$$ (3.1)
and extend this to negative n $n$ by setting ψ n = ψ n $\psi _n=-\psi _{-n}$ . These formulas are equivalent to the recurrence relation
ψ m + n ψ m n ψ r 2 = ψ m + r ψ m r ψ n 2 ψ n + r ψ n r ψ m 2 $$\begin{equation} \psi _{m+n}\psi _{m-n}\psi _r^2=\psi _{m+r}\psi _{m-r}\psi _n^2-\psi _{n+r}\psi _{n-r}\psi _m^2 \end{equation}$$ (3.2)
for any integers m , n , r $m,n,r$ . The sequence ψ n $\psi _n$ forms a divisibility sequence in Z [ x , y ] ${{\mathbb {Z}}}[x,y]$ , that is,  ψ n ψ m $\psi _n\mid \psi _m$ for n m $n\mid m$ . One notion of an EDS in a commutative ring would be a divisibility sequence satisfying (3.2). The study of EDS in Z ${{\mathbb {Z}}}$ , in this sense, was begun by Ward [34], and a modern exposition can be found in [12, Chapter 10]. We will use a slightly different kind of EDS considered by Verzobio [32], which is better suited to our purpose.
We can interpret x , y $x,y$ and each ψ n $\psi _n$ as rational functions on E ( Q ) $E({{\mathbb {Q}}})$ . By [26, Example III.3.7], multiplication by n $n$ is given as a rational map by
[ n ] ( P ) = x ( P ) ψ n 2 ψ n 1 ψ n + 1 ψ n 2 , ψ n 1 2 ψ n + 2 ψ n 2 ψ n + 1 2 4 y ( P ) ψ n 3 . $$\begin{equation*} [n](P) = {\left(\frac{x(P)\psi _n^2-\psi _{n-1}\psi _{n+1}}{\psi _n^2}, \frac{\psi _{n-1}^2\psi _{n+2}-\psi _{n-2}\psi _{n+1}^2}{4y(P)\psi _n^3} \right)}. \end{equation*}$$
In particular ψ n $\psi _n$ is the square root of the denominator of the x $x$ -coordinate; the problem for us is that in general there may be some common factors between the numerator and denominator, so it will not be in lowest terms. We want to work with the genuine denominator as it has better p $p$ -adic properties (cf. Lemma 2.3).

Definition 3.1.Define the sequence e n $e_n$ by n P = ( a n / e n 2 , b n / e n 3 ) $nP=(a_n/e_n^2,b_n/e_n^3)$ with gcd ( a n b n , e n ) = 1 $\gcd (a_nb_n,e_n) = 1$ and e n > 0 $e_n &gt; 0$ . Writing sign ( t ) = t / | t | $\operatorname{sign}(t)=t/|t|$ for any t 0 $t\ne 0$ , set

β 0 = 0 , β n = sign ( ψ n ( P ) ) e n e 1 , ( n Z { 0 } ) . $$\begin{equation*} \beta _0 = 0, \qquad \beta _n = \operatorname{sign}(\psi _n(P)) \frac{e_n}{e_1}, \qquad (n\in {{\mathbb {Z}}}\setminus \lbrace 0\rbrace ). \end{equation*}$$

The sequence β n $\beta _n$ is not in general an EDS in the traditional sense, since it need not satisfy the recurrence relation (3.2); differences can occur if P $P$ admits primes of bad reduction. In [32] Verzobio calls such sequences EDSB, as opposed to sequences of the form ψ n ( P ) $\psi _n(P)$ which he terms EDSA. He shows in [32, Theorem 1.9] that the following weakened version of (3.2) does hold for an EDSB.

Proposition 3.2. (Verzobio)Set

M = M ( P ) = lcm { ord ( P + E 0 ( Q p ) ) : p prime } , $$\begin{equation*} M=M(P)=\operatorname{lcm}\lbrace \operatorname{ord}(P + E_0({{\mathbb {Q}}}_p)) : p \text{ prime}\rbrace , \end{equation*}$$
where ord ( P + E 0 ( Q p ) ) $\operatorname{ord}(P + E_0({{\mathbb {Q}}}_p))$ denotes the order of the image of P $P$ in the finite group E ( Q p ) / E 0 ( Q p ) $E({{\mathbb {Q}}}_p)/E_0({{\mathbb {Q}}}_p)$ . Let n , m , r Z $n, m, r\in {{\mathbb {Z}}}$ of which two are multiples of M ( P ) $M(P)$ . Then
β n + m β n m β r 2 = β m + r β m r β n 2 β n + r β n r β m 2 . $$\begin{equation} \beta _{n+m}\beta _{n-m}\beta _r^2 = \beta _{m+r}\beta _{m-r}\beta _n^2 -\beta _{n+r}\beta _{n-r}\beta _m^2. \end{equation}$$ (3.3)

Remark 3.3.Here M $M$ is the least positive integer such that M P $MP$ has everywhere good reduction. It divides p # ( E ( Q p ) / E 0 ( Q p ) ) $\prod _p\#(E({{\mathbb {Q}}}_p)/E_0({{\mathbb {Q}}}_p))$ , which is the product of the Tamagawa numbers of E $E$ if the model is globally minimal.

Verzobio defines β n $\beta _n$ for n 0 $n\geqslant 0$ and proves the theorem under the assumption n m r > 0 $n\geqslant m\geqslant r&gt;0$ ; in our notation this can be removed by using β n = β n $\beta _{-n}=-\beta _n$ and permuting the variables as appropriate.

To illustrate some of the nice p $p$ -adic properties of this sequence, we prove that it is a strong divisibility sequence. We first make explicit Lemma 2.3.

Lemma 3.4.For all primes p $p$ we have v p ( β n ) = v p ( n P ) v p ( P ) $v_p(\beta _n) = v_p(nP) - v_p(P)$ .

Proof.Immediate from the definition and Lemma 2.3. $\Box$

Lemma 3.5.For all n , m Z $n,m\in {{\mathbb {Z}}}$ we have gcd ( β m , β n ) = | β gcd ( m , n ) | $\gcd (\beta _m,\beta _n)=|\beta _{\gcd (m,n)}|$ .

Proof.By Lemma 3.4, for any prime p $p$ and any V N $V\in {{\mathbb {N}}}$ we have

{ n Z : v p ( β n ) V } = { n Z : n P E V + v p ( P ) ( Q p ) } = q Z $$\begin{equation*} \lbrace n\in {{\mathbb {Z}}}: v_p(\beta _n) \geqslant V\rbrace =\lbrace n \in {{\mathbb {Z}}}:nP\in E_{V+v_p(P)}({{\mathbb {Q}}}_p)\rbrace =q{{\mathbb {Z}}} \end{equation*}$$
for some q N $q\in {{\mathbb {N}}}$ . In particular p V β n $p^V\mid \beta _n$ if and only if q n $q\mid n$ . Therefore
p V gcd ( β m , β n ) q gcd ( m , n ) p V β gcd ( m , n ) . $$\begin{equation*} p^V \mid \gcd (\beta _m,\beta _n) \iff q\mid \gcd (m,n) \iff p^V \mid \beta _{\gcd (m,n)}. \end{equation*}$$
$\Box$

We emphasise that an EDSA need not be a strong divisibility sequence if P $P$ admits primes of bad reduction. The elegance of Verzobio's EDSB is that it has both good p $p$ -adic properties and comes within a whisker of satisfying the recurrence relation.

3.2 Symmetry law

A central part of Ward's work on EDSs is a symmetry law [34, Theorem 8.1] (see [1, Theorem 1.11] for a modern formulation). This says that an integral EDSA modulo a prime forms a periodic sequence of a certain form. We prove a version of this for EDSBs for general prime powers.

Proposition 3.6.Let M $M$ be as in Proposition 3.2. Let n , r Z $n,r\in {{\mathbb {Z}}}$ with M r $M\mid r$ . Let p $p$ be a prime and let k N $k\in {{\mathbb {N}}}$ . Suppose that p k $p^k$ divides β r / gcd ( β r , β M ) $\beta _r/\gcd (\beta _r,\beta _M)$ . Then for all Z $\ell \in {{\mathbb {Z}}}$ we have

β n + r β M + r β M r β M 2 ( 1 ) 2 ( β n + r β n 1 ) β n mod p k , if p k β n , 0 mod p k , if p k β n , $$\begin{equation*} \beta _{n+\ell r} \equiv {\begin{cases} {\left(\beta _{M+r}\beta _{M-r}\beta _M^{-2}\right)}^{\frac{\ell (\ell -1)}{2}} (\beta _{n+r}\beta _{n}^{-1})^\ell \beta _{n} \bmod p^k, & \text{if }p^k\nmid \beta _n, \\ 0 \bmod p^k, & \text{if }p^k\mid \beta _n, \end{cases}} \end{equation*}$$
where in the first case the quotients β M + r β M r / β M 2 $\beta _{M+r}\beta _{M-r}/\beta _M^2$ and β n + r / β n $\beta _{n+r}/\beta _n$ are p $p$ -adic units.

Proof.Lemma 3.5 gives us

| β gcd ( n , r ) | = gcd ( β n + r , β r ) = gcd ( β n , β r ) $$\begin{equation} |\beta _{\gcd (n,r)}|= \gcd (\beta _{n+\ell r},\beta _r)= \gcd (\beta _{n},\beta _r) \end{equation}$$ (3.4)
for every Z $\ell \in {{\mathbb {Z}}}$ . This proves the proposition if p k β n $p^k\mid \beta _n$ , so assume that p k β n $p^k\nmid \beta _n$ .

Taking m = M $m = M$ in Proposition 3.2, and replacing n $n$ by n + r $n+\ell r$ , we obtain

β M + r β M r β n + r 2 β n + ( + 1 ) r β n + ( 1 ) r β M 2 mod β r 2 , $$\begin{equation} \beta _{M+r}\beta _{M-r}\beta _{n+\ell r}^2 \equiv \beta _{n+(\ell +1)r}\beta _{n+(\ell -1)r}\beta _M^2 \bmod \beta _r^2, \end{equation}$$ (3.5)
for any Z $\ell \in {{\mathbb {Z}}}$ . We want to combine this with Lemma 3.5. Let
C = β M + r β M r β M 2 , a = β n + r gcd ( β n , β r ) . $$\begin{align} C&=\frac{\beta _{M+r}\beta _{M-r}}{\beta _M^2}, \quad a_\ell =\frac{ \beta _{n+\ell r}}{\gcd (\beta _n,\beta _r)}. \end{align}$$ (3.6)
Since M r $M\mid r$ , Lemma 3.5 shows that C Z $C\in {{\mathbb {Z}}}$ . Also (3.4) shows that a $a_\ell$ is an integer coprime to β r / gcd ( β n , β r ) $\beta _r/\gcd (\beta _n,\beta _r)$ . Hence, dividing both sides of (3.5) by β n + r 2 β M 2 $\beta _{n+\ell r}^2\beta _M^2$ gives
C a + 1 a 1 a 2 mod β r 2 gcd ( β n β M , β r ) 2 for all Z , $$\begin{equation} C \equiv \frac{a_{\ell +1}a_{\ell -1}}{a_{\ell }^2} \bmod \frac{\beta _r^2}{\gcd (\beta _n\beta _M,\beta _r)^2} \quad \text{for all }\ell \in {{\mathbb {Z}}}, \end{equation}$$ (3.7)
where every a $a_{\ell }$ is coprime to the modulus. It follows by induction on $\ell$ from (3.7) that
a C ( 1 ) 2 a 1 a 0 1 mod β r 2 gcd ( β n β M , β r ) 2 . $$\begin{equation*} a_\ell \equiv C^{\frac{\ell (\ell -1)}{2}} a_1^\ell a_0^{1-\ell } \bmod \frac{\beta _r^2}{ \gcd (\beta _n\beta _M,\beta _r)^2}. \end{equation*}$$
Multiplying by gcd ( β n , β r ) $\gcd (\beta _n,\beta _r)$ we obtain
a gcd ( β n , β r ) C ( 1 ) 2 ( a 1 a 0 1 ) a 0 gcd ( β n , β r ) mod gcd ( β n , β r ) β r 2 gcd ( β n β M , β r ) 2 . $$\begin{equation*} a_\ell \gcd (\beta _n,\beta _r) \equiv C^{\frac{\ell (\ell -1)}{2}} (a_1a_0^{-1})^\ell a_0\gcd (\beta _n,\beta _r) \bmod \frac{\gcd (\beta _n,\beta _r)\beta _r^2}{ \gcd (\beta _n\beta _M,\beta _r)^2}. \end{equation*}$$
Here β r / gcd ( β M , β r ) $\beta _r/\gcd (\beta _M,\beta _r)$ divides the modulus, and so the congruence holds modulo p k $p^k$ . Inserting the definitions (3.6) proves the first case in the proposition.

Finally, since p k β r / gcd ( β M , β r ) $p^k\mid \beta _r/\gcd (\beta _M,\beta _r)$ and p k β n $p^k\nmid \beta _n$ , we see that p $p$ divides the modulus in (3.7). Since every a $a_\ell$ is coprime to the modulus, we see that C $C$ and β n + r β n 1 = a 1 a 0 1 $\beta _{n+r}\beta _n^{-1}=a_1a_0^{-1}$ are p $p$ -adic units, as claimed in the final part of the proposition. $\Box$

3.3 Periodicity

We now use the symmetry law to prove that β n $\beta _n$ is periodic modulo any prime power, and hence modulo any integer. Versions of this appear in the literature for differing definitions of EDS. Ward proved eventual periodicity modulo any prime in [34, Theorem 11.1]. Shipsey proved a version modulo p 2 $p^2$ for primes of good reduction [23, Theorem 3.5.4]. Ayad proved it modulo any integer, but assuming good reduction and avoiding p = 2 $p=2$ or ‘rank of apparition 2’ [3, Theorem D]. Silverman proved a version over finite fields [25, Theorem 1] as well as a version modulo prime powers whenever the curve has good ordinary reduction [25, Theorem 3]. Our version (Proposition 3.7) contains none of these technical assumptions and is a general version of periodicity, for Verzobio's arguably more elegant EDSB.

Our result is the following, which shows periodicity modulo an arbitrary prime power and gives an upper bound for the period. Note that the Chinese Remainder theorem then easily shows periodicity modulo an arbitrary integer.

Proposition 3.7.Let M $M$ be as in Proposition 3.2, let k N $k\in {{\mathbb {N}}}$ and let p $p$ be a prime. Let

r ( p k ) = M ord ( M P mod p k + v p ( M P ) ) $$\begin{equation} r(p^k)= M\operatorname{ord}(MP\bmod p^{k+v_p(MP)}) \end{equation}$$ (3.8)
and
π ( p k ) = ( p 1 ) p k 1 r ( p k ) , if p 2 and β M + r ( p k ) β M r ( p k ) p = 1 , 2 ( p 1 ) p k 1 r ( p k ) , otherwise . $$\begin{equation} \pi (p^k) = {\begin{cases} (p-1)p^{k-1}r(p^k),&\text{if }p\ne 2\text{ and } {\left(\frac{\beta _{M+r(p^k)}\beta _{M-r(p^k)}}{p}\right)}=1, \\ 2(p-1)p^{k-1}r(p^k),&\text{otherwise}. \end{cases}} \end{equation}$$ (3.9)
Then for every m Z $m \in {{\mathbb {Z}}}$ we have
m n mod π ( p k ) β m β n mod p k . $$\begin{equation*} m\equiv n \bmod \pi (p^k) {\hspace{5.0pt}}\Rightarrow {\hspace{5.0pt}} \beta _m \equiv \beta _n \bmod p^k. \end{equation*}$$
In other words the sequence β m mod p k $\beta _m \bmod p^k$ is periodic with period dividing π ( p k ) $\pi (p^k)$ .

We could slightly simplify the proof by defining π ( p k ) = 2 ( p 1 ) p k 1 r ( p k ) $\pi (p^k)=2(p-1)p^{k-1}r(p^k)$ in all cases. However it is of some interest to find cases in which 4 π ( p k ) $4\nmid \pi (p^k)$ , because this allows us to remove the condition P E ( R ) 0 $P\in E({{\mathbb {R}}})^0$ in some of our results (see Theorem 1.6). This is our reason to include the first case in (3.9).

Proof.For ease of notation we write r = r ( p k ) $r=r(p^k)$ throughout the proof.

We first observe that r P O mod p k + v p ( M P ) $rP \equiv O \bmod p^{k+v_p(MP)}$ by (3.8). That is we have k + v p ( M P ) v p ( r P ) $k+v_p(MP)\leqslant v_p(rP)$ , and hence by Lemma 3.4 and (3.8) we have

p k divides β r gcd ( β M , β r ) and M r . $$\begin{equation} p^k \text{ divides }\frac{\beta _r}{ \gcd (\beta _M,\beta _r)} \quad \text{ and }\quad M\mid r. \end{equation}$$ (3.10)
Let n Z $n\in {{\mathbb {Z}}}$ . By (3.10), the hypotheses of Proposition 3.6 are satisfied. If p k β n $p^k\mid \beta _n$ then the result follows immediately; suppose therefore that p k β n $p^k\nmid \beta _n$ . Proposition 3.6 shows that
β n + r β M + r β M r β M 2 ( 1 ) 2 ( β n + r β n 1 ) β n mod p k , $$\begin{equation*} \beta _{n+\ell r} \equiv {\left(\beta _{M+r}\beta _{M-r}\beta _M^{-2}\right)}^{\frac{\ell (\ell -1)}{2}} (\beta _{n+r}\beta _{n}^{-1})^\ell \beta _{n} \quad \bmod p^k, \end{equation*}$$
for every Z $\ell \in {{\mathbb {Z}}}$ , where β M + r β M r β M 2 , β n + r β n 1 $\beta _{M+r}\beta _{M-r}\beta _M^{-2}, \beta _{n+r}\beta _{n}^{-1}$ are p $p$ -adic units. In particular gcd ( β n , p k ) = gcd ( β n + r , p k ) = gcd ( β n , β r , p k ) $\gcd (\beta _n,p^k)=\gcd (\beta _{n+\ell r},p^k)=\gcd (\beta _n,\beta _r,p^k)$ .

Now # ( Z / p k Z ) × = ( p 1 ) p k 1 $\#({{\mathbb {Z}}}/p^k{{\mathbb {Z}}})^\times =(p-1)p^{k-1}$ , and so if u Z p × $u\in {{\mathbb {Z}}}_p^\times$ then

2 ( p 1 ) p k 1 u ( 1 ) 2 1 mod p k . $$\begin{equation*} 2(p-1)p^{k-1}\mid \ell \Rightarrow u^{\frac{\ell (\ell -1)}{2}} \equiv 1 \bmod p^k. \end{equation*}$$
Moreover if p 2 $p\ne 2$ and ( u p ) = 1 $(\frac{u}{p})=1$ then u = v 2 $u=v^2$ for v Z p × $v\in {{\mathbb {Z}}}_p^\times$ . So
( p 1 ) p k 1 , p 2 , u p = 1 u ( 1 ) 2 1 mod p k . $$\begin{equation*} (p-1)p^{k-1}\mid \ell ,\,p\ne 2, \,{\left(\frac{u}{p}\right)}=1 \Rightarrow u^{\frac{\ell (\ell -1)}{2}} \equiv 1 \bmod p^k. \end{equation*}$$
Thus by definition of π ( p k ) $\pi (p^k)$ , if π ( p k ) r $\pi (p^k) \mid \ell r$ then
β M + r β M r β M 2 ( 1 ) 2 ( β n + r β n 1 ) 1 mod p k , $$\begin{equation*} {\left(\beta _{M+r}\beta _{M-r}\beta _M^{-2}\right)}^{\frac{\ell (\ell -1)}{2}} (\beta _{n+r}\beta _{n}^{-1})^\ell \equiv 1\bmod p^k, \end{equation*}$$
which implies
β n + r β n mod p k . $$\begin{equation*} \beta _{n+\ell r} \equiv \beta _{n} \bmod p^k. \end{equation*}$$
Writing m = n + r $m=n+\ell r$ completes the proof. $\Box$

There is a simpler, but slightly weaker, bound for the period.

Lemma 3.8.Let M $M$ be as in Proposition 3.2, let k N $k\in {{\mathbb {N}}}$ , and let p $p$ be a prime. Then the period of β n mod p k $\beta _n\bmod p^k$ divides

2 M ( p 1 ) p 2 ( k 1 ) ord ( M P mod p ) , if v p ( M P ) = 0 , 2 M ( p 1 ) p 2 k 1 , otherwise . $$\begin{equation*} {\begin{cases} 2M(p-1)p^{2(k-1)} \operatorname{ord}(MP \bmod p), &\text{if } v_p(MP) = 0, \\ 2M(p-1)p^{2k-1}, \quad &\text{otherwise}. \end{cases}} \end{equation*}$$

Proof.Let Q = M P $Q = MP$ . By Proposition 3.7, it suffices to show that

ord ( Q mod p k + v p ( Q ) ) divides r 1 ( p k ) : = p k 1 ord ( Q mod p ) , if v p ( Q ) = 0 , p k , otherwise . $$\begin{equation*} \operatorname{ord}(Q\bmod p^{k+v_p(Q)}) \,\text{ divides }\, r_1(p^k) := {\begin{cases} p^{k-1} \operatorname{ord}(Q \bmod p), &\text{if } v_p(Q) = 0, \\ p^{k}, \quad &\text{otherwise}. \end{cases}} \end{equation*}$$
If v p ( Q ) = 0 $v_p(Q) = 0$ then Lemma 2.4 implies that v p ( p k 1 ord ( Q mod p ) Q ) k 1 + v p ( ord ( Q mod p ) Q ) k $v_p(p^{k-1}\operatorname{ord}(Q \bmod p)Q) \geqslant k-1 + v_p(\operatorname{ord}(Q \bmod p)Q) \geqslant k$ . If v p ( Q ) > 0 $v_p(Q) &gt; 0$ then Lemma 2.4 yields v p ( p k Q ) k + v p ( Q ) $v_p(p^kQ) \geqslant k + v_p(Q)$ . In both cases r 1 ( p k ) Q 0 mod p k + v p ( Q ) $r_1(p^k)Q \equiv 0 \bmod p^{k+ v_p(Q)}$ , as required. $\Box$

Remark 3.9.By definition M $M$ divides p | E ( Q p ) / E 0 ( Q p ) | $\prod _{p} |E({{\mathbb {Q}}}_p)/E_0({{\mathbb {Q}}}_p)|$ , hence is bounded uniformly with respect to P $P$ . Moreover ord ( M P mod p ) $\operatorname{ord}(MP \bmod p)$ divides | E 0 ( Q p ) / E q ( Q p ) | $|E_0({{\mathbb {Q}}}_p) / E_q({{\mathbb {Q}}}_p)|$ . Thus Lemma 3.8 shows that the period of β n mod N $\beta _n \bmod N$ can be bounded independently of P $P$ for all N N $N \in {{\mathbb {N}}}$ , with the bound only depending on E $E$ and N $N$ .

3.4 Signs

Recall from Definition 3.1 that the sign of β n $\beta _n$ is the sign of the sequence ψ n ( P ) $\psi _n(P)$ . The following is [27, Theorem 4] (see also [2] for a generalisation.)

Proposition 3.10. (Silverman–Stephens)There is a sign σ { ± 1 } $\sigma \in \lbrace \pm 1\rbrace $ and an irrational number β $\beta$ such that for all n N $n \in {{\mathbb {N}}}$ we have

σ n 1 sign ( β n ) = ( 1 ) n β , if P E ( R ) 0 , ( 1 ) n β + n 2 , if P E ( R ) 0 and n is even, ( 1 ) n 1 2 , if P E ( R ) 0 and n is odd. $$\begin{equation*} \sigma ^{n-1} \operatorname{sign}(\beta _n)= {\begin{cases} (-1)^{\lfloor n\beta \rfloor }, &\text{if}\nobreakspace P\in E(\mathbb {R})^{0}, \\ (-1)^{\lfloor n\beta \rfloor +\frac{n}{2}}, &\text{if}\nobreakspace P\notin E(\mathbb {R})^{0}\nobreakspace \text{and}\nobreakspace n\nobreakspace \text{is\nobreakspace even,} \\ (-1)^{\frac{n-1}{2}}, &\text{if}\nobreakspace P\notin E(\mathbb {R})^{0}\nobreakspace \text{and}\nobreakspace n\nobreakspace \text{is\nobreakspace odd.} \end{cases}} \end{equation*}$$
If P E ( R ) 0 $P\in E(\mathbb {R})^{0}$ then β $\beta$ is defined as follows. We fix an R ${{\mathbb {R}}}$ -analytic group isomorphism ψ : E ( R ) 0 R > 0 * / e Z $\psi :E({{\mathbb {R}}})^0\rightarrow {{\mathbb {R}}}^\ast _{&gt;0}/e^{{{\mathbb {Z}}}}$ . Then let β = log u $\beta = \log u$ where u $u$ is a representative of ψ ( P ) $\psi (P)$ in R > 0 * ${{\mathbb {R}}}^\ast _{&gt;0}$ with e 1 < u < 1 $e^{-1}&lt;u&lt;1$ .

In Silverman and Stephens' original statement of the theorem there is an isomorphism E ( R ) R * / q Z $E({{\mathbb {R}}})\rightarrow {{\mathbb {R}}}^\ast /q^{{\mathbb {Z}}}$ , which maps E ( R ) 0 $E({{\mathbb {R}}})^0$ to either R > 0 * / q Z ${{\mathbb {R}}}^\ast _{&gt;0}/q^{{\mathbb {Z}}}$ if q > 0 $q&gt;0$ or R > 0 * / q 2 Z ${{\mathbb {R}}}^\ast _{&gt;0}/q^{2{{\mathbb {Z}}}}$ otherwise. Without loss of generality we can assume that E ( R ) 0 $E({{\mathbb {R}}})^0$ is mapped to R > 0 * / e Z ${{\mathbb {R}}}^\ast _{&gt;0}/e^{{{\mathbb {Z}}}}$ , or else we can compose our isomorphism with v v 1 / log q $v\mapsto v^{-1/\log q}$ or v 1 / 2 log q $v^{-1/2\log q}$ . When P E ( R ) 0 $P\in E(\mathbb {R})^{0}$ their choice of u $u$ then satisfies e 1 < u < 1 $e^{-1}&lt;u&lt;1$ as above.

We want to say something about the Diophantine approximation properties of the irrational number β $\beta$ from the theorem. Let exp E : C E ( C ) $\exp _E:{{\mathbb {C}}}\rightarrow E({{\mathbb {C}}})$ be the usual parametrisation of E $E$ using the Weierstrass $\wp$ -function, see for example [26, Corollary VI.5.1.1]. Bosser and Gaudron [6, Theorem 1.2] proved:

Proposition 3.11. (Bosser–Gaudron)Let z C $z\in {{\mathbb {C}}}$ such that exp E ( z ) E ( Q ) { O } $\exp _E(z)\in E({{\mathbb {Q}}})\setminus \lbrace O\rbrace$ . Then we have

log | z | E 1 h ̂ ( exp E z ) , $$\begin{equation*} \log |z| \gg _E -1-\widehat{h}(\exp _E z), \end{equation*}$$
where h ̂ $\widehat{h}$ is the canonical height, as in Section 1.6.

A remark about the definition of exp E : C E ( C ) $\exp _E:{{\mathbb {C}}}\rightarrow E({{\mathbb {C}}})$ may be helpful. The usual convention would be to normalise this map so that in a certain sense the derivative of exp E $\exp _E$ at the origin is the identity. This is not necessary for our purposes, and the naive parametrisation familiar from a first course on elliptic curves suffices. We use only the fact that exp E $\exp _E$ is a fixed R ${{\mathbb {R}}}$ -analytic surjective additive group homomorphism, and Proposition 3.11 which holds regardless of normalisation. We use these to prove

Lemma 3.12.Suppose the point P $P$ from the start of this section satisfies P E ( R ) 0 $P\in E({{\mathbb {R}}})^0$ . Let β $\beta$ be as in Proposition 3.10, and let N Z { 0 } $N\in {{\mathbb {Z}}}\setminus \lbrace 0\rbrace$ . Then

min M Z log | N β M | E 1 h ̂ ( N P ) . $$\begin{equation*} \min _{M\in {{\mathbb {Z}}}}\log |N\beta -M| \gg _E -1- \widehat{h}(NP). \end{equation*}$$

Proof.Let w C $w\in {{\mathbb {C}}}^*$ such that exp E ( w ) = P $\exp _E(w)=P$ , so that exp E ( w R ) = E ( R ) 0 $\exp _E(w{{\mathbb {R}}})=E({{\mathbb {R}}})^0$ and ψ ( exp E ( t w ) ) = e t β + Z $\psi (\exp _E(tw))=e^{t\beta +{{\mathbb {Z}}}}$ for any t R $t\in {{\mathbb {R}}}$ . For any M Z $M\in {{\mathbb {Z}}}$ we deduce that

exp E ( N + β 1 M ) = ϕ 1 ( e N β + M + Z ) . $$\begin{equation*} \exp _E(N+\beta ^{-1}M) = \phi ^{-1}(e^{N\beta +M+{{\mathbb {Z}}}}). \end{equation*}$$
By the definition of β $\beta$ we deduce exp E ( N + β 1 M ) = ϕ 1 ( u N e Z ) $\exp _E(N+\beta ^{-1}M) = \phi ^{-1}(u^Ne^{{\mathbb {Z}}})$ which is N P $NP$ by definition of u $u$ . That is,
exp E 1 ( N P ) { ( N + β 1 M ) w : M Z } . $$\begin{equation*} \exp _E^{-1}(NP) \supseteq \lbrace (N+\beta ^{-1}M)w : M\in {{\mathbb {Z}}}\rbrace. \end{equation*}$$
Now by Proposition 3.11, any t $t$ such that t w exp E 1 ( N P ) $tw\in \exp _E^{-1}(NP)$ has log | t | E 1 h ̂ ( N P ) $\log |t|\gg _E -1- \widehat{h}(NP)$ , and so
min M Z log | N β M | E 1 h ̂ ( N P ) . $$\begin{equation*} \min _{M\in {{\mathbb {Z}}}}\log |N\beta -M| \gg _E -1- \widehat{h}(NP). \end{equation*}$$
$\Box$

3.5 Main result

We now provide the main technical input required for the results stated in the introduction (Proposition 3.19). Under certain assumptions, it stipulates the existence of many prime-numbered elements of the sequence β n $\beta _n$ which are divisible by primes to a certain valuation that are non-trivial with respect to a given Dirichlet character. We require an effective version of uniform distribution modulo 1 for primes in an arithmetic progression multiplied by an irrational. This is deduced from an exponential sum estimate. To begin, we quote two standard results on exponential sums in primes from Vaughan [31, Theorem 3.1, Lemma 3.1].

Lemma 3.13. (Vinogradov)If α R , a Z , q N $\alpha \in {{\mathbb {R}}}, a\in {{\mathbb {Z}}},q\in {{\mathbb {N}}}$ with gcd ( a , q ) = 1 , q y $\gcd (a,q)=1, q\leqslant y$ , and | α a / q | q 2 $|\alpha -a/q| \leqslant q^{-2}$ then

p y p prime ( log p ) e ( α p ) ( log y ) 4 ( y q 1 / 2 + y 4 / 5 + y 1 / 2 q 1 / 2 ) . $$\begin{equation*} \sum _{\substack{p\leqslant y\\ p \text{ prime}}} (\log p) e(\alpha p) \ll (\log y)^4 (y q^{-1/2}+y^{4/5}+y^{1/2}q^{1/2}). \end{equation*}$$

Lemma 3.14. (Siegel–Walfisz for linear exponential sums)Let B > 0 $B&gt;0$ . If α R , a Z , q N $\alpha \in {{\mathbb {R}}}, a\in {{\mathbb {Z}}},q\in {{\mathbb {N}}}$ with gcd ( a , q ) = 1 , q ( log y ) B $\gcd (a,q)=1, q\leqslant (\log y)^B$ and | α a / q | ( log y ) B / y $ |\alpha -a/q| \leqslant (\log y)^B/y$ then there is C B > 0 $C_B&gt;0$ such that

p y p prime e ( α p ) = μ ( q ) φ ( q ) m = 1 y e ( ( α a / q ) m ) + O B ( y exp ( C B log y ) ) , $$\begin{equation*} \sum _{\substack{p\leqslant y\\ p \text{ prime}}} e(\alpha p) =\frac{\mu (q)}{\varphi (q)}\sum _{m =1}^y e((\alpha -a/q)m) +O_B(y\exp (-C_B{\sqrt {\log y}})), \end{equation*}$$
where μ $\mu$ is the Möbius function and φ $\varphi$ is the Euler totient function.

We use these to estimate sums of the form y , prime ( log ) e ( α ) $\sum _{\ell \leqslant y,\ell \text{ prime}} (\log \ell ) e(\alpha \ell )$ .

Lemma 3.15.Suppose that we have α R , a Z , q N , y R $\alpha \in {{\mathbb {R}}}, a\in {{\mathbb {Z}}},q\in {{\mathbb {N}}},y\in {{\mathbb {R}}}$ such that

gcd ( a , q ) = 1 , q y | α a / q | 1 / q y . $$\begin{align} \gcd (a,q)&=1,\quad q\leqslant y\quad |\alpha -a/q| \leqslant 1/qy. \end{align}$$ (3.11)
Then
y prime ( log ) e ( α ) y ( log y ) 1 + y log log q q . $$\begin{equation} \sum _{\substack{\ell \leqslant y\\ \ell \text{prime}}} (\log \ell ) e(\alpha \ell ) \ll y(\log y)^{-1} + \frac{y\log \log q}{q}. \end{equation}$$ (3.12)

Proof.If q ( log y ) 5 $q\leqslant (\log y)^5$ we apply Lemma 3.14; otherwise we apply Lemma 3.13. Recalling the standard bound φ ( q ) q log log q $\varphi (q)\gg \frac{q}{\log \log q}$ gives the result. $\Box$

From this simple estimate we pass to a more difficult exponential sum.

Lemma 3.16.Let s , t N $s,t \in {{\mathbb {N}}}$ with gcd ( s , t ) = 1 $\gcd (s,t) = 1$ and let β $\beta$ be as in Proposition 3.10. For all y e e e , j N $y\geqslant e^{e^e},j\in {{\mathbb {N}}}$ we have

y s mod t prime ( log ) e ( j β / 2 ) E y t j h ̂ ( P ) log y log log log y . $$\begin{equation} \sum _{\substack{\ell \leqslant y \\ \ell \equiv s \bmod t\\ \ell \text{ prime}}}(\log \ell ) e(j \ell \beta /2) \ll _E ytj \sqrt {\frac{\widehat{h}(P)}{\log y}} \log \log \log y. \end{equation}$$ (3.13)

Proof.Fix y 1 , j N $y\geqslant 1,j\in {{\mathbb {N}}}$ . We use the formula

1 t m Z / t Z e ( m ( n s ) / t ) = 1 , n s mod t , 0 , otherwise , $$\begin{equation*} \frac{1}{t} \sum _{ m \in {{\mathbb {Z}}}/t{{\mathbb {Z}}}} e(m(n-s)/t) = {\begin{cases} 1, & n \equiv s \bmod t, \\ 0, & \text{otherwise}, \end{cases}} \end{equation*}$$
which is valid for all integers n $n$ . This gives
y s mod t ( log ) e ( j β / 2 ) = 1 t m Z / t Z e ( m s / t ) y ( log ) e ( ( m / t + j β / 2 ) ) . $$\begin{align*} \sum _{\substack{\ell \leqslant y \\ \ell \equiv s \bmod t}} (\log \ell )e(j \ell \beta /2) & = \frac{1}{t} \sum _{m \in {{\mathbb {Z}}}/t{{\mathbb {Z}}}} e(-ms/t) \sum _{\substack{\ell \leqslant y}} (\log \ell ) e((m/t + j \beta /2)\ell ). \end{align*}$$
We apply Lemma 3.15 to estimate the final sum in (3.13). We set
α = m / t + j β / 2 . $$\begin{equation} \alpha = m/t+j\beta /2. \end{equation}$$ (3.14)
We fix a Z , q N $a\in {{\mathbb {Z}}},q\in {{\mathbb {N}}}$ satisfying (3.11), noting that the existence of such ( a , q ) $(a,q)$ is guaranteed by Dirichlet's approximation theorem. Then (3.12) becomes
y ( log ) e ( ( m / t + j β / 2 ) ) y log y + y log log q q . $$\begin{equation} \sum _{\ell \leqslant y} (\log \ell ) e((m/t+j\beta /2) \ell ) \ll \frac{y}{\log y} + \frac{y\log \log q}{q}. \end{equation}$$ (3.15)
For this to be useful we need to deduce from (3.11) and (3.14) a lower bound on q $q$ . We are going to prove that
either y t 3 / 2 or q 1 t j log y h ̂ ( P ) . $$\begin{equation} \text{either}\quad y\leqslant t^{3/2}\quad \text{or}\quad q\geqslant \frac{1}{tj}\sqrt {\frac{\log y}{\widehat{h}(P)}}. \end{equation}$$ (3.16)

We start with the final condition in (3.11), and multiply it by 2 t q $2tq$ to get

log ( 2 t / y ) min M Z log | 2 t q α M | . $$\begin{equation*} \log (2t/y)\geqslant \min _{M\in {{\mathbb {Z}}}}\log |2tq\alpha -M|. \end{equation*}$$
The definition (3.14) gives
min M Z log | 2 t q α M | = min M Z log | t j q β M | , $$\begin{equation*} \min _{M\in {{\mathbb {Z}}}}\log |2tq\alpha -M|= \min _{M\in {{\mathbb {Z}}}}\log |tjq\beta -M|, \end{equation*}$$
and by Lemma 3.12 with N = t j q $N=tjq$ we have
min M Z log | t j q β M | E 1 h ̂ ( t j q P ) . $$\begin{equation*} \min _{M\in {{\mathbb {Z}}}}\log |tjq\beta -M| \gg _E -1- \widehat{h}(tjqP). \end{equation*}$$
Putting the last three displays together gives us
h ̂ ( t j q P ) + 1 E log ( y / 2 t ) . $$\begin{equation*} \widehat{h}(tjqP)+1\gg _E \log (y/2t). \end{equation*}$$
Recalling that h ̂ $\widehat{h}$ is a quadratic form on E ( Q ) R $E({{\mathbb {Q}}}) \otimes {{\mathbb {R}}}$ , we have h ̂ ( t j q P ) = ( t j q ) 2 h ̂ ( P ) $ \widehat{h}(tjqP)= (tjq)^2 \widehat{h}(P)$ and so
1 + log ( y / t ) E ( t j q ) 2 h ̂ ( P ) . $$\begin{equation*} -1+\log (y/t)\ll _E (tjq)^2 \widehat{h}(P). \end{equation*}$$
Provided y t 3 / 2 $y\geqslant t^{3/2}$ , this implies that log y E ( t j q ) 2 h ̂ ( P ) $\log y \ll _E (tjq)^2 \widehat{h}(P)$ , which is (3.16).

We substitute (3.16) into (3.15) and recall that log log log y 1 $ \log \log \log y \geqslant 1$ by assumption, to show that either

y ( log ) e ( ( m / t + j β / 2 ) ) E y ( log y ) 1 + y t j h ̂ ( P ) log y log log log y or y t 3 / 2 . $$\begin{equation*} \sum _{\ell \leqslant y} (\log \ell ) e(\ell (m/t+j\beta /2) ) \ll _{E} y(\log y)^{-1} +ytj \sqrt {\frac{\widehat{h}(P)}{\log y}} \log \log \log y \quad \text{or}\quad y \leqslant t^{3/2}. \end{equation*}$$
In the latter case we have y ( log ) e ( ( m / t + j β / 2 ) ) y y 1 / 3 t $\sum _{\ell \leqslant y} (\log \ell ) e(\ell (m/t+j\beta /2))\ll y\leqslant y^{1/3}t$ by the Prime Number theorem. So in either case
y ( log ) e ( ( m / t + j β / 2 ) ) E y 1 / 3 t + y ( log y ) 1 + y t j h ̂ ( P ) log y log log log y . $$\begin{equation*} \sum _{\ell \leqslant y} (\log \ell ) e(\ell (m/t+j\beta /2) ) \ll _{E} y^{1/3}t+ y(\log y)^{-1} +ytj \sqrt {\frac{\widehat{h}(P)}{\log y}} \log \log \log y. \end{equation*}$$
Since h ̂ ( P ) E 1 $\widehat{h}(P)\gg _E 1$ this implies (3.13). $\Box$

To apply the previous lemma we turn to the Erdős–Turán inequality [10, Theorem III]:

Lemma 3.17. (Erdős–Turán)For any 0 a < b 1 $0 \leqslant a &lt; b \leqslant 1$ , any real sequence t m $t_m$ , any M N $M\in {{\mathbb {N}}}$ and any H > 0 $H&gt;0$ we have

( b a ) M m = 1 M 1 { t m } [ a , b ) M H + 1 j H 1 j m = 1 M e ( j t m ) , $$\begin{equation*} {\left|(b-a)M - \sum _{m=1}^M \mathbf {1}_{\lbrace t_m\rbrace \in [a,b)} \right|}\ll \frac{M}{H} + \sum _{1\leqslant j \leqslant H} \frac{1}{j} {\left|\sum _{m=1}^M e(jt_m) \right|}, \end{equation*}$$
where we write { · } $\lbrace \,\cdot \,\rbrace$ for the fractional part, and 1 { t m } [ a , b ) = 1 $\mathbf {1}_{\lbrace t_m\rbrace \in [a,b)}=1$ if { t m } [ a , b ) ${\lbrace t_m\rbrace \in [a,b)}$ and 0 $\hskip.001pt 0$  otherwise.

We are now ready to prove our equidistribution result.

Proposition 3.18.Suppose the point P $P$ from the start of this section satisfies P E ( R ) 0 $P\in E({{\mathbb {R}}})^0$ . Let s , t N $s,t \in {{\mathbb {N}}}$ with gcd ( s , t ) = 1 $\gcd (s,t) = 1$ and let β $\beta$ be as in Proposition 3.10. For any 0 a < b 1 $0\leqslant a &lt; b \leqslant 1$ and any ε > 0 $\epsilon &gt;0$ we have

# { primes x : s mod t , { β / 2 } [ a , b ) } = b a φ ( t ) + O E , ε t ε ( log log log x ) 2 h ̂ ( P ) log x 1 / 4 x log x , $$\begin{eqnarray*} &&\#\lbrace \text{primes } \ell \leqslant x : \ell \equiv s \bmod t, \lbrace \ell \beta /2\rbrace \in [a,b) \rbrace \\ &&\quad = {\left(\frac{b-a}{\varphi (t)}+O_{E,\epsilon } {\left(\frac{t^\epsilon (\log \log \log x)^2{\widehat{h}(P)}}{\log x}\right)}^{1/4} \right)} \frac{x}{\log x}, \end{eqnarray*}$$
where we write { · } $\lbrace \,\cdot \,\rbrace$ for the fractional part. In particular for any ε > 0 $\epsilon &gt;0$ and any σ { ± 1 } $\sigma \in \lbrace \pm 1\rbrace$ we have
# { primes x : s mod t , ( 1 ) β = σ } = 1 2 φ ( t ) + O E , ε t ε ( log log log x ) 2 h ̂ ( P ) log x 1 / 4 x log x . $$\begin{eqnarray*} &&\#\lbrace \text{primes } \ell \leqslant x : \ell \equiv s \bmod t, (-1)^{\lfloor \ell \beta \rfloor } = {\sigma }\rbrace \\ &&\quad = {\left(\frac{1}{2\varphi (t)}+O_{E,\epsilon } {\left(\frac{t^\epsilon (\log \log \log x)^2{\widehat{h}(P)}}{\log x}\right)}^{1/4} \right)} \frac{x}{\log x}. \end{eqnarray*}$$

Proof.For the second part, we note that ( 1 ) β = 1 $(-1)^{\lfloor \ell \beta \rfloor } =1$ if and only if 0 { ( β / 2 ) } < 1 / 2 $0 \leqslant \lbrace \ell (\beta /2)\rbrace &lt; 1/2$ . So it suffices to prove the first claim in the proposition.

During the proof we will repeatedly use the fact that h ̂ ( P ) E 1 $\widehat{h}(P)\gg _E 1$ , which holds by for example [26, Theorem VIII.9.10(a)]. If t > ( log x ) 1 / ε $t&gt;(\log x)^{1/\epsilon }$ then the bound follows at once from this last result and the Prime Number theorem. We will assume from now on that t ( log x ) 1 / ε $t\leqslant (\log x)^{1/\epsilon }$ .

Throughout the proof we write e ( y ) = exp ( 2 π i y ) $e(y) = \exp (2 \pi i y)$ .

We first apply Lemma 3.17 with M , t m $M,t_m$ as follows. Denote the primes s mod t , x $\ell \equiv s \bmod t, \ell \leqslant x$ by 1 , , M $\ell _1,\ldots\,,\ell _M$ and let t m = { m β / 2 } $t_m=\lbrace \ell _m\beta /2\rbrace$ . There is c > 0 $c&gt;0$ such that for each B > 0 $B&gt;0$ with t ( log x ) B $t\leqslant (\log x)^B$ , we have

M = x φ ( t ) log x + O x φ ( t ) ( log x ) 2 + O B ( x exp ( c log x ) ) , $$\begin{equation*} M = \frac{x}{\varphi (t)\log x} +O{\left(\frac{x}{\varphi (t)(\log x)^2}\right)}+ O_B(x\exp (-c\sqrt {\log x})), \end{equation*}$$
by the Siegel–Walfisz theorem [20, Corollary 11.21, see also p. 5]. In particular, setting B = 1 / ε $B=1/\epsilon$ and recalling that t ( log x ) 1 / ε $t\leqslant (\log x)^{1/\epsilon }$ and thus exp ( c log x ) ε t ( log x ) 2 $ \exp (c\sqrt {\log x}) \gg _\epsilon t (\log x)^2$ , it follows that
M = x φ ( t ) log x + O x φ ( t ) ( log x ) 2 , $$\begin{equation*} {M} = \frac{x}{\varphi (t)\log x} + O{\left(\frac{x}{\varphi (t)(\log x)^{2}}\right)}, \end{equation*}$$
We substitute this into Lemma 3.17 to obtain
# { m M : { m β / 2 } [ a , b ) } ( b a ) x φ ( t ) log x x H φ ( t ) log x + x φ ( t ) ( log x ) 2 + 1 j H 1 j x s mod t e ( j β / 2 ) . $$\begin{eqnarray} &&\#\big \lbrace m\leqslant M: \lbrace \ell _m\beta /2\rbrace \in [a,b)\big \rbrace -\frac{(b-a)x}{\varphi (t)\log x} \\ &&\quad \ll \frac{x}{H\varphi (t)\log x} +\frac{x}{\varphi (t)(\log x)^{2}} + \sum _{1\leqslant j \leqslant H} \frac{1}{j}{\left|\sum _{\substack{\ell \leqslant x \\ \ell \equiv s \bmod t}} e(j\ell \beta /2)\right|}. \end{eqnarray}$$ (3.17)
Our goal is to estimate the last sum above. As often happens it is convenient to count primes weighted by the von Mangoldt function. By partial summation,
x s mod t prime e ( j β / 2 ) = 1 log x x s mod t prime ( log ) e ( j β / 2 ) + 1 x 1 y ( log y ) 2 y s mod t prime ( log ) e ( j β / 2 ) d y . $$\begin{eqnarray} &&\sum _{\substack{\ell \leqslant x \\ \ell \equiv s \bmod t \\ \ell \text{ prime}}} e(j \ell \beta /2) = \frac{1}{\log x}\sum _{\substack{\ell \leqslant x \\ \ell \equiv s \bmod t\\ \ell \text{ prime}}} (\log \ell ) e(j \ell \beta /2) \, \nonumber\\ &&\quad + \int _{1}^{x}\frac{1}{y(\log y )^2} \sum _{\substack{\ell \leqslant y \\ \ell \equiv s \bmod t\\ \ell \text{ prime}}}(\log \ell ) e(j \ell \beta /2)\,\mathrm{d} y. \end{eqnarray}$$ (3.18)
It follows from Lemma 3.16 and (3.18) that for each non-zero integer j $j$ we have
x s mod t prime e ( j β / 2 ) E x log x · t j h ̂ ( P ) log x log log log x . $$\begin{equation*} \sum _{\substack{\ell \leqslant x \\ \ell \equiv s \bmod t\\ \ell \text{ prime}}} e(j \ell \beta /2) \ll _E \frac{x}{\log x}\cdot tj\sqrt {\frac{\widehat{h}(P)}{\log x}}\log \log \log x. \end{equation*}$$
Together with (3.17) and the choice
H = log x h ̂ ( P ) 1 / 4 t φ ( t ) log log log x 1 / 2 , $$\begin{equation*} H = {\left(\frac{\log x}{\widehat{h}(P)}\right)}^{1/4} {\left(t\varphi (t) \log \log \log x\right)}^{-1/2}, \end{equation*}$$
this implies that
# { m M : { m β / 2 } [ a , b ) } ( b a ) x φ ( t ) log x x log x · 1 φ ( t ) log x + h ̂ ( P ) log x 1 / 4 t log log log x φ ( t ) 1 / 2 . $$\begin{eqnarray*} &&\#\big \lbrace m\leqslant M: \lbrace \ell _m\beta /2\rbrace \in [a,b)\big \rbrace -\frac{(b-a)x}{\varphi (t)\log x} \nonumber\\ &&\quad \ll \frac{x}{\log x}\cdot {\left(\frac{1}{\varphi (t)\log x} + {\left(\frac{\widehat{h}(P)}{\log x}\right)}^{1/4} {\left(\frac{t\log \log \log x}{ \varphi (t) }\right)}^{1/2} \right)}. \end{eqnarray*}$$
The result follows since h ̂ ( P ) E 1 $\widehat{h}(P)\gg _E 1$ and φ ( t ) ε t 1 ε $\varphi (t)\gg _\epsilon t^{1-\epsilon }$ . $\Box$

Our main result is now as follows. In the statement ord ( χ ( p ) ) $\operatorname{ord}(\chi (p))$ denotes the multiplicative order of the root of unity χ ( p ) $\chi (p)$ .

Proposition 3.19.Let χ $\chi$ be a Dirichlet character with modulus q ( χ ) $q(\chi )$ . Let π $\pi$ be the period of β n mod q ( χ ) $\beta _n\bmod q(\chi )$ . Suppose that there exists α N $\alpha \in {{\mathbb {N}}}$ such that

gcd ( α , π ) = 1 $$\begin{equation} \gcd (\alpha , \pi )=1 \end{equation}$$ (3.19)
and such that one of the following holds:
χ ( | β α | ) 0 , 1 , or χ ( | β α | ) 0 , 1 and 4 π , or χ ( | β α | ) 0 , 1 and P E ( R ) 0 . $$\begin{align} & \chi (|\beta _\alpha |) \ne 0,1, \quad \text{or} \nonumber \\ & \chi (-|\beta _\alpha |)\ne 0,1\text{ and } 4\nmid \pi , \quad \text{or} \\ & \chi (-|\beta _\alpha |)\ne 0,1 \text{ and }P\in E(\mathbb {R})^{0}. \nonumber\end{align}$$ (3.20)
Then for any ε > 0 $\epsilon &gt;0$ and x > exp ( π 2 ) $x&gt;\exp (\pi ^2)$ we have
# primes x : ord ( χ ( p ) ) v p ( β ) for some prime p q ( χ ) 1 2 φ ( π ) + O E , ε π ε ( log log log x ) 2 h ̂ ( P ) log x 1 / 4 x log x . $$\begin{multline*} \#{\left\lbrace \text{primes } \ell \leqslant x : \operatorname{ord}(\chi (p)) \nmid v_p(\beta _\ell ) \text{ for some prime }p \nmid q(\chi ) \right\rbrace} \\ \geqslant {\left(\frac{1}{2\varphi (\pi )}+O_{E,\epsilon } {\left(\frac{\pi ^\epsilon (\log \log \log x)^2{\widehat{h}(P)}}{\log x}\right)}^{1/4} \right)} \frac{x}{\log x}. \end{multline*}$$

Proof.From (3.20), there is τ { ± 1 } $\tau \in \lbrace \pm 1\rbrace$ such that χ ( τ | β α | ) 0 , 1 $\chi (\tau |\beta _\alpha |)\ne 0,1$ . We separate into two cases depending on the real properties of P $P$ .

Case 1. P E ( R ) 0 $P\in E(\mathbb {R})^{0}$ : From Proposition 3.10 we have sign ( β n ) = σ n 1 ( 1 ) n β $\operatorname{sign}(\beta _n) = \sigma ^{n-1}(-1)^{\lfloor n \beta \rfloor }$ for some σ { ± 1 } $\sigma \in \lbrace \pm 1\rbrace$ and some irrational number β $\beta$ . Now consider the set of primes

Λ = { prime : α mod π , sign ( β ) = τ sign ( β α ) } } . $$\begin{equation*} \Lambda =\lbrace \ell \text{ prime} : \ell \equiv \alpha \bmod \pi ,\,\operatorname{sign}(\beta _\ell ) =\tau \operatorname{sign}(\beta _\alpha ) \rbrace \rbrace. \end{equation*}$$
Let Λ $\ell \in \Lambda$ . Then by periodicity we have β β α mod q ( χ ) $\beta _\ell \equiv \beta _\alpha \bmod q(\chi )$ , so χ ( β ) = χ ( β α ) $\chi (\beta _\ell ) = \chi (\beta _\alpha )$ as χ $\chi$ is periodic modulo q ( χ ) $q(\chi )$ . Moreover, we have arranged signs so that χ ( | β | ) = χ ( τ | β α | ) 0 , 1 $\chi (|\beta _\ell |)=\chi (\tau |\beta _\alpha |)\ne 0,1$ . Hence as χ $\chi$ is multiplicative we deduce the existence of a prime factor p $p$ of | β | $|\beta _\ell |$ with p q ( χ ) $p\nmid q(\chi )$ and ord ( χ ( p ) ) v p ( β ) $\operatorname{ord}(\chi (p)) \nmid v_p(\beta _\ell )$ . It thus suffices to note that { Λ : x } $\lbrace \ell \in \Lambda :\ell \leqslant x\rbrace$ satisfies the required lower bound by (3.19) and Proposition 3.18.

Case 2. P E ( R ) 0 $P\not\in E(\mathbb {R})^{0}$ : In order to handle a number of subcases simultaneously, we show that there is ι { 0 , 1 , 2 , 3 } $\iota \in \lbrace 0,1,2,3\rbrace$ such that α + ι π $\alpha +\iota \pi$ is odd and

( 1 ) ( α + ι π 1 ) / 2 = τ sign ( β α ) , if α is even , τ ( 1 ) ( α 1 ) / 2 , if α is odd . $$\begin{equation} (-1)^{(\alpha +\iota \pi -1)/2}= {\begin{cases} \tau \operatorname{sign}(\beta _\alpha ),& \text{if }\alpha \text{ is even}, \\ \tau (-1)^{(\alpha -1)/2},& \text{if }\alpha \text{ is odd}. \end{cases}} \end{equation}$$ (3.21)

Case 2.1. 2 α $2 \mid \alpha$ . Here π $\pi$ is odd by (3.19). Choosing ι { 1 , 3 } $\iota \in \lbrace 1,3\rbrace$ we can arrange for α + ι π 1 2 $\frac{\alpha +\iota \pi -1}{2}$ to be odd or even, and hence ( 1 ) ( α + ι π 1 ) / 2 = 1 $(-1)^{(\alpha +\iota \pi -1)/2}= -1$ or 1 to satisfy (3.21).

Case 2.2. 2 α $2\nmid \alpha$ and 4 π $4\mid \pi$ . Here we have τ = 1 $\tau =1$ by (3.20). Let ι = 0 $\iota =0$ and then ( 1 ) ( α 1 ) / 2 = τ ( 1 ) ( α 1 ) / 2 $(-1)^{(\alpha -1)/2}=\tau (-1)^{(\alpha -1)/2}$ as required for (3.21).

Case 2.3. 2 α $2\nmid \alpha$ and 4 π $4\nmid \pi$ . We can choose ι { 0 , 2 } $\iota \in \lbrace 0,2\rbrace$ so that ι π / 2 $\iota \pi /2$ is odd or even as needed. So we arrange ( 1 ) ι π / 2 = τ $(-1)^{\iota \pi /2}=\tau$ which gives (3.21).

We now let q = lcm ( 4 , π ) $q=\operatorname{lcm}(4,\pi )$ and consider primes $\ell$ of the form α + ι π mod q $\ell \equiv \alpha +\iota \pi \bmod q$ . By Proposition 3.10 and (3.21) we then have sign ( β ) = τ sign ( β α ) $\operatorname{sign}(\beta _\ell ) = \tau \operatorname{sign}(\beta _\alpha )$ . But β β α mod q ( χ ) $\beta _\ell \equiv \beta _\alpha \bmod q(\chi )$ by periodicity, so χ ( | β | ) = χ ( τ | β α | ) 0 , 1 $\chi (|\beta _\ell |)=\chi (\tau |\beta _\alpha |)\ne 0,1$ by (3.20) as χ $\chi$ is periodic modulo q ( χ ) $q(\chi )$ . We are now in a similar situation to Case 1. Here (3.19) and the fact that α + ι π $\alpha + \iota \pi$ is odd implies that gcd ( α + ι π , q ) = 1 $\gcd (\alpha +\iota \pi , q) = 1$ . Together with the assumption that π < log x $\pi &lt; \sqrt {\log x}$ in the proposition, this allows us to apply the Siegel–Walfisz theorem [20, Corollary 11.21] to show that the set under consideration has size

x φ ( q ) log x + O q x ( log x ) 2 1 2 φ ( π ) + O π log x x log x . $$\begin{equation*} \frac{x}{\varphi (q) \log x}+O{\left(\frac{qx}{(\log x)^2}\right)}\geqslant {\left(\frac{1}{2\varphi (\pi )} + O{\left(\frac{\pi }{\log x}\right)}\right)}\frac{x}{\log x}. \end{equation*}$$
Using again the fact that π < log x $\pi &lt; \sqrt {\log x}$ , the claim follows. $\Box$

Remark 3.20.Aside from finitely many exceptions, we expect that the primes p $p$ constructed in Proposition 3.19 satisfy the stronger condition v p ( β ) = 1 $v_p(\beta _{\ell }) = 1$ . This is the condition referred to in Section 1.3 as being a ‘non-Wieferich prime for base P E $P\in E$ ’.

4 BRAUER GROUPS

The aim of this section is to prove Theorem 1.6. We begin with some preliminaries on Brauer groups.

4.1 Recap of Brauer groups

For a scheme X $X$ we denote by Br X = H 2 ( X , G m ) $\operatorname{Br}X = {{\mathrm{H}}}^2(X,\mathbb {G}_\mathrm{m})$ its (cohomological) Brauer group. If X $X$ is regular and integral and D X $D \subset X$ is a regular integral divisor, then there is an associated residue map
D : Br ( X D ) [ ] H 1 ( D , Q / Z ) , $$\begin{equation*} \operatorname{\partial }_D : \operatorname{Br}(X \setminus D) [\ell ^\infty ] \rightarrow {{\mathrm{H}}}^1(D, {{\mathbb {Q}}}/{{\mathbb {Z}}}), \end{equation*}$$
where