Uniform boundedness in terms of ramification
 Álvaro LozanoRobledo^{1}Email authorView ORCID ID profile
Received: 12 December 2016
Accepted: 21 September 2017
Published: 31 January 2018
Abstract
Let \(d\ge 1\) be fixed. Let F be a number field of degree d, and let E / F be an elliptic curve. Let \(E(F)_{\text {tors}}\) be the torsion subgroup of E(F). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a constant B(d), which depends on d but not on the chosen field F or on the curve E / F, such that the size of \(E(F)_{\text {tors}}\) is bounded by B(d). Moreover, Merel gave a bound (exponential in d) for the largest prime that may be a divisor of the order of \(E(F)_{\text {tors}}\). In 1996, Parent proved a bound (also exponential in d) for the largest ppower order of a torsion point that may appear in \(E(F)_{\text {tors}}\). It has been conjectured, however, that there is a bound for the size of \(E(F)_{\text {tors}}\) that is polynomial in d. In this article we show that under certain hypotheses there is a linear bound for the largest ppower order of a torsion point defined over F, which in fact is linear in the maximum ramification index of a prime ideal of the ring of integers F over (p).
Mathematics Subject Classification
1 Introduction
Let F be a number field, and let E / F be an elliptic curve defined over F. The Mordell–Weil theorem states that E(F), the set of Frational points on E, can be given the structure of a finitely generated abelian group. In particular, the torsion subgroup of E(F), henceforth denoted by \(E(F)_{\text {tors}}\), is a finite group. In 1996, Merel proved that there is a uniform bound for the size of \(E(F)_{\text {tors}}\), which is independent of the chosen curve E / F and, in fact, the bound only depends on the degree of \(F/\mathbb {Q}\). The bounds were improved by Oesterlé, and later by Parent in 1999.
Definition 1.1
For each \(n\ge 1\), we define \(S^n(d)\) as the set of primes p for which there exists a number field F of degree \(\le d\) and an elliptic curve E / F such that E(F) contains a point of exact order \(p^n\). We also define T(d) as the supremum of \(E(F)_{\text {tors}}\), over all F and E as above. Finally, we define \(S_{\text {nonCM}}^n(d)\) (resp. \(S_{\text {CM}}^n(d)\)) as before, except that we only consider elliptic curves E / F without CM (resp. with CM).
We remark that \(S^{n+1}(d)\subseteq S^n(d)\) for all \(n\ge 1\), and if \(p\in S^{n}(d)\), then \(p^n\le T(d)\). Mazur [31] has shown that \(S^1(d)=\{2,3,5,7\}\) and \(T(1)=16\). Results of Kenku, Kamienny, and Momose imply that \(S^1(2)=\{2,3,5,7,11,13\}\) and \(T(2)=24\). Parent determined \(S^1(3)=S^1(2)\). In addition, Derickx et al. [6] have shown that \(S^1(4) = S^1(3)\cup \{17\}\), \(S^1(5)=S^1(4)\cup \{19\}\), \(S^1(6)= S^1(5)\cup \{37\}\), and \(S^1(7)\subseteq \{p\le 23 \}\cup \{37, 43, 59, 61, 67, 71, 73, 113, 127\}\). Let us cite Merel, Oesterlé, and Parent’s work more precisely (Oesterlé’s bound is unpublished, but appears in [6]).
Theorem 1.2
It is a “folklore” conjecture that T(d) should be subexponentially bounded (see for instance [12, 15]). We reproduce an explicit version of the conjecture, as in Conjecture 1 of [3].
Conjecture 1.3
There is a constant \(C_1\) such that \(T(d)\le C_1 \cdot d \log \log d\), for all \(d\ge 1\).
Flexor and Oesterlé [12] have shown that if E / F has at least one place of additive reduction, then \(E(F)_{\text {tors}}\le 48d\), and if it has at least two places of additive reduction, then \(E(F)_{\text {tors}}\le 12\). Hindry and Silverman ([15, Théorème 1]) show that if E / F has everywhere good reduction then \(E(F)_{\text {tors}}\le 1977408\cdot d\log d\). Turning our attention once again to \(S^n(d)\), we propose the following conjecture. Here \(\varphi (\cdot )\) is the Euler phi function.
Conjecture 1.4
There is a constant \(C_2\) such that if \(p\in S^n(d)\), then \(\varphi (p^n)\le C_2\cdot d\), for all \(d\ge 1\).
If we restrict our attention to CM curves, then Conjecture 1.4 follows from work of Silverberg [42], and Prasad and Yogananda ([38]; see also [3]), and the constant is \(\le 6\), i.e., if \(p\in S_\text {CM}^n(d)\), then \(\varphi (p^n)\le 6d\). See Theorem 6.9 below for a precise statement. In addition, in [28], the author has shown that Conjecture 1.4 holds (with \(C_2=24\)) when E / F has potential supersingular reduction at a prime above p.
Theorem 1.5
In this article, we restrict our study of \(E(F)_{\text {tors}}\) to the simpler case of elliptic curves E / F that arise from elliptic curves defined over a fixed number field L (contained in F), whose base field has been extended to F.
Definition 1.6
Let L be a fixed number field, let d be an integer with \(d\ge [L:\mathbb {Q}]\), and let \(S_L^n(d)\) be the set of pairs (p, F), where p is a prime for which there exists a finite extension F / L of number fields with \([F:\mathbb {Q}]\le d\), and an elliptic curve E / L (either without CM, or with CM by a maximal order), such that \(E(F)_{\text {tors}}\) contains a point of exact order \(p^n\). If \(\Sigma \subseteq L\) is specified, then \(S_L^n(d,\Sigma )\) is as before, except that we only consider elliptic curves E with \(j(E)\not \in \Sigma \). Finally, we define \(S_{L,\text {maxCM}}^n(d)\) when we restrict to curves E / L with CM by a maximal order.
In [27], we showed that if \(p\in S_\mathbb {Q}^1(d)\) with \(p\ge 11\) and \(p\ne 13\), then \(\varphi (p)\le 3d\), and if \(p\ne 37\), then \(\varphi (p)\le 2d\). Moreover, we gave a conjectural formula for \(S_\mathbb {Q}^1(d)\), and showed that the formula holds for all \(1\le d \le 42\). Our theorems here provide bounds in terms of certain ramification indices that we define next. In the rest of the paper, if \(\mathbb {F}\) is a number field or a local field, then \(\mathcal {O}_\mathbb {F}\) denotes its ring of integers.
Definition 1.7
Let p be a prime, and let F / L be an extension of number fields. We define \(e_\text {min}(p,F/L)\) (resp. \(e_\text {max}(p,F/L)\)) as the smallest (resp. largest) ramification index \(e(\mathfrak {P}\wp )\) for a prime \(\mathfrak {P}\) of \(\mathcal {O}_F\) over a prime \(\wp \) of \(\mathcal {O}_L\) lying above the rational prime p.
Now we can state our main theorems.
Theorem 1.8
Theorem 1.9
The finite set \(\Sigma _L\) is computable (or decidable) in the sense that given \(j_0\in L\), there is an algorithm to check whether \(j_0\) belongs to \(\Sigma _L\). We emphasize here that the notation \(S_L^n(d)\), as in Definition 1.6, excludes elliptic curves with CM by nonmaximal orders for technical reasons (that we hope to address in future work). However, there are only finitely many elliptic curves with CM by nonmaximal orders defined over L, so such jinvariants could be included in \(\Sigma _L\), and the second bound in Theorem 1.9 would apply to all elliptic curves E defined over L with j(E) not in the finite set \(\Sigma _L\).
When \(L=\mathbb {Q}\), the set \(\Sigma _L\) can be made explicit (it is formed by the six jinvariants without CM of Table 1 of Sect. 3), and our methods yield an improved bound.
Theorem 1.10
If \(p>2\) and \((p,F)\in S_\mathbb {Q}^n(d)\), then \(\varphi (p^n)\le 222\cdot e_\text {max}(p,F/\mathbb {Q})\le 222\cdot d\).
In light of Theorems 1.5, 1.8, 1.9, and 1.10, we revisit Conjecture 1.4 and propose the following stronger version.
Conjecture 1.11
Theorem 1.5 shows Conjecture 1.11 when E / F has a prime of potential supersingular reduction above p, with \(C_3=24\). When E / F has at least one prime \(\mathfrak {P}\) of additive reduction, then Conjecture 1.11 follows from the aforementioned work of Flexor and Oesterlé ([12, Théorèm 2 and Remarque 2]), for they in fact show that \(E(F)_{\text {tors}}\le 48e(\mathfrak {P}p)\), where \(e(\mathfrak {P}p)\) denotes the ramification index of \(\mathfrak {P}\) over (p) in \(F/\mathbb {Q}\).
Our theorems follow from explicit lower bounds (divisibility properties, in fact) on the ramification of primes above p, in the extensions generated by points of ppower order, and recent work of Larson and Vaintrob on isogenies [23]. In Sect. 2 we state our refined bound (Theorem 2.1), we specialize the bounds to elliptic curves over \(\mathbb {Q}\) in Theorem 2.2 (which proves Theorem 1.10). The proof of Theorem 1.8 will be delayed to Sect. 6.3 (see Theorem 6.10), and we put everything together to prove Theorem 1.9 in Sect. 8.
2 Refined bounds
Let L be a number field, let p be a prime, let \(n\ge 1\), and let \(\zeta =\zeta _{p^n}\) be a primitive \(p^n\)th root of unity. Let \(\wp \) be a prime ideal of the ring of integers \(\mathcal {O}_L\) of L lying above p. The ramification index of the primes above \(\wp \) in the extension \(L(\zeta )/L\) is a divisor of \(\varphi (p^n)\), and it is divisible by \(\varphi (p^n)/\gcd (\varphi (p^n),e(\wp p))\). In this article we study the ramification above p in the extension L(R) / L, where R is a torsion point of exact order \(p^n\) in an elliptic curve E defined over L. We show the following:
Theorem 2.1
 (1)
E / L does not admit an Lrational isogeny of degree \(p^a\), or
 (2)
Let \(L_\wp ^{\text {nr}}\) be the maximal unramified extension of \(L_\wp \), the completion of L at \(\wp \), and let \(K/L_\wp ^{\text {nr}}\) is the smallest extension such that E / K has good or multiplicative reduction. If E / L admits an Lrational isogeny \(\phi \) of degree p, such that \(\ker (\phi )=\langle S\rangle \subset E[p]\), then the ramification index of K(S) / K is \(>1\), or the ramification index of \(\wp \) in the Galois extension L(S) / L satisfies that the quotient \(e(\wp ,L(S)/L)/\gcd (e(\wp ,L(S)/L),e(K/L_\wp ^{\text {nr}}))>1\). If so, let \(a=1\).
Proof

The case of potential multiplicative reduction is treated in Sect. 5. In particular, if E / L satisfies hypothesis (1) or (2), then Theorem 5.1, parts (e) and (f), imply that there is a prime \(\Omega _R\) of L(R) above \(\wp \) such that the ramification index \(e(\Omega _R\wp )\) is divisible either byand the theorem follows in this case.$$\begin{aligned} \varphi (p^n)/\gcd (\varphi (p^n),2e(\wp p)p^{a1}), \text { or }\ p^{na+1}, \end{aligned}$$

The case of potential good ordinary reduction is treated in Sect. 6.1. In particular, if E / L satisfies hypotheses (1) or (2), then Theorem 6.3 implies that there is a prime \(\Omega _R\) of L(R) above \(\wp \) such that the ramification index \(e(\Omega _R\wp )\) is divisible either bywhere \(e=e(K/\mathbb {Q}_p)\) is the ramification index of \(K/\mathbb {Q}_p\). If \(p=3\), then \(\varphi (p^n)=2\cdot 3^{n1}\) and e is a divisor of \(12e(\wp p)\), so if \(e(\wp p)=1\) then \(\varphi (p^n)/\gcd (\varphi (p^n),e\cdot p^{a1})\) is divisible by \(\varphi (p^n)/\gcd (\varphi (p^n),t\cdot p^{a1})\) with \(t=9\) or 6. If \(p>3\), then e is a divisor of either \(4e(\wp p)\) or \(6e(\wp p)\), and the theorem follows in this case.$$\begin{aligned} \varphi (p^n)/\gcd (\varphi (p^n),e\cdot p^{a1}) \text { or } p^{na+1}, \end{aligned}$$

The case of potential good supersingular reduction is treated in Sect. 6.2, where we quote our results from previous works ([26, 28]). Theorem 6.7 implies that there is a number \(c=c(E/L,R,\wp )\) with \(1\le c\le 12e(\wp p)\) (with \(c\le 6e(\wp p)\) if \(p>3\)), such that the ramification index \(e(\Omega _R\wp )\) of any prime \(\Omega _R\) above \(\wp \) in the extension L(R) / L is divisible by \(\varphi (p^n)/\gcd (c,\varphi (p^n))\). Moreover, if \(e(\wp p)=1\) and \(p>3\), then \(e(\Omega _R\wp )\) is divisible by \((p^21)p^{2(n1)}/6\), or \((p1)p^{2(n1)}/\gcd (p1,4)\), therefore it is also divisible by \(\varphi (p^n)/\gcd (\varphi (p^n),t)\) with \(t=4\) or 6. If \(d=1\) and \(p=3\), then \(e(\Omega _R\wp )\) is divisible by \(\varphi (3^n)/\gcd (\varphi (3^n),t)\) with \(t=6\) or 9. \(\square \)
When \(L=\mathbb {Q}\), the previous theorem can be improved because we have a complete classification of noncuspidal \(\mathbb {Q}\)points on the modular curves \(X_0(N)\), which correspond to all possible \(\mathbb {Q}\)rational isogenies of elliptic curves over \(\mathbb {Q}\), as discussed in Sect. 3.
Theorem 2.2
p  3  5  7  13  37  else 

b(p)  3  3  2  2  \({\left\{ \begin{array}{ll} 2,&{} if\;j(E)=7\cdot 11^3 \\ 1,&{} otherwise \\ \end{array}\right. }\)  1 
Proof
p  3  5  7  11  13  17  19  37  43  67  163  else 

\(a(\mathbb {Q},p)\)  4  3  2  2  2  2  2  2  2  2  2  1. 
Let \((j_0,p)\) be any of the jinvariants that are listed in Table 1, with \(p\ne 37\), and let \(E/\mathbb {Q}\) be an elliptic curve with \(j(E)=j_0\). Then \(E/\mathbb {Q}\) has potential supersingular reduction at p (see Sect. 6.2, Table 2). Let \(R\in E[p^n]\) be a point of exact order \(p^n\). Theorem 6.8 shows that the ramification index of any prime \(\Omega _R\) that lies above p in the extension \(\mathbb {Q}(R)/\mathbb {Q}\) is divisible by \((p1)p^{2n2}/2\) if \(p>3\) and \(n\ge 1\), and by \(3^{2n4}\) if \(p=3\) and \(n\ge 3\). In particular, \(e(\Omega _Rp)\) is divisible by \(\varphi (p^n)/2\) for \(p>3\) (which in turn is divisible by \(\varphi (p^n)/\gcd (\varphi (p^n),t)\) for \(t=4\) or \(t=6\) as claimed), and when \(p=3\), it is divisible by \(3^{n2}\), which is divisible by \(\varphi (3^n)/\gcd (\varphi (3^n),t\cdot 3^{b(3)1})\), for \(t=6\) or 9, because \(b(3)=3\).

Let \(R\in E\) be a point of exact order \(37^n\), for \(n\ge 2\). Then, there is a prime \(\Omega _R\) of \(\mathbb {Q}(R)\) over (37) such that \(e(\Omega _R37)\) is divisible by \(\varphi (37^n)/37=\varphi (37^{n1})\), or \(f\cdot 37^{n1}\).

Let \(R\in E'\) be a point of exact order \(37^n\), for \(n\ge 1\). Then, there is a prime \(\Omega _R\) of \(\mathbb {Q}(R)\) over (37) such that \(e(\Omega _R37)\) is divisible by \(\varphi (37^n)\), or \(f'\cdot 37^{n}\).
Remark 2.3
3 Rational points on the modular curve \(X_0(N)\)
Theorem 3.1
Using the formula for the genus of \(X_0(p^a)\), we can find all those with genus 1 and 2.
Corollary 3.2
 (1)
\(a_1(p)=1\) for all \(p\ge 17\), and \(a_2(p)=1\) for all \(p\ge 23\).
 (2)Moreover, the values of \(a_1(p)\) and \(a_2(p)\) are given by the following table.
p
2
3
5
7
11
13
17
19
else
\(a_1(p)\)
5
3
3
2
1
2
1
1
1
\(a_2(p)\)
6
4
3
3
2
2
2
2
1
The \(\mathbb {Q}\)rational points on \(X_0(N)\) have been described completely in the literature, for all \(N\ge 1\). Certainly, one decisive step in their classification was [31], where Mazur dealt with the case when N is prime. The complete classification of \(\mathbb {Q}\)rational points on \(X_0(N)\), for any N, was completed due to work of Fricke, Kenku, Klein, Kubert, Ligozat, Mazur and Ogg, among others (see the references at the bottom of Table 1).
Theorem 3.3
 (1)
\(N\le 10\), or \(N= 12,13, 16,18\) or 25. In this case \(X_0(N)\) is a curve of genus 0 and its \(\mathbb {Q}\)rational points form an infinite 1parameter family; or
 (2)
\(N=11,14,15,17,21\), or 37. In this case \(X_0(N)\) is a curve of genus \(\ge 1\) and there exist a finite number of nonCM noncuspidal \(\mathbb {Q}\)rational points on the curve.
All noncuspidal rational points on \(X_0(p^n)\), genus \(>0\) case
\({\varvec{N}}\) , genus ( \({\varvec{X}}_{\varvec{0}}{\varvec{(N)}}\) )  \({\varvec{j}}\) invariants  Examples  Conductor  CM? 

11, \(g=1\)  \(j = \,11\cdot 131^3\)  121A1, 121C2  \(11^2\)  No 
\(j = \,2^{15}\)  121B1, 121B2  \(11^2\)  \(\,11\)  
\(j = \,11^2\)  121A2, 121C1  \(11^2\)  No  
17, \(g=1\)  \(j = \,17^2 \cdot 101^3/2\)  14450P1  \(2 \cdot 5^2 \cdot 17^2\)  No 
\(j = \,17 \cdot 373^3/2^{17}\)  14450P2  \(2 \cdot 5^2 \cdot 17^2 \)  No  
19, \(g=1\)  \(j = \,2^{15}\cdot 3^3\)  361A1, 361A2  \(19^2\)  \(19\) 
27, \(g=1\)  \(j = \,2^{15} \cdot 3 \cdot 5^3\)  27A2, 27A4  \(3^3\)  \(\,27\) 
37, \(g=2\)  \(j = \,7 \cdot 11^3\)  1225H1  \(5^2\cdot 7^2\)  No 
\(j = \,7 \cdot 137^3 \cdot 2083^3\)  1225H2  \(5^2\cdot 7^2\)  No  
43, \(g=3\)  \(j = \,2^{18} \cdot 3^3 \cdot 5^3\)  1849A1, 1849A2  \(43^2\)  \(\,43\) 
67, \(g=5\)  \(j = \,2^{15}\cdot 3^3\cdot 5^3\cdot 11^3\)  4489A1, 4489A2  \(67^2\)  \(\,67\) 
163, \(g=13\)  \(j = \,2^{18}\cdot 3^3\cdot 5^3\cdot 23^3\cdot 29^3\)  26569A1, 26569A2  \(163^2\)  \(\,163\) 
4 Borel subgroups
In order to prove Theorem 2.1 in the cases of potential multiplicative reduction, or potential good ordinary reduction, we shall need results about ramification indices when the image of \(\rho _{E,p}:{\text {Gal}}(\overline{K}/K)\rightarrow {\text {Aut}}(E[p^n])\) is a Borel subgroup of \({\text {GL}}(2,\mathbb {Z}/p^n\mathbb {Z})\), for some finite extension K of \(L_\wp ^{\text {nr}}\). In this section we study Borel subgroups in general.
Definition 4.1
Lemma 4.2
 (1)
B and \(B'\) are conjugates, more precisely \(B'=h^{1}Bh\) with \(h=\left( \begin{array}{c@{\quad }c} 1 &{} b/(ca) \\ 0 &{} 1\end{array}\right) \).
 (2)
\(B' = B_d'B_1'\), i.e., for every \(M\in B'\) there is \(U\in B_d'\) and \(V\in B_1'\) such that \(M=UV\).
 (3)
\([B,B]=B_1\) and \([B',B']=B_1'\). In particular, [B, B] and \([B',B']\) are cyclic groups whose order is \(p^s\), for some \(0\le s\le n\).
 (4)
\(B/[B,B]\cong B'/[B',B']\) is isomorphic to a subgroup of \((\mathbb {Z}/p^n\mathbb {Z})^\times \times (\mathbb {Z}/p^n\mathbb {Z})^\times \).
 (5)
If \(n=1\), then \(B\cong B'=B_d'B_1'\) for any Borel subgroup \(B\le {\text {GL}}(2,\mathbb {Z}/p\mathbb {Z})\).
Proof
Finally, if \(B\le {\text {GL}}(2,\mathbb {Z}_p)\) is a closed Borel subgroup, and there is a \(g=\left( \begin{array}{c@{\quad }c} a &{} b \\ 0 &{} c\end{array}\right) \in B\) with \(a\not \equiv c \bmod p\), we can set h as in (1), and \(B'=h^{1}Bh\). Then, \(B_n=B \bmod p^n\) and \(B'_n=B'\bmod p^n\) satisfy properties (1)–(4) as subgroups of \({\text {GL}}(2,\mathbb {Z}/p^n\mathbb {Z})\). Thus, \(B'\) requires all the required properties because B and therefore \(B'\) are closed subgroups of \({\text {GL}}(2,\mathbb {Z}_p)\).
\(\square \)
Remark 4.3
Lemma 4.4
 (1)If \(\lambda \not \equiv 0 \bmod p\), and \(\nu _p(\mu ) = t\) for some \(1\le t\le n\), then$$\begin{aligned} B_R = \left\{ \left( \begin{array}{c@{\quad }c} 1bp^t/\lambda &{} b \\ 0 &{} c\end{array}\right) : b\in \mathbb {Z}/p^n\mathbb {Z},\ c\equiv 1 \bmod p^{nt} \right\} \cap B, \end{aligned}$$
 (2)If \(\mu \not \equiv 0 \bmod p\), and \(\nu _p(\lambda ) = t \) for some \(0\le t\le n\), then$$\begin{aligned} B_R = \left\{ \left( \begin{array}{c@{\quad }c} a &{} (1a)p^t/\mu \\ 0 &{} 1\end{array}\right) : a\in (\mathbb {Z}/p^n\mathbb {Z})^\times \right\} \cap B. \end{aligned}$$
Proof
Notice that \(A\in {\text {GL}}(V)\) fixes R if and only if A fixes \(\delta R\), for all \(\delta \in (\mathbb {Z}/p^n\mathbb {Z})^\times \). Hence, \(B_R = B_{\delta R}\), for all \(\delta \in (\mathbb {Z}/p^n\mathbb {Z})^\times \).
Lemma 4.5
Proof
Finally, if J is even, then there exists \(m\in J\) such that \(m\equiv 1\bmod p^n\), and there is a bijection \(J_{1,b}\rightarrow J_{1,b}\) given by \(a\mapsto m\cdot a\). \(\square \)
Remark 4.6
Let \(p>2\) be a prime, let \(J\subseteq (\mathbb {Z}/p^n\mathbb {Z})^\times \) be a subgroup, and let \(\psi :J\rightarrow \{\pm 1\}\) be a quadratic character (note that we assume here that the word quadratic implies nontrivial). Then, \({\text {Ker}}(\psi )=J^2\) and \(\psi (a)=1\) if and only if \(a\in J\) is a quadratic nonresidue mod \(p^n\), if and only if a is a quadratic nonresidue mod p. In particular, if \(a\psi (a)\equiv 1 \bmod p\), then \(a\equiv \psi (a)\bmod p\) and so either \(a\equiv \psi (a)\equiv 1 \bmod p\) or \(a\equiv \psi (a)\equiv 1 \bmod p\), and therefore \(1\) is a quadratic nonresidue and \(p\equiv 3\bmod 4\). Thus, if \(p\equiv 1 \bmod 4\) and \(a\psi (a)\equiv 1 \bmod p\), then we must necessarily have \(a\equiv \psi (a)\equiv 1 \bmod p\).
Lemma 4.7
 (1)Suppose \(R=\lambda P +\mu Q\), with \(\lambda \not \equiv 0 \bmod p\) and \(\nu _p(\mu )=t\), for some \(1\le t\le n\).
 (a)If \(\psi \) is trivial, or \(1\le t \le n1\), or \(p\equiv 1 \bmod 4\), then(Note: in the second case, \(nm\nu _p(J)t>0\).)$$\begin{aligned} I/I_R ={\left\{ \begin{array}{ll} J &{}, \text { if } n\nu _p(J)\le t+m \le n,\\ Jp^{nm\nu _p(J)t} &{}, \text { if } 1\le t+m< n\nu _p(J).\end{array}\right. } \end{aligned}$$
 (b)
Otherwise, if \(\psi \) is nontrivial, \(t=n\) and \(p\equiv 3\bmod 4\), then \(I/I_R=J/2\).
 (a)
 (2)Suppose that \(R=\lambda P +\mu Q\), with \(\mu \not \equiv 0 \bmod p\), and \(\nu _p(\lambda )=t\):
 (a)
If \(t=n\le m\) or \(0\le m\le t\le n\), then \(I/I_R=f\cdot \max \{1,p^{nm}\}\). In particular, if \(n\le m\) and \(R\in \langle Q\rangle \), then \(I/I_R=f\).
 (b)If \(0\le t <\min \{m,n\}\), thenIn particular, \(I/I_R\) is divisible by \((J/p^{\nu _p(J)})p^{nm}\) if \(m<n\), and it is divisible by \(J/p^{\nu _p(J)}\) if \(n\le m\).$$\begin{aligned} I/I_R = {\left\{ \begin{array}{ll} \frac{J}{p^t} &{}, \text { if } n\nu _p(J)\le \min \{m,n\}t \le n,\\ \frac{J\cdot p^n}{p^{\min \{m,n\}+\nu _p(J)}} &{}, \text { if } 1\le \min \{m,n\}t< n\nu _p(J). \end{array}\right. } \end{aligned}$$
 (a)
Proof
 (1)Suppose first that \(\lambda \not \equiv 0 \bmod p\) and \(\mu \equiv 0\bmod p^t\), for some \(1\le t \le n\). By Lemma 4.4, the subgroup of B that fixes R isIf a matrix \(\left( \begin{array}{c@{\quad }c} 1bp^t/\lambda &{} b \\ 0 &{} c\end{array}\right) \) is in \(I_R\), then \(b\equiv 0\bmod p^m\), and then \(a=(1bp^t/\lambda )\in J\) is congruent to \(1\bmod p^{t+m}\). Let \(J_{1,b}\) be those elements of J that are congruent to \(1\bmod p^{b}\), so that \(a\in J_{1,t+m}\). If \(\psi \) is trivial, or \(1\le t\le n1\), or \(p\equiv 1 \bmod 4\) (see Remark 4.6), then \(I\cap B_R\) is given by:$$\begin{aligned} I_R = I\cap B_R=I\cap \left\{ \left( \begin{array}{c@{\quad }c} 1bp^t/\lambda &{} b \\ 0 &{} c\end{array}\right) : b\in \mathbb {Z}/p^n\mathbb {Z},\ c\equiv 1 \bmod p^{nt} \right\} . \end{aligned}$$Thus, Lemma 4.5 implies that$$\begin{aligned} I_R&= \left\{ \left( \begin{array}{c@{\quad }c} 1bp^t/\lambda &{} b \\ 0 &{} 1\end{array}\right) : b\equiv 0 \bmod p^m, 1bp^t/\lambda \in J_{1,t+m} \right\} \\&= \left\{ \left( \begin{array}{c@{\quad }c} 1\delta p^{t+m} &{} (\delta +\tau )p^m\lambda \\ 0 &{} 1\end{array}\right) : \tau \in (p^{nt}\mathbb {Z}/p^n\mathbb {Z}), 1\delta p^{t+m}\in J_{1,t+m} \right\} . \end{aligned}$$If we put \(N=N(m,n)=\max \{1,p^{nm}\}\), then$$\begin{aligned} I_R = J_{1,t+m}\cdot p^t = {\left\{ \begin{array}{ll} p^{n(t+m)}p^t =p^{nm} ,&{} \text { if } n\nu _p(J)\le t+m \le n,\\ p^{\nu _p(J)}p^t = p^{\nu _p(J)+t} , &{}\text { if } 1\le t+m< n\nu _p(J). \end{array}\right. } \end{aligned}$$Notice that in the second case the quantity \(nm\nu _p(J)t\) is greater than 0, and \(nm>t+\nu _p(J)\ge 1\), so \(N=p^{nm}\). Thus, in both cases, \(I/I_R\) is divisible by J$$\begin{aligned} I/I_R= & {} \frac{J\cdot N}{I_R}\\= & {} {\left\{ \begin{array}{ll} J\cdot N/N = J , &{} \text { if } n\nu _p(J)\le t+m \le n,\\ J\cdot N/p^{\nu _p(J)+t}=Jp^{nm\nu _p(J)t} , &{} \text { if } 1\le t+m< n\nu _p(J).\end{array}\right. } \end{aligned}$$Otherwise, if \(\psi \) is nontrivial, \(t=n\) and \(p\equiv 3 \bmod 4\) (again, Remark 4.6 plays a role here), then it is given by:Hence, \(I_R=I\cap B_R=2N\). It follows that \(I/I_R = J\cdot N/2N=J/2\). This shows part (1).$$\begin{aligned} I\cap B_R = \left\{ \left( \begin{array}{c@{\quad }c} 1 &{} b \\ 0 &{} \pm 1\end{array}\right) : b\in p^m\mathbb {Z}/p^n\mathbb {Z}\right\} . \end{aligned}$$
 (2)Now, suppose that \(\mu \not \equiv 0\bmod p\) and \(\lambda \equiv 0 \bmod p^t\), for some \(0\le t \le n\). By Lemma 4.4, the subgroup of I that fixes R isThus, \(I_R\) is given by$$\begin{aligned} I_R = I\cap \left\{ \left( \begin{array}{c@{\quad }c} a &{} (a1)p^t/\mu \\ 0 &{} 1\end{array}\right) : a\in (\mathbb {Z}/p^n\mathbb {Z})^\times \right\} . \end{aligned}$$Notice that if \(\psi \) is trivial, then \({\text {Ker}}(\psi )=\{a\in J : \psi (a)=1\}\) has size J. Otherwise, if \(\psi \) is quadratic, then \({\text {Ker}}(\psi )\) has size J / 2. Next, we distinguish two cases according to whether \(m\le t\).$$\begin{aligned} I_R = \left\{ \left( \begin{array}{c@{\quad }c} a &{} (a1)p^t/\mu \\ 0 &{} 1\end{array}\right) : a\in J,\ \psi (a)=1,\ (a1)p^t\equiv 0 \bmod p^{\min \{n,m\}} \right\} . \end{aligned}$$This shows part (2), and concludes the proof of the lemma.

If \(0\le m\le t\le n\) or \(t=n\le m\), then \(I_R = {\text {Ker}}(\psi )\) and \(I/I_R=N=\max \{1,p^{nm}\}\) or 2N depending on whether \(\psi \) is trivial or quadratic, respectively. In particular, if \(n\le m\) and \(R=\mu Q\) with \(\psi \) trivial, then \(I/I_R=1\).

If \(0\le t<\min \{m,n\}\), thenSince \(\min \{m,n\}>t\), any \(a\equiv 1 \bmod p^{\min \{m,n\}t}\) satisfies \(a\equiv 1 \bmod p\) and \(\psi (a)=1\) is automatic. Thus, we have \(I_R = J_{1,\min \{m,n\}t}\). It follows from Lemma 4.5 that$$\begin{aligned} I_R = \left\{ \left( \begin{array}{c@{\quad }c} a &{} (a1)p^t/\mu \\ 0 &{} 1\end{array}\right) : a\in J,\ \psi (a)=1,\ a\equiv 1 \bmod p^{\min \{m,n\}t} \right\} . \end{aligned}$$If \(n\nu _p(J)\le mt\) and \(m\le n\), then \(t\le \nu _p(J)+mn\) and \(I/I_R=J/p^t\) is divisible by \((J/p^{\nu _p(J)})p^{nm}\). If \(m\ge n\), then \(n\nu _p(J)\le nt\) implies \(t\le \nu _p(J)\) and \(I/I_R\) is divisible by \(J/\nu _p(J)\).$$\begin{aligned} I/I_R = {\left\{ \begin{array}{ll} \frac{J\cdot p^{n\min \{m,n\}}}{p^{n(\min \{m,n\}t)}}=\frac{J}{p^t} , &{} \text { if } n\nu _p(J)\le \min \{m,n\}t \le n,\\ \frac{J\cdot p^{n\min \{m,n\}}}{p^{\nu _p(J)}} = \frac{J\cdot p^n}{p^{\min \{m,n\}+\nu _p(J)}} , &{} \text { if } 1\le \min \{m,n\}t< n\nu _p(J). \end{array}\right. } \end{aligned}$$

Remark 4.8
Let L be a number field with ring of integers \(\mathcal {O}_L\), and let \(\wp \) be a prime ideal of \(\mathcal {O}_L\) lying above a rational prime p. Let E / L be an elliptic curve, and let \(R\in E(\overline{L})[p^n]\) be a point of exact order \(p^n\). Let \(\iota :\overline{L}\hookrightarrow \overline{L}_\wp \) be a fixed embedding. Let \(F=L(R)\) and let \(\Omega _R\) be the prime of F above \(\wp \) associated to the embedding \(\iota \). Let K be a finite Galois extension of \(L_\wp ^{\text {nr}}\), such that the ramification index of K over \(\mathbb {Q}_p\) is e. Let \(\tilde{E}/K\) be a curve isomorphic to E over K, and let \(T\in \tilde{E}(K)[p^n]\) be the point that corresponds to \(\iota (R)\) on \(E(\overline{L}_\wp )\). Suppose that the degree of the extension K(T) / K is g. Since \(K/L_\wp ^{\text {nr}}\) is of degree \(e/e(\wp p)\), it follows that the degree of \(K(T)/L_\wp ^{\text {nr}}\) is \(eg/e(\wp p)\).
Let \(\mathcal {F} = \iota (F)\subseteq \overline{L}_\wp \). Since E and \(\tilde{E}\) are isomorphic over K, it follows that \(K(T)=K\mathcal {F}\) and, therefore, the degree of the extension \(K\mathcal {F}/L_\wp ^{\text {nr}}\) is \(eg/e(\wp p)\). Since \(K/L_\wp ^{\text {nr}}\) is Galois by assumption, it follows that \(g=[K(T):K]=[\mathcal {F}L_\wp ^{\text {nr}}:K\cap \mathcal {F}L_\wp ^{\text {nr}}]\), so the degree of \([\mathcal {F}L_\wp ^{\text {nr}}:L_\wp ^{\text {nr}}]\) equals \(g\cdot k\) where \(k= [K\cap \mathcal {F}L_\wp ^{\text {nr}}:L_\wp ^{\text {nr}}]\). Hence, the degree of \(\mathcal {F}/L_\wp \) is divisible by gk and, in particular, the ramification index of the prime ideal \(\Omega _R\) over \(\wp \) in the extension L(R) / L is divisible by gk, where \(g=[K(T):K]\).
Moreover, let \(\Omega \) be the prime of \(L(E[p^n])\), lying above \(\wp \), associated to the embedding \(\iota \), and let \(G={\text {Gal}}(L(E[p^n])/L)\). Let \(I_\Omega \subset D_\Omega \subset G\) be the inertia and decomposition groups associated to \(\Omega \). It follows that \(I_\Omega \cong {\text {Gal}}(L_\wp ^{\text {nr}}(\iota (E[p^n]))/L_\wp ^{\text {nr}})\). Let K and \(\tilde{E}\) be as before. Then, by the same argument as above, we have that \(I_\Omega \) has a subgroup \(I_{K,\Omega }\) isomorphic to \({\text {Gal}}(K(\tilde{E}[p^n])/K)\) such that \(\sigma \in I_{K,\Omega }\) acts on \(R\in E[p^n]\) just like \(\sigma \in {\text {Gal}}(K(\tilde{E}[p^n])/K)\) acts on \(\iota (R)\in \tilde{E}[p^n]\).
Theorem 4.9
 (i)
There exists a point \(R\in E[p^n]\) of exact order \(p^n\) with \(e(\Omega _R\wp )\) divisible by \(f(\psi )\max \{1,p^{nm}\}\) (and equality if \(I=I_\Omega \)). If \(m\ge n\), \(I=I_\Omega \), L(R) / L is Galois and \(\psi \) is unramified (over \(\wp \)), then the extension is unramified at \(\wp \). Otherwise, there is another prime \(\Omega '\) of \(L(E[p^n])\) over \(\wp \) such that \(e(\Omega _R'\wp )\) is either \(\chi _n/\delta (\psi )\) or divisible by \(\chi _n/p^{\nu _p(\chi _n)}\).
 (ii)
If \(0\le m<n\), then the ramification index of any prime ideal \(\Omega _R\) over \(\wp \) in the extension L(R) / L is divisible by \(\chi _n/\delta (\psi )\) or \(f(\psi )\cdot p^{nm}\), or \((\chi _n/p^{\nu _p(\chi _n)})p^{nm}\), for any point \(R\in E\) of exact order \(p^n\).
 (iii)
If R is a point of exact order \(p^n\), the subgroup \(\langle [p^{na}]R \rangle \subset E[p^a]\) is not \({\text {Gal}}(\overline{L}/L)\)stable for some \(a\ge 1\), then there is a prime \(\Omega _R''\) of L(R) over \(\wp \) such that \(e(\Omega _R''\wp )\) is divisible by \(\chi _n/\delta (\psi )\), or \(f(\psi )p^{na+1}\), or \(\chi _n/p^{\min \{a1,\nu _p(\chi _n)\}}\).
 (iv)
If R is a point of exact order \(p^n\), and the subgroup \(\langle [p^{na}]Q \rangle \subset E[p^a]\) is not \({\text {Gal}}(\overline{L}/L)\)stable for some \(a\ge 1\), then the same conclusion as in part (iii) holds for L(R).
 (v)
If \(\langle [p^{na}]Q \rangle \subset E[p^a]\) is \({\text {Gal}}(\overline{L}/L)\)stable for some \(1\le a\le n\), then \(m\ge a\). Equivalently, if \(m\le a1\) for some \(1\le a \le n\), then \(\langle [p^{na}]Q \rangle \) is not \({\text {Gal}}(\overline{L}/L)\)stable.
Proof
First, by Lemma 4.7 part (2a), if \(R\in \langle Q \rangle \), then the ramification index \(e(\Omega _R\wp )\) of L(R) / L is divisible by \(f(\psi )\max \{1,p^{nm}\}\). If L(R) / L is Galois, \(m\ge n\), \(I=I_\Omega \), and \(\psi \) is unramified over \(\wp \), then all the primes above \(\wp \) in L(R) / L would be unramified. On the other hand, suppose that \(m\ge n\), \(I=I_\Omega \), and the prime \(\Omega _R\) of L(R) / L is unramified (or ramification index of 2 for some \(R\in \langle Q\rangle \) of exact order \(p^n\), if \(\psi \) is ramified) but the extension is not Galois. This implies that \(R\in \langle Q \rangle \), and there is a \(\sigma \in {\text {Gal}}(L(E[p^n])/L)\) such that \(L(\sigma (R))\not \subset L(R)\). Thus, \(\sigma (R)\not \in \langle R\rangle = \langle Q \rangle \). When \(m\ge n\), our Lemma 4.7 implies that \(e(\Omega _{\sigma (R)}\wp )\) is either \(\chi _n/\delta (\psi )\), or divisible by \(\chi _n/p^{\nu _p(\chi _n)}\), in all cases. Hence, the ramification index of \(\Omega _R'=\sigma ^{1}(\Omega _{\sigma (R)})\) in L(R) / L is either \(\chi _n/\delta (\psi )\), or divisible by \(\chi _n/p^{\nu _p(\chi _n)}\), by Remark 4.8. This completes the proof of (i).
Part (ii) follows directly from Lemma 4.7.

(Case (1)): \(\chi _n/\delta (\psi )\), if \(\nu _p(\lambda )=0\) and \(1\le \nu _p(\mu )\le n\).

(Case (2a)): \(f(\psi )p^{nm}\ge f(\psi )p^{na+1}\), if \(\nu _p(\mu )=0\) and \(0\le m\le \nu _p(\lambda )< a \le n\) (notice that in the case (2.a) we cannot have \(\nu _p(\lambda )=n\le m\) because \(\nu _p(\lambda )<a\le n\)).

(Case (2b.i)): \(\chi _n/p^{\nu _p(\lambda )}\), if \(\nu _p(\mu )=0\), and \(0\le \nu _p(\lambda )<\min \{m,n\}\), and \(n\nu _p(\chi _n)\le \min \{m,n\}\nu _p(\lambda )\le n\). In this case, \(\nu _p(\chi _n)\nu _p(\lambda )\ge n\min \{m,n\}\ge 0\), so \(\chi _n/p^{\nu _p(\lambda )}\in \mathbb {Z}\). Since \(\nu _p(\lambda )<a\), then \(e(\Omega _{\tau (R)}\wp )\) is divisible by \(\chi _n/p^{\min \{a1,\nu _p(\chi _n)\}}\).

(Case (2b.ii)): \((\chi _n/p^{\nu _p(\chi _n)})p^{n\min \{m,n\}}\), if \(\nu _p(\mu )=0\), and \(0\le \nu _p(\lambda )<\min \{m,n\}\), and \(n\nu _p(\chi _n) > \min \{m,n\}\nu _p(\lambda )\ge 1\). Notice that in this case, \(n\min \{m,n\} > \nu _p(\chi _n)\nu _p(\lambda )\), and also \(n\min \{m,n\}\ge 0\). Thus,Thus, \((\chi _n/p^{\nu _p(\chi _n)})p^{n\min \{m,n\}}\) is divisible by \(\chi _n/p^{\min \{a1,\nu _p(\chi _n)\}}.\)$$\begin{aligned} n\min \{m,n\}&\ge \max \{0,\nu _p(\chi _n)\nu _p(\lambda )\}\\&\ge \max \{0, \nu _p(\chi _n)(a1)\}\\&= \nu _p(\chi _n)\min \{a1,\nu _p(\chi _n)\}. \end{aligned}$$
Finally, for (iv), suppose that \(\langle [p^{na}]Q \rangle \) is not Galoisstable, for some \(1\le a \le n\). Let \(P_a=[p^{na}]P\) and \(Q_a=[p^{na}]Q\), so that \(\{P_a,Q_a\}\) is a basis of \(E[p^a]\). Since \(\langle [p^{na}]Q \rangle \) is not Galoisstable, it follows that there is \(\tau \in {\text {Gal}}(L(E[p^a])/L)\) such that \(\tau ([p^{na}]R)\not \in \langle [p^{na}]Q\rangle =\langle Q_a\rangle \), i.e., \([p^{na}](\tau (R))\not \in \langle Q_a\rangle \). The rest of the argument can now proceed as in (iii).
Part (v) is clear, as \(\langle [p^{ns}]Q \rangle \) is not stable under \(I\subseteq I_\Omega \) as long as \(s>m\). This concludes the proof of the theorem. \(\square \)
Remark 4.10
Corollary 4.11
 (1)
Suppose that either E / L does not admit a Lrational isogeny of degree \(p^{a}\), or \(m\le a1\). If \(n\ge a\), then there is a prime \(\Omega _R\) of L(R) over \(\wp \) such that \(e(\Omega _R\wp )\) is divisible by \(\chi _n/\delta (\psi )\), or \(f(\psi )p^{na+1}\), or \(\chi _n/p^{\min \{a1,\nu _p(\chi _n)\}}\). In particular, \(e(\Omega _R\wp )\) is divisible by \(\varphi (p^n)/\gcd (\varphi (p^n),\delta (\psi )ep^{a1})\) or \(f(\psi )p^{na+1}\).
 (2)Suppose that \(m\ge 1\), and if E / L admits a Lrational isogeny \(\phi \) of degree p, such that \(\ker (\phi )=\langle S\rangle \subset E[p]\), then the ramification index of K(S) / K is \(>f(\psi )\), or the ramification index of \(\wp \) in the Galois extension L(S) / L satisfiesThen, there is a prime \(\Omega _R\) of L(R) over \(\wp \) such that \(e(\Omega _R\wp )\) is divisible by \(\chi _n/\delta (\psi )\), or \(f(\psi )p^{n}\), or \(\chi _n\). In particular, \(e(\Omega _R\wp )\) is divisible by \(\varphi (p^n)/\gcd (\varphi (p^n),\delta (\psi )e)\) or \(f(\psi )p^{n}\).$$\begin{aligned} e(\wp ,L(S)/L)/\gcd (e(\wp ,L(S)/L),e(K/L_\wp ^{\text {nr}}))>f(\psi ). \end{aligned}$$
 (3)
The size of \(I\subseteq I_\Omega \subseteq {\text {Gal}}(L(E[p^n])/L)\) is exactly \(\chi _n\), for \(n\le m\), and \(\chi _np^{nm}\), for all \(n> m\). In particular, \(e(\Omega \wp )\) is divisible by \(\varphi (p^n)/\gcd (\varphi (p^n),e)\) for \(n\le m\), and by \(\varphi (p^n)p^{nm}/\gcd (\varphi (p^n),e)\) if \(n> m\).
 (4)
Let \(n\ge 2\), let \(\{P,Q\}\) be the basis defined in Theorem 4.9, let \(P_1=[p^{n1}]P\), and let \(H_n=\langle R,E[p^{n1}]\rangle \subset E[p^n]\), where \([p^{n1}]R=P_1\). Then, the ramification index in the extension \(K(H_n)/K\) is \(\chi _n\) if \(n\le m\), and \(\chi _np^{nm1}\) if \(n>m\). In particular, the ramification index of a prime above \(\wp \) in \(L(H_n)/L\) is divisible by \(\varphi (p^n)/\gcd (\varphi (p^n),e)\) for \(n\le m\), and by \(\varphi (p^n)p^{nm1}/\gcd (\varphi (p^n),e)\) if \(n> m\).
 (5)
Let \(n\ge 2\) and \(p>2\), assume that \(\nu _p(\chi _n)\ge 1\), let \(\{P,Q\}\) be as in (3), let \(Q_1=[p^{n1}]Q\), and let \(H_n=\langle R,E[p^{n1}]\rangle \subset E[p^n]\), where \([p^{n1}]R=Q_1\). Then, the ramification index in the extension \(K(H_n)/K\) is \(\chi _{n}/p\) if \(n\le m\) and \(\chi _np^{nm1}\) if \(n>m\). In particular, the ramification index of a prime above \(\wp \) in \(L(H_n)/L\) is divisible by \(\varphi (p^{n1})/\gcd (\varphi (p^n),e)\) for \(n\le m\), and by \(\varphi (p^n)p^{nm1}/\gcd (\varphi (p^n),e)\) if \(n> m\).
Proof
We show (1) first. If E / L does not admit a Lrational isogeny of degree \(p^{a}\), then \(\langle S \rangle \subset E[p^{a}]\) is not \({\text {Gal}}(\overline{L}/L)\)stable for any \(S\in E[p^{a}]\). In particular, \(\langle [p^{na}]R \rangle \) is not \({\text {Gal}}(\overline{L}/L)\)stable. Now we can apply Theorem 4.9, part (iii). If \(m\le a1\), then \(\langle [p^{na}]Q\rangle \) is not \({\text {Gal}}(\overline{L}/L)\)stable (by Thm. 4.9, part (v)). Now we can apply Theorem 4.9, part (iv). The last piece of (1) follows from Remark 4.10, because \(\chi _n\) is divisible by \(\varphi (p^n)/\gcd (\varphi (p^n),e)\).
It is clear from our assumptions on the shape of I that \(I=\left\{ \left( \begin{array}{c@{\quad }c} \chi _n\psi &{} 0 \\ 0 &{} \psi ^{1}\end{array}\right) \right\} \) for \(n\le m\), and \(I=\left\{ \left( \begin{array}{c@{\quad }c} \chi _n\psi &{} b \\ 0 &{} \psi ^{1}\end{array}\right) : b\equiv 0 \bmod p^m\right\} \) for \(n\ge m\). Thus, \(I=\chi _n\cdot \max \{1,p^{nm}\}\) for all \(n\ge 1\), which shows (3).
Lemma 4.12
 (1)
E and \(E'\) are isomorphic over F or \(E'\) is a quadratic twist of E.
 (2)
For all \(R\in E(\overline{F})\), we have \(F(x(R))=F(x(\phi (R)))\).
 (3)
Moreover, if F(R) / F is Galois, cyclic, and [F(x(R)) : F] is even, then the quotient \([F(\phi (R)):F]/[F(R):F]=1\) or 2.
Proof
Let E and \(E'\), respectively, be given by Weierstrass equations \(y^2=x^3+Ax+B\) and \(y^2=x^3+A'x+B'\), with coefficients in F. Since \(j(E)=j(E')\ne 0,1728\), none of the coefficients is zero. By [43, Ch. III, Prop. 3.1(b)], the isomorphism \(\phi :E \rightarrow E'\) is given by \((x,y)\mapsto (u^2x,u^3y)\) for some \(u\in \overline{F}\setminus \{0\}\). Hence \(A'=u^4A\) and \(B'=u^6B\), and so \(u^2\in F\). Thus, either \(E\cong _F E'\), or \(E'\) is the quadratic twist of E by u. This shows (1).
Let \(R\in E(\overline{F})\). If \(E\cong _\mathbb {Q}E'\) then \(F(R)=F(\phi (R))\) and the same holds for the subfields of the xcoordinates, so (2) and (3) are immediate. Let us assume for the rest of the proof that \(E'\) is the quadratic twist of E by \(\sqrt{d}\), for some \(d\in F\setminus F^2\). It follows that \(\phi ((x,y))=(dx,d\sqrt{d}\cdot y)\) and, therefore, \(F(x(\phi (R)))=F(d\cdot x(R))=F(x(R))\). This proves (2).

If \(F(x)=F(x,y)=F(R)\), then \(y\in F(x)\) and \(F(x,\sqrt{d}\cdot y)=F(x,\sqrt{d})\). Thus, we have \([F(\phi (R)):F]=[F(x,\sqrt{d}):F(x)]\cdot [F(x):F]\) and hence \([F(\phi (R)):F]/[F(R):F]=1\) or 2.

Suppose F(x, y) / F(x) is quadratic. If \(F(x,\sqrt{d}\cdot y)/F(x)\) is also quadratic, then we have \([F(\phi (R)):F]/[F(R):F]=1\). Otherwise, assume that \(F(x,\sqrt{d}\cdot y)=F(x)\) and we will reach a contradiction. Indeed, in this case \(\sqrt{d}\cdot y \in F(x)\). Hence, there is \(z\in F(x)\) such that \(y=\sqrt{d}\cdot z\) and we may conclude that \(F(x,y)=F(x,\sqrt{d})\). It follows that \(\sqrt{d}\in F(R)\). Let \(K=F(\sqrt{d})\subseteq F(R)\). Since F(R) / F is Galois and cyclic, K is the unique quadratic extension of F contained in F(R). Moreover, F(x) / F is of even degree by assumption, and Galois, cyclic because \(F(x)\subseteq F(R)\). Thus, \(K=F(\sqrt{d})\subseteq F(x)\). It would follow that \(F(x,y)=F(x,\sqrt{d})=F(x)\) which is a contradiction, since we have assumed that F(R) / F(x) is quadratic.
5 Potential multiplicative reduction
Let L be a number field, and let E / L be an elliptic curve. We say that E / L has potential multiplicative reduction at a prime ideal \(\wp \) of \(\mathcal {O}_L\) if there is an extension of number fields F / L and a prime \(\mathfrak {P}\) of \(\mathcal {O}_F\) lying above \(\wp \) such that E / F has multiplicative reduction at \(\mathfrak {P}\). The curve E / F has bad multiplicative reduction at \(\mathfrak {P}\) if and only if \(\nu _\mathfrak {P}(c_4)=0\) and \(\nu _{\mathfrak {P}}(\Delta )>0\) (for a minimal model at \(\mathfrak {P}\)), if and only if \(\nu _\mathfrak {P}(j)<0\) (because \(j=c_4^3/\Delta \); see [43, Proposition 5.1, Ch. VII.5]). If E is defined over \(L\subseteq F\), then \(\nu _\mathfrak {P}(j)<0\) if and only if \(\nu _\wp (j)<0\). Thus, E / L has potential multiplicative reduction at \(\wp \) if and only if \(\nu _\wp (j)<0\).
Theorem 5.1
 (a)
There is a \(q\in L_\wp ^*\) such that \(\nu _\wp (q)>0\), and \(E/L_\wp \) is a twist of \(E_q/L_\wp \) by a trivial or quadratic character \(\psi _\wp :{\text {Gal}}(\overline{L}_\wp /L_\wp )\rightarrow \{\pm 1\}\).
 (b)Let \(n\ge 1\) be fixed, let \(\Omega \) be a prime of \(L(E[p^n])\) lying above \(\wp \), associated to the embedding \(\iota \), and let \(I_{\Omega }\) be the associated inertia subgroup in \({\text {Gal}}(L(E[p^n])/L)\). Then, there is a \(\mathbb {Z}/p^n\mathbb {Z}\)basis \(\{P,Q\}\) of \(E[p^n]\) such that the inertia subgroup \(I_\Omega \) is of the formfor some \(m \ge 0\), where \(\chi _n:I_\Omega \rightarrow (\mathbb {Z}/p^n\mathbb {Z})^\times \) is the \(p^n\)th cyclotomic character, and \(\psi :I_\Omega \rightarrow \{\pm 1 \}\) is induced by the character \(\psi _\wp \) of part (1).$$\begin{aligned} I_\Omega =\left\{ \left( \begin{array}{c@{\quad }c} \chi _n\psi &{} *\\ 0 &{} \psi ^{1}\end{array}\right) : *\equiv 0\bmod p^m \right\} , \end{aligned}$$
 (c)
The reduction of E / L at \(\wp \) is bad multiplicative if and only if the character \(\psi _\wp \) is unramified at \(\wp \) (i.e., \(\psi \) is trivial on \(I_\Omega \)), if and only if \(f(\psi )=1\) (in the notation of Theorem 4.9).
 (d)
The number m that appears in part (b) satisfies \(m\le \nu _p(\nu _\wp (j)).\)
 (e)Let \(\chi _n\), \(\delta (\psi )\), and \(f(\psi )\) be as in Theorem 4.9. Suppose that there is a number \(a\ge 1\) such thatThen, the conclusions of Cor. 4.11 hold, for every \(R\in E[p^n]\) of exact order \(p^n\), with \(n\ge a\). Then there is a prime \(\Omega _R\) of L(R) over \(\wp \) such that \(e(\Omega _R\wp )\) is divisible by \(\chi _n/\delta (\psi )\), or \(f(\psi )p^{na+1}\), or \(\chi _n/p^{\min \{a1,\nu _p(\chi _n)\}}\). In particular, \(e(\Omega _R\wp )\) is divisible by

E / L does not admit a Lrational isogeny of degree \(p^{a}\), or

\(m\le \nu _p(\nu _\wp (j))\le a1\), or

Suppose that \(m\ge 1\) (where m is as in (b)), and if E / L admits a Lrational isogeny \(\phi \) of degree p, such that \(\ker (\phi )=\langle S\rangle \subset E[p]\), then the ramification index of \(\wp \) in the Galois extension L(S) / L satisfies \(e(\wp ,L(S)/L)>f(\psi )\). If so, here set \(a=1\).
$$\begin{aligned} \varphi (p^n)/\gcd (\varphi (p^n),\delta (\psi )e(\wp p)p^{a1}) \text { or } f(\psi )p^{na+1}. \end{aligned}$$ 
 (f)Suppose that \(m\ge 1\) (where m is as in (b)), and if E / L admits a Lrational isogeny \(\phi \) of degree p, such that \(\ker (\phi )=\langle S\rangle \subset E[p]\), then the ramification index of K(S) / K is \(>1\), or the ramification index of \(\wp \) in the Galois extension L(S) / L satisfieswhere \(K/L_\wp ^{\text {nr}}\) is the smallest extension such that E / K has multiplicative reduction. Then, there is a prime \(\Omega _R\) of L(R) over \(\wp \) such that \(e(\Omega _R\wp )\) is divisible by$$\begin{aligned} e(\wp ,L(S)/L)/\gcd (e(\wp ,L(S)/L),e(K/L_\wp ^{\text {nr}}))>1, \end{aligned}$$where \([K:L_\wp ^{\text {nr}}]=1\) or 2.$$\begin{aligned} \varphi (p^n)/\gcd (\varphi (p^n),[K:L_\wp ^{\text {nr}}]e(\wp p)) \text { or } p^{n}, \end{aligned}$$
Proof
Let L, \(\wp \), E / L, and \(\iota \) be as in the statement of the theorem and let \(j_0=j(E)\). By assumption, \(\nu _\wp (j_0)<0\). By the theory of Tate curves (see our remarks at the beginning of this section), there is a \(q\in L_\wp ^*\) such that \(\nu _\wp (q)>0\) and \(j(E_q)=j_0\). By Lemma 4.12, part (1), the curves \(E/L_\wp \) and \(E_q/L_\wp \) are either isomorphic over \(L_\wp \), or they are a quadratic twist of each other. This proves (a).
Part (c) follows from the fact that E and \(E_q\) are isomorphic over \(L_\wp \) if and only if E / L has split multiplicative reduction at \(\wp \) ([44, Ch. V, Theorem 5.3(b)]). If E / L has nonsplit multiplicative reduction, then E and \(E_q\) are isomorphic over a quadratic unramified extension of \(L_\wp \), and so in this case \(\psi _\wp \) is nontrivial, but its restriction to inertia is trivial (i.e., \(\psi \) is unramified). Finally, if E / L has additive reduction (potential multiplicative), then E and \(E_q\) are isomorphic over a quadratic ramified extension of \(L_\wp \), and in this case \(\psi \) is quadratic, nontrivial, and ramified.
We have seen that m is the largest nonnegative integer such that \(q\in (L_\wp ^*)^{p^m}\). Since \(q\in L_\wp ^*\) and \(\nu _\wp (q)>0\), it follows that \(\nu _\wp (q)\) is a positive multiple of \(p^m\). Hence, \(\nu _\wp (j)=\nu _\wp (q)\) is a multiple of \(p^m\) or, in other words, \(\nu _p(\nu _\wp (j))\ge m\). This shows (d).
Suppose that one of the three conditions listed in (e) is satisfied for \(a\ge 1\). In order to apply Corollary 4.11, we let \(K=L_\wp ^{\text {nr}}\) and the needed hypothesis is that \(e(L_\wp ^{\text {nr}}(S)/L_\wp ^{\text {nr}})=e(\wp ,L(S)/L)>f(\psi )\). Thus, the results cited in (e) follow from Cor. 4.11.
6 Potential good reduction
Let L be a number field with ring of integers \(\mathcal {O}_{L}\), let \(p\ge 2\) be a prime, let \(\wp \) be a prime ideal of \(\mathcal {O}_{L}\) lying above p, and let \(L_\wp \) be the completion of L at \(\wp \). Let E be an elliptic curve defined over L with potential good (ordinary or supersingular) reduction at \(\wp \). Let us fix an embedding \(\iota :\overline{L}\hookrightarrow \overline{L}_\wp \). Via \(\iota \), we may regard E as defined over \(L_\wp \). Let \(L_\wp ^{\text {nr}}\) be the maximal unramified extension of \(L_\wp \).
 (1)
\(E/K_E\) has good (ordinary or supersingular) reduction.
 (2)
\(K_E\) is the smallest extension of \(L_\wp ^{\text {nr}}\) such that \(E/K_E\) has good reduction, i.e., if \(K'/L_\wp ^{\text {nr}}\) is another extension such that \(E/K'\) has good reduction, then \(K_E\subseteq K'\).
 (3)\(K_E/L_\wp ^{\text {nr}}\) is finite and Galois. Moreover (see [39], §5.6, p. 312 when \(L=\mathbb {Q}\), but the same reasoning holds over number fields, as the work of Néron is valid for any local field, [35] pp. 124–125):

If \(p>3\), then \(K_E/L_\wp ^{\text {nr}}\) is cyclic of degree 1, 2, 3, 4, or 6.

If \(p=3\), the degree of \(K_E/L_\wp ^{\text {nr}}\) is a divisor of 12.

If \(p=2\), the degree of \(K_E/L_\wp ^{\text {nr}}\) is 2, 3, 4, 6, 8, or 24.

Theorem 6.1

\(e/e(\wp p)=2\) if and only if \(\nu _\wp (\Delta _L)\equiv 6 \bmod 12\),

\(e/e(\wp p)=3\) if and only if \(\nu _\wp (\Delta _L)\equiv 4\) or \(8 \bmod 12\),

\(e/e(\wp p)=4\) if and only if \(\nu _\wp (\Delta _L)\equiv 3\) or \(9 \bmod 12\),

\(e/e(\wp p)=6\) if and only if \(\nu _\wp (\Delta _L)\equiv 2\) or \(10 \bmod 12\).
6.1 Good ordinary reduction
Lemma 6.2
Proof
By Lemma 6.2, the inertia subgroup \(I_K\) is a Borel (with trivial character \(\psi \) as in the notation of Lemma 4.7, so \(\delta (\psi )=f(\psi )=1\)), and we can use the machinery of Sect. 4. In particular, Theorem 4.9 and Corollary 4.11, together with our previous remarks in this section, imply the following result.
Theorem 6.3
 (1)
E / L does not admit a Lrational isogeny of degree \(p^{a}\), or
 (2)
\(m\le a1\), or
 (3)If E / L admits a Lrational isogeny \(\phi \) of degree p, with \(\ker (\phi )=\langle S\rangle \subset E[p]\), then the ramification index of K(S) / K is \(>1\), or the ramification index of \(\wp \) in the Galois extension L(S) / L satisfiesIn this case, the conclusions below work with \(a=1\).$$\begin{aligned} e(\wp ,L(S)/L)/\gcd (e(\wp ,L(S)/L),e(K/L_\wp ^{\text {nr}}))>1. \end{aligned}$$
We apply Theorem 6.3 to study elliptic curves over \(\mathbb {Q}\) with potential good ordinary reduction.
Proposition 6.4
 (1)
E (resp. \(E'\)) is a quadratic twist of \(E_1/\mathbb {Q}\) (resp. \(E_1'/\mathbb {Q}\)), the curve with Cremona label “1225h1” (resp. “1225h2”) and good ordinary reduction at \(p=37\).
 (2)
E and \(E'\) admit a \(\mathbb {Q}\)rational isogeny of degree 37, but do not admit one of degree \(37^2\).
 (3)
There is a point \(R\in E\) of exact order 37 such that the ramification index of the primes above 37 in \(\mathbb {Q}(R)/\mathbb {Q}\) is \(f(\psi )\), where E (resp. \(E'\)) is a quadratic twist of \(E_1/\mathbb {Q}\) (resp. \(E_1'/\mathbb {Q}\)) by the character \(\psi \).
 (4)
Let \(R\in E\) be a point of exact order \(37^n\), for \(n\ge 2\). Then, there is a prime \(\Omega _R\) of \(\mathbb {Q}(R)\) over (37) such that \(e(\Omega _R37)\) is divisible by \(\varphi (37^n)/37=\varphi (37^{n1})\), or \(f\cdot 37^{n1}\).
 (5)
Let \(R\in E'\) be a point of exact order \(37^n\), for \(n\ge 1\). Then, there is a prime \(\Omega _R\) of \(\mathbb {Q}(R)\) over (37) such that \(e(\Omega _R37)\) is divisible by \(\varphi (37^n)\), or \(f'\cdot 37^{n}\).
Proof
Let \(E_1/\mathbb {Q}\) and \(E_1'/\mathbb {Q}\) be the elliptic curves with Cremona labels “1225h1” and “1225h2”, respectively. Then, \(j(E_1)=7\cdot 11^3\) and \(j(E_1')=7\cdot 137^3\cdot 2083^3\). By Lemma 4.12, the curves E and \(E'\) are, respectively, quadratic twists of \(E_1\) and \(E_1'\) associated to some characters \(\psi _1\) and \(\psi _2\). Notice that \(f=f(\psi )\) and \(f'=f(\psi ')\), where \(f(\psi )\) is defined in Theorem 4.9, i.e., \(f(\psi )=1\) if \(\psi \) is unramified above 37, and \(=2\) otherwise. Note that \(\delta (\psi )=\delta (\psi ')=1\) because \(p=37\equiv 1 \bmod 4\). The fact that the elliptic curves with \(j=7\cdot 11^3\) and \(j=7\cdot 137^3\cdot 2083^3\) have a \(\mathbb {Q}\)rational isogeny of degree 37 was discussed in Sect. 3. The classification of rational isogenies also implies that no elliptic curve over \(\mathbb {Q}\) admits an isogeny of degree \(37^2\).
Remark 6.5
Suppose E / L has potential good ordinary reduction, but E is not a CM curve. There is a criterion of Gross to find m such that \(I_{K,m}\) is diagonalizable, but \(I_{K,m+1}\) is not. We give a version here for curves over \(\mathbb {Q}\).
Theorem 6.6
(Gross; see [11], p. 514; see also §14–15) Let p be a prime, and let \(E/\mathbb {Q}\) be an elliptic curve with ordinary good reduction at p, with \(j\ne 0, 1728\), and assume that E[p] is an irreducible \({\text {Gal}}(\overline{\mathbb {Q}}/\mathbb {Q})\)module. Let \(D_n\le {\text {Gal}}(\mathbb {Q}(E[p^n])/\mathbb {Q}) \le {\text {GL}}(2,\mathbb {Z}/p^n\mathbb {Z})\) be a decomposition group at p. Let \(j_E=j(E)\) be the jinvariant of E and let \(j_0\) be the jinvariant of the “canonical lifting” of the reduction of j(E) modulo p, i.e., \(j_0\) is the jinvariant of the unique elliptic curve \(E_0/\mathbb {Q}_p\) which satisfies \(E_0\equiv E \bmod p\) and \({\text {End}}_{\mathbb {Q}_p}(E_0)\equiv {\text {End}}_{\mathbb {F}_p}(E)\). Then, \(D_{n}\) is diagonalizable if and only if \(j_E\equiv j_0 \bmod p^{n+1}\) if p is odd, and \(j_E\equiv j_0\bmod 2^{n+2}\) if \(p=2\).
6.2 Good supersingular reduction
The bounds on the ramification indices of extensions generated by torsion points, in the case of potential supersingular reduction were studied separately by the author in the articles [26] and [28] (see §1 of [26], or §2 of [28] for the definition of \(e_1\)). Here we simply quote the two theorems that are needed to show Theorems 2.1 and 2.2.
Theorem 6.7
 (1)
There is a constant \(f(\eta )\), which depends only on \(\eta \), such that \(cf(\eta )\). Moreover \(f(\eta )\) is a divisor of \(F(\eta )={\text {lcm}}(\{n: 1\le n < 24\eta ,\ \gcd (n,6)\ne 1\}).\) If \(p>3\), then \(f(\eta )\) is a divisor of \(F_0(\eta )={\text {lcm}}(\{n: 1\le n < 6\eta ,\ \gcd (n,6)\ne 1\}).\)
 (2)
Let \(\sigma \) be the smallest nonnegative integer such that \(8\eta \le 2^\sigma \) (or such that \(\eta \le 5^\sigma \), if \(p>3\)). If \(n>\sigma +1\), then \(e(\mathfrak {P}\wp )\) is divisible by \((p1)p^{2(n1)\sigma }/\gcd ((p1)p^{2(n1)\sigma },c)\).
 (3)
If \(p>3\eta \), then \(e(\mathfrak {P}\wp )\) is divisible by \((p1)p^{n1}/\gcd (p1,c)\).
 (4)
If \(\eta =1\) and \(p>3\), then \(e(\mathfrak {P}\wp )\) is divisible by \((p^21)p^{2(n1)}/6\), or \((p1)p^{2(n1)}/\gcd (p1,4)\). If \(\eta =1\) and \(p=3\), then \(e(\mathfrak {P}\wp )\) is divisible by \(\varphi (3^n)/\gcd (\varphi (3^n),t)\) with \(t=6\) or 9.
In Table 2, we give a list of every noncuspidal \(\mathbb {Q}\)rational point on the modular curves \(X_0(p^n)\) of genus \(\ge 1\), which correspond to elliptic curves with potential supersingular reduction at the prime p, together with Cremona labels for curves with the given jinvariant and least conductor. See Sect. 6 of [28]. We also give the values of e and \(e_1\) for each j, which we define next.
Elliptic curves with potential supersingular reduction on \(X_0(p^n)\)
\({\varvec{j}}\) invariant  \({\varvec{p}}\)  Examples  Good reduction over  \({\varvec{e}}\)  \({\varvec{e}}_{\varvec{1}}\) 

\(j = 2^{15} \cdot 3 \cdot 5^3\)  3  27A2, 27A4  \(\mathbb {Q}(\root 4 \of {3},\beta ^3120\beta +506=0)\)  12  2 
\(j = 11\cdot 131^3\)  121C2  \(\mathbb {Q}(\root 3 \of {11})\)  3  1  
\(j = 2^{15}\)  11  121B1, 121B2  \(\mathbb {Q}(\root 4 \of {11})\)  4  2 
\(j = 11^2\)  121C1  \(\mathbb {Q}(\root 3 \of {11})\)  3  2  
\(j = 17^2 \cdot 101^3/2\)  17  14450P1  \(\mathbb {Q}(\root 3 \of {17})\)  3  2 
\(j = 17 \cdot 373^3/2^{17}\)  14450P2  \(\mathbb {Q}(\root 3 \of {17})\)  3  1  
\(j = 2^{15}\cdot 3^3\)  19  361A1, 361A2  \(\mathbb {Q}(\root 4 \of {19})\)  4  2 
\(j = 2^{18} \cdot 3^3 \cdot 5^3\)  43  1849A1, 1849A2  \(\mathbb {Q}(\root 4 \of {43})\)  4  2 
\(j = 2^{15}\cdot 3^3\cdot 5^3\cdot 11^3\)  67  4489A1, 4489A2  \(\mathbb {Q}(\root 4 \of {67})\)  4  2 
\(j = 2^{18}\cdot 3^3\cdot 5^3\cdot 23^3\cdot 29^3\)  163  26569A1, 26569A2  \(\mathbb {Q}(\root 4 \of {163})\)  4  2 
Theorem 6.8
([28, Theorem 6.1]) Let \((j_0,p)\) be any of the jinvariants that are listed in Table 2, together with the corresponding prime p of potential supersingular reduction. Let \(E/\mathbb {Q}\) be an elliptic curve with \(j(E)=j_0\), and let \(T_n\in E[p^n]\) be a point of exact order \(p^n\). Then, the ramification index of any prime \(\wp \) that lies above p in the extension \(\mathbb {Q}(T_n)/\mathbb {Q}\) is divisible by \((p1)p^{2n2}/2\) if \(p>3\) and \(n\ge 1\), and by \(3^{2n4}\) if \(p=3\) and \(n\ge 3\).
6.3 CM curves
The goal of this section is to show Theorem 1.8. We begin by citing some work of Silverberg, Prasad, and Yogananda (see also [3] for related work).
Theorem 6.9
 (1)
\(\varphi (\theta )\le w\cdot d\).
 (2)
If \(k\subseteq L\), then \(\varphi (\theta )\le \frac{w}{2}d\).
 (3)
If L does not contain k, then \(\varphi (\#E(L)_{\text {tors}})\le w\cdot d\).
An elliptic curve with CM has integral jinvariant and therefore potential good reduction everywhere. Thus, we can apply our results from Sect. 6 to prove the following theorem, which is analogous to (1) of Theorem 6.9, except that the bound here is in terms of ramification.
Theorem 6.10
Proof
Let E / L be an elliptic curve with CM by the maximal order \(\mathcal {O}_F\) of an imaginary quadratic field F, let p be a prime, and let \(R\in E(\overline{L})\) be a point of exact order \(p^n\), for some \(n\ge 1\). We distinguish two cases, according to whether p splits in \(F/\mathbb {Q}\), or p is inert or ramified in \(F/\mathbb {Q}\).
Thus, it only remains to show the following lemma.
Lemma 6.11
Proof
First, let us show that we may increase the base field K by a finite extension if we need to. Suppose \(K'/K\) is a finite extension of local fields, baseextend E to be defined over \(K'\), let \(\mathcal {M}'\) be the maximal ideal of \(K'\) above \(\mathcal {M}\) of K (which in turn is a finite extension of \(L_\wp ^{\text {nr}}\)). Since E / K has good (ordinary) reduction at \(\mathcal {M}\), the curve \(E/K'\) has good (ordinary) reduction at \(\mathcal {M}'\). Now suppose that the kernel of reduction \(\bmod \mathcal {M}'\) of \(E(\overline{K}')[p^n]\) is \(X_{p^n}'=E[\mathfrak {p}^n]\). Since E is originally defined over K, we have that \(E(\overline{K})[p^n]\cong E(\overline{K}')[p^n]\), and if a point R reduces to the origin modulo \(\mathcal {M}\), then it also reduces to the origin modulo \(\mathcal {M}'\), because \(\mathcal {M}'\) divides \(\mathcal {M}\). Hence, \(X_{p^n}'\subseteq X_{p^n}\). Since \(X_{p^n}'=X_{p^n}=p^n\), we conclude \(X_{p^n}'=X_{p^n}=E[\mathfrak {p}]\) as desired.
This concludes the proof of 6.10. \(\square \)
Remark 6.12
The proof of Theorem 6.10 carries over to elliptic curves with CM by a nonmaximal order \(\mathcal {O}\), except, perhaps, for the case when p splits in \(F/\mathbb {Q}\) but p divides the conductor of the order \(\mathcal {O}\). This remaining case will be dealt with in future work.
We refer the reader to Sect. 6.2 for the definition of \(e_1\), a quantity that appears in the next two results (see also §1 of [26], or §2 of [28]). Moreover, we remark that by Corollary 4.8 of [28], if \(p>3e(\wp p)\), then \(e_1\) is not divisible by p.
Theorem 6.13
 (1)
If p is inert or ramified in \(\mathcal {O}_F\), and \(\wp \) is a prime of L above p, let \(K=K_E\) be an extension of \(L_\wp ^{\text {nr}}\) such that \(E/K_E\) has good reduction, and let \(I_K\subseteq {\text {Gal}}(\overline{K}/K)\) be the inertia subgroup. Assume that \(e_1\) is not divisible by p. Then, \(\rho _\phi \) restricted to I is either \(\theta _{p1}^{ee_1}\), or \(\theta _{p^21}^e\), where \(\theta _{q1}:I\rightarrow \mathbb {F}_q^\times \) with \(q=p^h\) is a fundamental character of level h, and e and \(e_1\) are the usual quantities as defined in Sect. 6.2. If \(\rho _\phi _I=\theta _{p^21}^e\), then \(p+1e\) and the values are in \(\mathbb {F}_p^\times \).
 (2)If p is split, i.e., \(p\mathcal {O}_F=\mathfrak {p}\overline{\mathfrak {p}}\), then either
 (a)
\(\langle S \rangle = E[\mathfrak {p}]\) or \(E[\overline{\mathfrak {p}}]\), and if \(\wp \) is a prime of L above \(\mathfrak {p}\) (resp. \(\wp '\) above \(\mathfrak {p}\)) and \(K_E/L_\wp ^{\text {nr}}\) and \(I_K=I_{K,\wp }\) are as before (resp. \(K_E/L_{\wp '}^{\text {nr}}\) and \(I_{K,\wp '}\)), then \(\rho _\phi \) restricted to \(I_{K,\wp }\) (resp. \(I_{K,\wp '}\)) is given by \(\theta _{p1}^e\), or
 (b)
The character \(\theta _{p1}^e:I_{K,\wp }\rightarrow \mathbb {F}_p^\times \) is trivial, for any prime \(\wp \) of \(\mathcal {O}_L\) above p.
 (a)
Proof

If \(pe/(p+1)>e_1\), then there is an \(\mathbb {F}_p\)basis \(\{P,Q\}\) of E[p] such that the action of \(I_K\) in on E[p] is given by a Borel subgroup B of \({\text {GL}}(2,\mathbb {F}_p)\) such that the diagonal characters are \(\theta _{p1}^{ee_1}\) and \(\theta _{p1}^{e_1}\). Moreover, since \(e_1\) is not divisible by p, the ramification in the extension \(K_E(E[p])/K_E\) is divisible by p (by Proposition 5.6 of [28]) and therefore the upper right hand corner of the Borel B is nontrivial. It follows that the only inertiastable subspace of E[p] is \(\langle P \rangle \), and the action is given by \(\theta _{p1}^{ee_1}\). We conclude that \(\rho _\phi _{I_K} = \theta _{p1}^{ee_1}\) as claimed.

If \(pe/(p+1)\le e_1\), then the action of inertia \(I_K\) on E[p] is given by \(\theta _{p^21}^{e}\), and therefore the action in terms of a basis of E[p] is given by the eth power of a (full) nonsplit Cartan subgroup \(C_{\text {ns}}\) of \({\text {GL}}(2,\mathbb {F}_p)\). Since the eigenvalues of a nondiagonal matrix in \(C_{\text {ns}}\) are not in \(\mathbb {F}_p\), then E[p] has a 1dimensional \(\mathbb {F}_p\)submodule that is fixed by inertia if and only if \(C_{\text {ns}}^e\) only contains diagonal entries. In particular, \((p+1)e\) and \(\rho _\phi _{I_K}=\theta _{p^21}^e\).
Hence \(\langle S\rangle = E[\mathfrak {p}]\) or \(E[\overline{\mathfrak {p}}]\), unless all characters \(\theta _{p1}^e:I_{K,\wp }\rightarrow \mathbb {F}_p^\times \) are trivial for all \(\wp \) above p. \(\square \)
Corollary 6.14
Let \(p>2\), E / L with CM by the maximal order \(\mathcal {O}_F\subseteq F\), \(K=K_E\), \(\phi \), \(\langle S\rangle \), and \(\rho _\phi \) be as before. If p is ramified or inert in \(\mathcal {O}_F\), assume that \(e_1\) is not divisible by p. Let \(K_1\) be the subfield of K(S) fixed by the kernel of \(\rho _\phi ^{12}\). Then, either \(p1\) is a divisor of e (which in turn is a divisor of \(24e(\wp p)\)) for any \(\wp \) of \(\mathcal {O}_L\) above p, or there is a prime \(\wp \) of \(\mathcal {O}_L\) and a prime \(\mathfrak {P}\) of \(L_1\), such that the ramification index \(e(\mathfrak {P}\wp )\) is divisible by \((p1)/\gcd (p1,12t)\) for \(t=e\), or \(ee_1\), and the ramification in \(K_1/K\) is also divisible by \((p1)/\gcd (p1,12t)\).
Proof
By Theorem 6.13, either \(\theta _{p1}^e\) is trivial for all \(\wp \) over p, or \(\rho _\phi _{I_K} = \theta _{p1}^{ee_1}\) or \(\theta _{p^21}^e\) and \(p+1e\).
If \(\theta _{p1}^e\) is trivial for all \(\wp \) over p, and since the fundamental character \(\theta _{p1}\) is surjective ([39], §1.7), it follows that e is divisible by \(p1\).
If \(\rho _\phi ^{12}_{I_K} = \theta _{p1}^{12(ee_1)}\) or \(\theta _{p^21}^{12e}\), and \(K_1\) is the subfield of K(S) fixed by the image of \(\rho _\phi ^{12}_{I_K}\), then the ramification in the extension \(K_1/K\) is divisible by \((p1)/\gcd (p1,12(ee_1))\) or by \((p^21)/\gcd (p^21,12e)\). In particular, the ramification in \(K_1/K\) is divisible by \((p1)/\gcd (p1,12t)\) with \(t=ee_1\) or e. Hence, by Remark 4.8 there is a prime \(\mathfrak {P}\) of \(\mathcal {O}_{L(S)}\) above \(\wp \) such that \(e(\mathfrak {P}\wp )\) is divisible by \((p1)/\gcd (p1,12t)\), as claimed. \(\square \)
7 Auxiliary results for the proof of Theorem 1.9
In this section we collect a number of auxiliary results that will be used in the proof of Theorem 1.9 in Sect. 8. In order to apply Theorem 2.1 to all elliptic curves defined over a number field L, we need some control on those curves that admit Lrational isogenies of ppower order. For an elliptic curve E / L we denote by \(\rho _{E,p}\) the Galois representation \({\text {Gal}}(\overline{L}/L)\rightarrow {\text {Aut}}(T_p(E))\) associated to the natural action of Galois on the padic Tate module \(T_p(E)\) of the curve E.
Theorem 7.1
Let p be a prime, and let L be a number field. Fix an element \(j_0\in L\). Then, there is a number \(n=n(p,j_0)\) such that for any elliptic curve E / L without CM and with \(j(E)=j_0\) we have \(1+p^n\text {M}_2(\mathbb {Z}_p)\subseteq \rho _{E,p}({\text {Gal}}(\overline{L}/L)).\) In particular, E / L does not admit Lrational isogenies of degree \(p^{a(p,j_0)}\) with \(a(p,j_0)=n(p,j_0)+1\).
Proof
Momose has given a classification of isogenies of prime degree over number fields (see [34, Theorem A]), but here we use another classification recently shown by Larson and Vaintrob, which we cite next.
Theorem 7.2
 (1)There exists a CM elliptic curve \(E'\), which is defined over L and whose CMfield is contained in L, with a padic degree 1 associated character whose mod p reduction \(\overline{\psi }'\) satisfies$$\begin{aligned} \overline{\psi }^{12}=\left( \overline{\psi }'\right) ^{12}. \end{aligned}$$
 (2)The Generalized Riemann Hypothesis fails for \(L(\sqrt{p})\), andwhere \(\chi _p\) is the cyclotomic character. (Moreover, in this case we must have \(p\equiv 3 \bmod 4\) and the representation \(\rho _{E,p} \bmod p\) is already reducible.)$$\begin{aligned} \overline{\psi }^{12}=\overline{\chi _p}^6, \end{aligned}$$
For technical reasons, we need to strengthen Larson and Vaintrob’s result, so that the curve \(E'\) in (1) hash CM by a maximal order. Before we do that, we recall that if \(E'/L\) is an elliptic curve with CM by an order \(\mathcal {O}_f\) of conductor \(f\ge 1\) of an imaginary quadratic field F, such that \(F(j(E'))\subseteq L\), then there is an elliptic curve \(E''/L\) with CM by the full ring of integers \(\mathcal {O}_F\), and an Lrational isogeny \(E'\rightarrow E''\) that is cyclic of degree f. Indeed, the isogeny arises from the inclusion \(\mathcal {O}_f\subseteq \mathcal {O}_F\) which induces a map \(E'\cong \mathbb {C}/\mathcal {O}_f \rightarrow \mathbb {C}/\mathcal {O}_F\cong E''\), with kernel \(\mathcal {O}_F/\mathcal {O}_f\cong \mathbb {Z}/f\mathbb {Z}\). Note that the kernel is isomorphic to \( \mathcal {O}_F/\mathcal {O}_f\cong f\mathcal {O}_F/f\mathcal {O}_f \subseteq \mathcal {O}_f/f\mathcal {O}_f\), which is a cyclic group of order f that is invariant under the action of Galois (recall that \({\text {Gal}}(L(E'[f])/L)\hookrightarrow {\text {Aut}}_{\mathcal {O}_f/f\mathcal {O}_f}(E'[f])\cong (\mathcal {O}_f/f\mathcal {O}_f)^\times \) as long as \(F(j(E'))\subseteq L\); see [44, Ch 2., Theorem 2.3]). Hence, \(T=f\mathcal {O}_F/f\mathcal {O}_f\) is a cyclic Lrational subgroup of order f, the quotient \(E'' = E'/T\) is an elliptic curve defined over L, and the map \(E'\rightarrow E''\) is an Lrational isogeny.
Lemma 7.3
Let L be a number field, let \(E'/L\) be an elliptic curve with CM by an order \(\mathcal {O}_f\) of conductor \(f\ge 1\), contained in an imaginary quadratic field F, and suppose that the CMfield of \(E'\) is contained in L. Further, assume that \(E'\) has a pisogeny with associated modp character \(\psi '\) of degree 1, where p is a prime with \(\gcd (p,f)=1\). Then, there is an elliptic curve \(E''/L\) with CM by the maximal order \(\mathcal {O}_F\), such that \(F(j(E''))\subseteq L\) and \(E''\) has another pisogeny with the same associated modp character \(\psi '\) of degree 1.
Proof
Suppose E / L is an elliptic curve with CM by the order \(\mathcal {O}_f\) of conductor \(f\ge 1\) inside the quadratic imaginary field F, and such that \(F(j(E))\subseteq L\). Then, there exists an elliptic curve \(E''/L\) with CM by \(\mathcal {O}_F\) and a canonical Lrational isogeny \(\phi :E'\rightarrow E''\) that is cyclic of degree f (see the paragraph before the statement of the lemma). Since \(\gcd (p,f)=1\), the isogeny \(\phi \) induces an isomorphism \(\phi :E'[p]\cong E''[p]\) defined over L. Now suppose that \(E'\) has a pisogeny with kernel \(\langle P\rangle \) and its associated isogeny character is \(\psi ':{\text {Gal}}(\overline{L}/L)\rightarrow (\mathbb {Z}/p\mathbb {Z})^\times \), such that \(\sigma (P)=\psi '(\sigma )P\) for any \(\sigma \in {\text {Gal}}(\overline{L}/L)\). Then, the isomorphism \(E'[p]\cong E''[p]\) implies that \(E''\) also has a pisogeny with kernel \(\langle \phi (P)\rangle \) and \(\sigma (\phi (P))=\phi (\sigma (P))=\phi (\psi '(\sigma )P)=\psi '(\sigma )\phi (P)\), where the action of \(\sigma \) commutes with both \(\phi \) and \([\psi '(\sigma )]\) because both maps are defined over L. Hence, the isogeny character for \(E''\) is also \(\psi '\) as claimed.
Now we are ready to prove the following variant of Theorem 7.2.
Theorem 7.4
 (1)There exists an elliptic curve \(E''\) with CM by the full ring of integers \(\mathcal {O}_F\) of an imaginary quadratic field F, such that \(E''\) is defined over L, its CMfield is contained in L, and has a padic degree 1 associated character whose mod p reduction \(\overline{\psi }'\) satisfies$$\begin{aligned} \overline{\psi }^{12}=\left( \overline{\psi }'\right) ^{12}. \end{aligned}$$
 (2)The Generalized Riemann Hypothesis fails for \(L(\sqrt{p})\), andwhere \(\chi _p\) is the cyclotomic character. (Moreover, in this case we must have \(p\equiv 3 \bmod 4\) and the representation \(\rho _{E,p} \bmod p\) is already reducible.)$$\begin{aligned} \overline{\psi }^{12}=\overline{\chi _p}^6, \end{aligned}$$
Proof
Let L be a number field of degree d. It is well known that there are only finitely many imaginary quadratic fields with class number less or equal than a given bound d (see [10]). Moreover, the class number of an order contained in a maximal order grows with the conductor ([4, Theorem 7.24]). Hence, there are only finitely many jinvariants with CM defined over L (see also [5] for some bounds on the number of jinvariants defined over a number field), say \(\{j_1,\ldots ,j_n\}\), associated to orders \(\mathcal {O}_{f_1},\ldots ,\mathcal {O}_{f_n}\) with conductors \(f_1,\ldots ,f_n\ge 1\). Let \(S_L\) be the set of primes given by Theorem 7.2, and enlarge it by adding to \(S_L\) all the prime divisors of \(f_1,\ldots ,f_n\) (in particular, \(S_L\) is still a finite set of prime numbers).
Now, for a prime \(p\not \in S_L\), and an elliptic curve E / L for which \(E[p]\otimes \overline{\mathbb {F}}_p\) is reducible with degree 1 associated character \(\psi \), either (1) or (2) of Theorem 7.2 holds. If (2) holds, we are done. If (1) holds, then there exists a CM elliptic curve \(E'\), which is defined over L and whose CMfield is contained in L, with a padic degree 1 associated character whose mod p reduction \(\overline{\psi }'\) satisfies \(\overline{\psi }^{12}=\left( \overline{\psi }'\right) ^{12}.\) If \(E'\) has CM by an order \(\mathcal {O}_f\) of conductor \(f\ge 1\), and since \(F(j(E'))\subseteq L\) by assumption, we must have \(f=f_i\) for some \(1\le i \le n\). Since \(p\not \in S_L\), and \(S_L\) contains all prime divisors of f, we have \(\gcd (p,f)=1\). Hence, Lemma 7.3 applies, and there is an elliptic curve \(E''/L\) with CM by the maximal order \(\mathcal {O}_F\), such that \(F(j(E''))\subseteq L\) and \(E''\) has a pisogeny with the same associated modp character \(\psi '\) of degree 1, as claimed.
In order to apply Theorem 7.4 in the proof of Theorem 1.9, we need uniform bounds on the ramification in the fixed fields by kernels of powers of the cyclotomic character.
Lemma 7.5
Let \(p>2\), and let \(e(\wp p)\) be the ramification index in \(L/\mathbb {Q}\) of a prime \(\wp \) of \(\mathcal {O}_L\) lying above p. Then, the ramification index of any prime ideal of \(L(\zeta _{p^n})\) above \(\wp \) is divisible by the quantity \(\varphi (p^n)/\gcd (\varphi (p^n),e(\wp p))\).
More generally: let \(G_L={\text {Gal}}(\overline{L}/L)\), and let \(\chi _{p,n}:G_L\rightarrow (\mathbb {Z}/p^n\mathbb {Z})^\times \) be the \(p^n\)th cyclotomic character. Let \(s\ge 1\), and let \(L_{n,s}\subseteq L(\zeta _{p^n})\) be the fixed field by the kernel of \(\chi _{p,n}^s\). Then, the ramification index of any prime ideal of \(L_{n,s}\) above \(\wp \) is divisible by \(\varphi (p^n)/\gcd (\varphi (p^n),s\cdot e(\wp p))\).
Proof
8 Proof of Theorem 1.9
For each \(j_0\in \Sigma (L,p)\), we let \(a=a(p,j_0)\) be the least positive integer a such that any curve E / L with \(j(E)=j_0\) does not admit Lrational isogenies of degree \(p^a\). The existence of \(a(p,j_0)\) is guaranteed by Theorem 7.1 since \(\Sigma _L\) only contains nonCM jinvariants. Let \(A(L,p)=\max \{a(L,p), a(p,j_0): j_0\in \Sigma (L,p) \}\). The number A(L, p) is welldefined because \(\Sigma (L,p)\) is a finite set.
Next, suppose that E / L admits a Lrational isogeny \(\phi \) of degree \(p^a\), for some \(a\ge 1\). Let \(\psi :{\text {Gal}}(\overline{L}/L)\rightarrow (\mathbb {Z}/p^a\mathbb {Z})^\times \) be the character associated to the isogeny \(\phi \). Note that the existence of \(\phi \) implies the existence of a Lrational isogeny \(\phi _1\) of degree p, with associated character \(\psi _1=\overline{\psi }\), the mod p reduction of \(\psi \). Let \(\langle S \rangle \subseteq E[p]\) be the kernel of \(\phi _1\).

If we are in option (1), then there exists an elliptic curve \(E''\), with CM by a full ring of integers \(\mathcal {O}_k\), which is defined over L and whose CMfield is contained in L, with a padic degree 1 associated character whose mod p reduction \(\overline{\psi }'\) satisfiesLet K be the smallest extension of \(L_\wp ^{\text {nr}}\) such that \(E'/K\) has good reduction. Let \(K_1\) be the subfield of K(S) fixed by the kernel of \(\rho _{\phi _1}^{12}\), and similarly define \(L_1\). Since \(p\not \in S_L'\), \(p>3e(\wp p)\), and by [28, Corollary 4.8], if \(p>3e(\wp p)\), then \(e_1\) is not divisible by p (see §1 of [26], or §2 of [28] for the definition of \(e_1\)). Then, by Corollary 6.14, either \(p1\) is a divisor of \(e=e(K/\mathbb {Q}_p)\) (which in turn is a divisor of \(12e(\wp p)\)) for any \(\wp \) of \(\mathcal {O}_L\) above p, or the ramification index in \(K_1/K\) is divisible by \((p1)/\gcd (p1,12t)\) for \(t=e\), or \(ee_1\). Since \(p\not \in S_L'\), we have \(p1>12^2\cdot e(\wp p)\ge 12e\ge 12t\), and therefore the ramification in K(S) / K is \(>1\). It follows from Theorem 2.1 that there is a constant \(c=c(E/L,\wp )\) with \(1\le c \le 12e(\wp p)\), and a prime \(\Omega _R\) of L(R) above \(\wp \) such that the ramification index \(e(\Omega _R\wp )\) is divisible either by \(\varphi (p^n)/\gcd (\varphi (p^n),c), \text { or }\ p^{n}.\) In particular, \(\varphi (p^n)\le c\cdot e(\Omega _R\wp )\le 12 e(\wp p)e(\Omega _R\wp )\le 12e(\Omega _Rp)\), as before.$$\begin{aligned} \overline{\psi }^{12}=\left( \overline{\psi }'\right) ^{12}. \end{aligned}$$

Now consider option (2) of Theorem 7.4. In this case GRH fails for \(L(\sqrt{p})\) and \(\overline{\psi }^{12}=\overline{\chi }_p^6\), where \(\overline{\psi }\) and \(\overline{\chi }_p\) are, respectively, the mod p reductions of \(\psi \) and \(\chi _p\), the padic cyclotomic character. Let \(L_{1,6}\) be the fixed field of \(\overline{L}\) by the kernel of \(\overline{\chi }_p^6\). Then, \(L_{1,6}\subseteq L(S)\) and, by Lemma 7.5, the ramification index of any prime ideal of \(L_{1,6}\) above \(\wp \) is divisible by \((p1)/\gcd (p1,6\cdot e(\wp p))\). Thenis divisible by$$\begin{aligned} A=A(\wp ,L,S):=e(\wp ,L(S)/L)/\gcd (e(\wp ,L(S)/L),e(K/L_\wp ^{\text {nr}})) \end{aligned}$$Since \(p\ge 3\), then \(e(K/L_\wp ^{\text {nr}})\) is a divisor of 12. Hence, A is divisible by$$\begin{aligned}&\left( \frac{p1}{\gcd (p1,6\cdot e(\wp p))}\right) /\gcd \left( \frac{p1}{\gcd (p1,6\cdot e(\wp p))},e(K/L_\wp ^{\text {nr}})\right) \\&\quad =\frac{p1}{\gcd (p1,6\cdot e(\wp p)\cdot e(K/L_\wp ^{\text {nr}}))}. \end{aligned}$$and this quantity is \(>1\) because \(p\not \in S_L'\). Indeed, notice that if \(p1\ge 144e(\wp p)\), then \((p1)/\gcd (p1,72\cdot e(\wp p))\ge 2\) (because \(\gcd (p1,72\cdot e(\wp p))\le 72e(\wp p)\)).$$\begin{aligned} \frac{p1}{\gcd (p1,72\cdot e(\wp p))}, \end{aligned}$$Hence, \(A(\wp ,L,S)>1\) and by our Theorem 2.1, there is a constant \(c=c(E/L,\wp )\) with \(1\le c \le 12e(\wp p)\), and a prime \(\Omega _R\) of L(R) above \(\wp \) such that the ramification index \(e(\Omega _R\wp )\) is divisible either bySo, as before, \(\varphi (p^n)\le c\cdot e(\Omega _R\wp )\le 12 e(\wp p)e(\Omega _R\wp )\le 12e(\Omega _Rp)\).$$\begin{aligned} \varphi (p^n)/\gcd (\varphi (p^n),c), \text { or }\ p^{n}. \end{aligned}$$
Declarations
Author's contributions
The author would like to thank Kevin Buzzard, Pete Clark, Brian Conrad, Harris Daniels, Benjamin Lundell, Robert Pollack, James Stankewicz, Jeremy Teitelbaum, Ravi Ramakrishna, John Voight, Felipe Voloch and David Zywina for their helpful suggestions and comments. In addition, the author would like to express his gratitude to the anonymous referees for very detailed reports, and pointing out a crucial oversight in an earlier version of the paper.
Competing interests
The author declares that he has no competing interests.
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Authors’ Affiliations
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