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 Open Access
Selmer groups of twists of elliptic curves over K with Krational torsion points
 Jackson Salvatore Morrow^{1}Email author
 Received: 26 February 2016
 Accepted: 20 September 2016
 Published: 5 October 2016
Abstract
We generalize a result of Frey on Selmer groups of twists of elliptic curves over \(\mathbf Q \) with \(\mathbf Q \)rational torsion points to elliptic curves defined over number fields of small degree K with a Krational torsion point. We also provide examples of elliptic curves coming from Zywina that satisfy the conditions of our Corollary D.
Keywords
 Selmer groups
 Quadratic twists of elliptic curves
1 Introduction
Theorem 1.1
 1.
if \({{\mathrm{ord}}}_{\ell }(j_E) < 0\), then \(\left( \frac{d}{\ell } \right) = 1\);
 2.if pN(E) is an odd prime, then$$\begin{aligned} \left( \frac{d}{p} \right) = \left\{ \begin{array}{llll} 1 &{} &{}\quad \text { if }{{\mathrm{ord}}}_p(j_E) \ge 0; \\ 1 &{} &{}\quad \text { if }{{\mathrm{ord}}}_p(j_E) < 0 \; \text { and }\; E/\mathbf Q _p \text { is a Tate curve}; \\ 1 &{} &{}\quad \text { otherwise}. \end{array}\right. \end{aligned}$$
Remark 1.2
Frey actually proved a more explicit double divisibility statement [2, Theorem] concerning the \(\ell \)Selmer group of \(E^d\) and \(\ell \)torsion of ray class groups, when \(\widetilde{S}_E \ne \varnothing \).
In this paper, we generalize Frey’s results [2, Theorem, Corollary] to number fields K of small degree. We show that for specific quadratic twists \(E^d\), the order of the \(\ell \)torsion of some ray class group of \(K(\sqrt{d})\) divides the order of \({{\mathrm{Sel}}}_{\ell }(E^d,K)\), and the order of \({{\mathrm{Sel}}}_{\ell }(E^d,K)\) divides the order of the \(\ell \)torsion of a different ray class group of \(K(\sqrt{d})\) times the degree of some maximal abelian extension of exponent \(\ell \) with prescribed ramification and Galois conditions (cf. Theorems A, B for precise statements and Remark 3.1 for a colloquial statement). These results allow us to give explicit applications to elliptic curves defined over \(\mathbf Q \) (cf. Corollaries C, D), and we provide explicit examples of elliptic curves over \(\mathbf Q \) satisfying Corollary D in Sect. 6. Finally, in Corollary E, we generalize Theorem 1.1 to number fields of small degree.
Remark 1.3
Frey’s idea was to obtain information about \({{\mathrm{Sel}}}_{\ell }(E^d,\mathbf Q )\) when \(E(\mathbf Q )\) contains an element of order \(\ell \). In particular, he studied the behavior of E over local fields \(\mathbf Q _{\ell }\) and their algebraic closures \(\overline{\mathbf{Q }}_{\ell }\). His work illustrated a deep relationship between \(\ell \)ranks of Selmer groups and class groups of finite Galois extensions of exponent \(\ell \). In this paper, we investigate the \(\ell \)Selmer rank in families of quadratic twists of elliptic curves E / K with Krational points of odd prime order \(\ell \). We use Frey’s proof as a blueprint for our own, but the techniques we utilize come from class field theory. That being said, many of his arguments go through mutatis mutandis.
n  S(n)  References 

1  \({{\mathrm{\textsf {Primes}}}}(7)\)  [3] 
2  \({{\mathrm{\textsf {Primes}}}}(13)\)  [8] 
3  \({{\mathrm{\textsf {Primes}}}}(13)\)  [9] 
4  \({{\mathrm{\textsf {Primes}}}}(17)\)  [10] 
5  \({{\mathrm{\textsf {Primes}}}}(19)\)  [11] 
6  \(\subseteq {{\mathrm{\textsf {Primes}}}}(19) \cup \left\{ 37,73 \right\} \)  [7] 
1.1 Some remarks about the proofs
1.2 Organization of paper
In Sect. 2, we recall some classical facts from class field theory and algebraic number theory. In Sect. 3, we state our main results, Theorems A, B and Corollary E. In Sect. 4, we prove Theorem A, which yields a single divisibility statement. In Sect. 5, we prove the double divisibility statement of Theorem B by investigating the Galois structure of splitting fields of \(\ell \)covers of E / K and the splitting fields of elements \({{\mathrm{Sel}}}_{\ell }(E^d,K)\). Finally in Sect. 6, we provide explicit examples of elliptic curves over \(\mathbf Q \) coming from [13] that satisfy the Corollary D.
2 Background and notation
Let L / K be a Galois extension of K, with ring of integers \(\mathcal {O}_L \text { and }\mathcal {O}_K.\) For any finite prime \(\mathfrak {P}\in \mathcal {O}_L\) lying over a prime \(\mathfrak {p}\in \mathcal {O}_K,\) let \(D(\mathfrak {P})\) denote the decomposition group of \(\mathfrak {P},\) let \(I(\mathfrak {P})\) denote the inertia group of \(\mathfrak {P}\) and let \(\kappa ' := \mathcal {O}_L / \mathfrak {P}\text { and }\kappa = \mathcal {O}_K / \mathfrak {p}\) be the residue fields of characteristic \(q = p^n\). In this note, we need a specific result concerning the Artin symbol and ramification theory for quadratic extensions L / K. For the definition of the Artin symbol \(\left( \frac{L/K}{\mathfrak {p}} \right) \), we refer the reader to [14, Chapter IV].
Lemma 2.1
 1.
\(\mathfrak {p}\) is unramified and splits completely in L \(\Longleftrightarrow \) \(\left( \frac{L/K}{\mathfrak {p}} \right) = {{\mathrm{id}}}\),
 2.
\(\mathfrak {p}\) is unramified and nonsplit in L \(\Longleftrightarrow \) \(\left( \frac{L/K}{\mathfrak {p}} \right) = \delta \),
 3.
\(\mathfrak {p}\) is ramified in L \(\Longleftrightarrow \) \(\mathfrak {p} \Delta _{L/K}\) where \(\Delta _{L/K}\) denotes the relative discriminant of L / K.
Proof
In Theorem A, we use primitive Hecke characters to describe a subset of primes \(\mathfrak {p} N(E)\); we refer the reader to [14, Chapter VII, Section 6] for the definition of these characters.
Remark 2.2
2.1 Notation
Definition 2.3
If M / L little ramified outside S, then M / L is unramified at all divisors of primes \(\mathfrak {p}\notin S \cup \left\{ \mathfrak {l} \right\} \).
2.2 Notation
Remark 2.4
If \(S = \emptyset \), we see that \({{\mathrm{cl}}}_{\emptyset ,u}(L)\) is equal to the order of the subgroup of the divisor class group of L consisting of elements of order \(\ell \) which we denote by \({{\mathrm{cl}}}(L)[\ell ]\).
Case 1 Assume that \({{\mathrm{ord}}}_{\mathfrak {p}}(j_E) \ge 0\). Then there is a finite extension N / K such that E has good reduction modulo all \(\mathfrak {P}_N  \mathfrak {p}\) i.e., we find an elliptic curve \(\widetilde{E}\) such that \(\tilde{E}\) modulo \(\mathfrak {P}_N\) is an elliptic curve over the residue field of \(\mathfrak {P}_N\). \(\widetilde{E}(\overline{N_{\mathfrak {P}}})\) contains a subgroup \(\widetilde{E}_(N_{\mathfrak {P}})\) consisting of points \((\widetilde{x},\widetilde{y})\) with \({{\mathrm{ord}}}_{\mathfrak {P}_N}(\widetilde{x}) < 0\). \(\widetilde{E}_\) is the kernel of reduction modulo \(\mathfrak {P}_N\), and \({{\mathrm{ord}}}_{\mathfrak {P}_N}(\widetilde{x}/\widetilde{y})\) is the level of \((\widetilde{x},\widetilde{y})\). For ease of notation, we say that a point \((x,y) \in E(\overline{N_{\mathfrak {P}}})\) is in the kernel of the reduction modulo \(\mathfrak {P}_N\) if its image \((\widetilde{x},\widetilde{y}) \in \widetilde{E}_(\overline{N_{\mathfrak {P}}})\).
Definition 2.5
If F / K is a number field and \(\mathfrak {P}_F  \mathfrak {p}\) we say that a point \((x,y) \in E(F_{\mathfrak {P}})\) is in the connected component of the unity modulo \(\mathfrak {P}_F\) if it is of the form \(\tau (u)\) with u a \(\mathfrak {P}_F\)adic unit, and (x, y) is in the kernel of the reduction modulo \(\mathfrak {P}_F\) if \(u1 \in \mathfrak {P}_F\).
Remark 2.6
One should notice that if E is not a Tate curve over \(K_{\mathfrak {p}}\) but over an extension of degree 2 of \(K_{\mathfrak {p}}\), then for all points \(P \in E(K_{\mathfrak {p}})\), 2P is in the connected component of unity modulo \(\mathfrak {p}\).
3 Statement of results
As mentioned above, [2, Theorem] gives a double divisibility statement involving the \(\ell \)torsion of the Selmer group. First, we generalize his single divisibility to elliptic curves E / K defined over number fields K of finite degree with Krational points of odd, prime order \(\ell \). Recall that S(n) is the set of primes that can arise as the order of a rational point on an elliptic curve defined over a number field of degree n.
Theorem A
 1.
if \(\mathfrak {q} N(E)\), then \(\mathfrak {q} \Delta _{K(\sqrt{d})/K}\);
 2.
if \({{\mathrm{ord}}}_{\mathfrak {l}}(j_{E}) < 0\), then \(\left( \frac{K(\sqrt{d})/K}{\mathfrak {l}} \right) = \delta \);
 3.if \(\mathfrak {p}N(E)\) is a prime of K with \(\mathfrak {p}\notin S_E\), then

if \({{\mathrm{ord}}}_{\mathfrak {p}}(j_{E}) \ge 0\), then \(\left( \frac{K(\sqrt{d})/K}{\mathfrak {p}} \right) = \delta \);

if \({{\mathrm{ord}}}_{\mathfrak {p}}(j_{E}) < 0 \text { and }E/K_{\mathfrak {p}} \text { is a Tate curve},\) then \(\left( \frac{K(\sqrt{d})/K}{\mathfrak {p}} \right) = \delta \);

otherwise, \(\left( \frac{K(\sqrt{d})/K}{\mathfrak {p}} \right) = {{\mathrm{id}}}\).

We also prove a stronger, more explicit version of Theorem A in the form of a double divisibility statement, which completely generalizes [2, Theorem].
Theorem B
 1.
if \(\mathfrak {q} N(E)\), then \(\mathfrak {q} \Delta _{K(\sqrt{d})/K}\);
 2.
if \({{\mathrm{ord}}}_{\mathfrak {l}}(j_{E}) < 0\), then \(\left( \frac{K(\sqrt{d})/K}{\mathfrak {l}} \right) = \delta \);
 3.if \(\mathfrak {p}N(E)\) is a prime of K with \(\mathfrak {p}\notin S_E\), then

if \({{\mathrm{ord}}}_{\mathfrak {p}}(j_{E}) \ge 0\), then \(\left( \frac{K(\sqrt{d})/K}{\mathfrak {p}} \right) = \delta \);

if \({{\mathrm{ord}}}_{\mathfrak {p}}(j_{E}) < 0 \text { and }E/K_{\mathfrak {p}} \text { is a Tate curve},\) then \(\left( \frac{K(\sqrt{d})/K}{\mathfrak {p}} \right) = \delta \);

otherwise, \(\left( \frac{K(\sqrt{d})/K}{\mathfrak {p}} \right) = {{\mathrm{id}}}\).

Remark 3.1
In words, (2) states that the order of the \(\ell \)torsion of the \(S_E\)ray class group of \(K(\sqrt{d})\) divides the order of \({{\mathrm{Sel}}}_{\ell }(E^d,K)\), and the order of \({{\mathrm{Sel}}}_{\ell }(E^d,K)\) divides the order of the \(\ell \)torsion of the \(\widetilde{S}_E\)ray class group of \(K(\sqrt{d})\) times the degree of the maximal abelian extension \(K''\) of \(K'\) of exponent \(\ell \) unramified outside of \(S_E \cup \left\{ \mathfrak {l} \right\} \) such that the Galois group \({{\mathrm{Gal}}}(K'/K)\) acts on \({{\mathrm{Gal}}}(K''/K)\) by \(\chi _{\ell }\varepsilon _d\), where \(\varepsilon _d\) is the character prescribing the Galois action on \(\sqrt{d}\).
Once we have proved Theorems A, B, we can immediately extend the divisibility statements (1), (2) to elliptic curves E defined over \(\mathbf Q \) by considering the values of \(S_\mathbf{Q }(n)\).
Corollary C
Let \(E/\mathbf Q \) be an elliptic curve defined over \(\mathbf Q \). For some Galois number field K, suppose that \(E_K\) attains a Krational point P of order \(\ell \) where \(\ell \in S_\mathbf{Q }(n)\backslash \left\{ 2,3 \right\} \) such that \(\ell \not \mid {{\mathrm{cl}}}(K)\) and \(\zeta _{\ell } \notin K \). In keeping with the notation and assumptions of Theorem A, we can produce examples of quadratic twists \(E_K^d \) that satisfy the divisibility statement (1).
Corollary D
Let \(E/\mathbf Q \) be an elliptic curve defined over \(\mathbf Q \); let \(E_K\) denote the base change of this curve to a Galois number field of degree \(n\le 20\) such that \(N_{K/\mathbf Q }(\mathfrak {q}) = 2 \text { for all }\mathfrak {q}2\). Choose \(\ell \in S_\mathbf{Q }(n)\backslash \left\{ 2,3 \right\} \) such that \(\ell \not \mid {{\mathrm{cl}}}(K)\), \(\zeta _{\ell } \notin K \), and the ramification index \(e_{\mathfrak {l}}(K/\mathbf Q )\) satisfies \(1>e_{\mathfrak {l}}(K/\mathbf Q )/(\ell 1)  1\). Suppose that \(E_K\) attains a Krational point P of order \(\ell \), then in keeping with the notation and assumptions of Theorem B, we can produce examples of quadratic twists \(E_K^d \) that satisfy the double divisibility statement (2).
We can also generalize [2, Corollary], which we stated as Theorem 1.1.
Corollary E
Remark 3.2
In his Ph.D. thesis [16], Mailhot was able to recover and sharpen [2], Theorem] for elliptic curves defined over \(\mathbf Q \) using purely cohomological methods. His refinement comes from prescribing a splitting behavior of primes above \(K'\) instead of just a nonramified condition. We remark that our methods and results are disjoint, however, we believe that [16, Corollary 2.17] can be generalized to elliptic curves defined over number fields K, using Theorem B.
4 Proof of Theorem A
In this section, we prove the divisibility statement (1). Before we proceed, we make a remark about some of the prime assumptions of Theorem A.
Remark 4.1
(Prime assumptions) If \({{\mathrm{ord}}}_{\mathfrak {p}}(j_{E}) < 0\), then we have that \(E/{K_{\mathfrak {p}}}\) has a Tate parametrization. The second condition \({{\mathrm{ord}}}_{\mathfrak {p}}(\Delta _{E}) \not \equiv 0 \pmod \ell \) assists us in Lemma 4.2. In short, it allows us to understand ramification in the \(\ell \)division field of \(E_{K_{\mathfrak {p}}}\). The final condition \(\chi _{H}(\mathfrak {p}) \ne 0\) is used in Lemma 4.3 and is an analogue of Frey’s condition that \(p \equiv 1 \pmod \ell \). Moreover, this condition allows us to deduce, using Remark 2.2, that for a cyclic extension \(M_2/K\) of degree \(\ell \), \(\mathfrak {p}\) is unramified in \(M_2\).
The first step in the proof is to exhibit an element in \({{\mathrm{Sel}}}_{\ell }(E^d,K)\).
Lemma 4.2
Let \(\ell > 3\) be a rational prime; let M / K be a nonabelian Galois extension of degree \(2\ell \) containing \(K(\sqrt{d})\) that is unramified over this field outside of \(S_E \); let \(\alpha \) be a generator of \({{\mathrm{Gal}}}(M/K(\sqrt{d}))\); and let \(\phi \) the element in \(H^1({{\mathrm{Gal}}}(M/K),E^d(M)[\ell ])\) determined by \(\phi (\alpha ) = P\), where P is a Krational point of order \(\ell \). Then \(\phi \) is an element of \({{\mathrm{Sel}}}_{\ell }(E^d,K)\).
Proof
Assume that \(\left( \frac{K(\sqrt{d})/K}{\mathfrak {p}} \right) = \delta \). In this case, \(\mathfrak {P}_M\) is either fully ramified or decomposed (since M / K is nonabelian). So assume that \(\mathfrak {P}_M\) is fully ramified and divides \(\mathfrak {p}\). Then \(\mathfrak {p}\in S_E\) and in particular \(\mathfrak {p}\ne \mathfrak {l}\) and \({{\mathrm{ord}}}_{\mathfrak {p}}(\Delta _{E_K}) \ne 0 \pmod \ell \). We claim that \(E^d/K_{\mathfrak {p}}(\sqrt{d})\) is a Tate curve and that P is contained in the connected component of the unity over \(K_{\mathfrak {p}}(\sqrt{d})\) corresponding to an \(\ell ^{\text {th}}\) root of unity \(\zeta _{\ell }\) e.g., \(P = \tau (\zeta _{\ell }^{\alpha })\) where \(\tau \) is the Tate parametrization and \(\alpha \in \left\{ 1,\ldots , \ell  1 \right\} \).
Next assume that \(\left( \frac{K(\sqrt{d})/K}{\mathfrak {p}} \right) = {{\mathrm{id}}}\) and \(\mathfrak {p}\ne \mathfrak {l}\). Then \({{\mathrm{ord}}}_{\mathfrak {p}}(j_{E}) < 0\) and E is a Tate curve over \(K_{\mathfrak {p}}\), and so again P corresponds to some \(\ell ^{\text {th}}\) root of unity \(\zeta _{\ell }\) under the Tate parametrization of \(E = E^d\) over \(K_{\mathfrak {p}}(\zeta _{\ell })\) and hence \(\overline{\phi }\) is split by \(K_{\mathfrak {p}}(\zeta _{\ell })\) as seen above. But since the degree of \(K_{\mathfrak {p}}(\zeta _{\ell })\) over \(K_{\mathfrak {p}}\) is prime to \(\ell \), \(\overline{\phi }\) is split over \(K_{\mathfrak {p}}\) already, and thus \(\overline{\phi }\) is locally trivial.
Next, we look at the action of \(\delta \) on \(H_{S_E,u}(K(\sqrt{d}))\).
Lemma 4.3
The generator \(\langle \delta \rangle = {{\mathrm{Gal}}}(K(\sqrt{d})/K)\) acts as \({{\mathrm{id}}}\) on the Galois group \(H_{S_E,u}(K(\sqrt{d}))\).
Proof
Proof of Theorem A
The divisibility of \(\#{{\mathrm{Sel}}}_{\ell }(E^d,K)\) by \({{\mathrm{cl}}}_{S_E,u}(K(\sqrt{d}))[\ell ]\) follows from Lemmas 4.2, 4.3 since our element \(\phi \in {{\mathrm{Sel}}}_{\ell }(E^d,K)\) is induced by \(\alpha \in {{\mathrm{Gal}}}(M/K(\sqrt{d}))\) and the action of \(\langle \delta \rangle \) on \(H_{S_E,u}(K(\sqrt{d}))\) does not affect the order of \(\alpha \) when considered as an element of \(H_{S_E,u}(K(\sqrt{d})).\)
5 Proof of Theorem B
Before we proceed with a proof of Theorem B, we wish to shed some light onto our assumptions. In general, our hypotheses allow us to control the ramification in cyclic extensions of \(K(\sqrt{d})\).
Remark 5.1
(Field assumptions) We assume that our field K is a number field of degree \(n\le 5\) such that \(N_{K/\mathbf Q }(\mathfrak {q}) = 2 \text { for all }\mathfrak {q}2\) and that for some \(\ell \in S(n)\backslash \left\{ 2,3 \right\} \), \(\ell \not \mid {{\mathrm{cl}}}(K)\) and \(\zeta _{\ell } \notin K \). The degree and norm condition appear in Lemma 5.5 and allow us to deduce ramification conditions on prime divisors \(\mathfrak {Q}_M\mathfrak {q}\) where \(M_1/K\) is cyclic. The condition that \(\ell \not \mid {{\mathrm{cl}}}(K)\) implies that there does not exist an extension \(M_2/K\) of degree \(\ell \) contained in the Hilbert class field of K; once again this gives us a ramification consequence. The assumption that \(\zeta _{\ell } \notin K \) is subtle, but it allows for more ramification possibilities since Kummer theory does not restrict cyclic extensions. The final condition that \(e_{\mathfrak {l}}(K/\mathbf Q ) \ne 5\) when \([K{:}\mathbf Q ] = 5\) and \(\ell = 5\) is due to a deep result of Katz [17] concerning the injectivity of \(\ell \)torsion under the reduction map; the assumption \(1>e_{\mathfrak {l}}(K/\mathbf Q )/(\ell 1)  1\) from Theorem D is the general condition. This assumption allows us to use the fact that prime to 2 torsion will inject under the reduction map.
5.1 Galois structure of splitting fields of \(\ell \)covers of E
We want to determine the Galois group structure of splitting fields of elements in \(H^1(G_K,E(\overline{K})[\ell ])\) for elliptic curves having a Krational point P of order \(\ell \). Recall that \(\zeta _{\ell } \notin K\). Denote the \(\ell \)division field by \(K(E[\ell ])\); this is the field obtained by adjoining the x, y coordinates of all points of order \(\ell \) of E to K. Then \(K(E[\ell ])\) is a Galois extension of K containing \(K(\zeta _{\ell })\), and it is cyclic over \(K(\zeta _{\ell })\) of degree dividing \(\ell \). From this point on, we shall abbreviate \(E(\overline{K})[\ell ]\) with \(E[\ell ]\), and similarly for \(E^d[\ell ]\).
Lemma 5.2
The Galois group \(K(E[\ell ])/K\) is generated by two elements \(\overline{\gamma },\overline{\varepsilon }\) with \(\overline{\gamma }^{\ell  1} = {{\mathrm{id}}}\), \(\overline{\varepsilon }^{\ell } = {{\mathrm{id}}}\), \(\overline{ \gamma }  K(\zeta _{\ell })\) generates \(K(\zeta _{\ell })/K\), and \(\overline{\gamma }\overline{\varepsilon }\overline{\gamma }^{1}= \overline{\varepsilon }^{\chi _{\ell }(\overline{\gamma })^{1}}\).
Proof
Remark 5.3
The choice of \(\overline{\gamma }\) and \(\overline{\varepsilon }\) is closely related to the choice of base \( \left\{ P,Q \right\} \). In particular, we have \(\overline{\varepsilon }(Q) = P + Q\) if \(\overline{\varepsilon } \ne {{\mathrm{id}}}\) and \(\overline{\gamma }(Q) = \chi _{\ell }(\overline{\gamma }) Q\).
Lemma 5.4
The group \(H^1(G_K,E^d[\ell ])\) injects into \({{\mathrm{Hom}}}_{{{\mathrm{Gal}}}(L_d/K)}({{\mathrm{Gal}}}(\overline{K}/L_d),E^d[\ell ])\).
Proof
Now we distinguish between two cases:
Thus, \(\overline{M}_1(\phi )\) determines \(\langle \phi \rangle \) up to elements of first type, and in order to determine all elements in \(H^1(G_K,E^d[\ell ])\), it is enough to determine all dihedral extensions of K of degree \(2\ell \) containing \(K(\sqrt{d})\) and all extensions \(M_1\) of degree \(\ell \) over \(K'\) which are normal over K such that conjugation by \(\overline{\gamma }\) on \({{\mathrm{Gal}}}(\overline{M}_1,K')\) is equal to \(\chi _{\ell }(\overline{\gamma })\).
Therefore to prove the double divisibility, one has to show that for \(\phi \in {{\mathrm{Sel}}}_{\ell }(E^d,K)\), the field \(\overline{M}_2(\phi )\) is unramified over \(K(\sqrt{d})\) outside \(\widetilde{S}_E \), and \(\overline{M}_1(\phi )\) is unramifed over \(K'\) outside \(S_E\) and little ramified at divisors of \(\mathfrak {l}\).
5.2 Splitting fields of elements in \({{\mathrm{Sel}}}_{\ell }(E^d,K)\)
We shall continue to use the assumptions and the notations of the Theorem B and Sect. 5.1.
Lemma 5.5
Let \(\phi \) be an element in \({{\mathrm{Sel}}}_{\ell }(E^d,K)\). Then \(\overline{M}_1(\phi ) =: \overline{M}_1\) is unramified at \(\mathfrak {q}\) over \(K'\) and \(\overline{M}_2(\phi ) =: \overline{M}_2\) is unramified at \(\mathfrak {q}\) over \(K(\sqrt{d})\).
Proof
We first prove the latter statement. Since \(\mathfrak {q} \Delta _{K(\sqrt{d})/K}\,\), we have that \(K(\sqrt{d})\) and \(K'\) are ramified at \(\mathfrak {q}\) over K. Hence the norm of \(\mathfrak {Q}\mathfrak {q}\) in \(K(\sqrt{d})\) is equal to \(\mathfrak {q}\), and by assumption the norm of \(\mathfrak {Q}2\) is equal to 2. Suppose that \(K(\sqrt{d})\) had a cyclic extension of degree \(\ell \) in which \(\mathfrak {Q}\) is ramified. Then the completion \(K(\sqrt{d})_{\mathfrak {Q}}\) admits a cyclic extension of degree \(\ell \) ramified at \(\mathfrak {Q}\). Since \(\ell \) is odd and \(\mathfrak {Q}\) has residue characteristic two, this extension is tamely ramified. By local class field theory, the tamely ramified cyclic extensions of a local field \(K(\sqrt{d})_{\mathfrak {Q}}\) all have degree dividing \(\kappa ^{\times }\), where \(\kappa \) is the residue field. Since \(\kappa = \mathbf F _2\), we have that there are no tamely ramified and ramified extensions of \(K(\sqrt{d})_{\mathfrak {Q}}\). Thus, \(K(\sqrt{d})\) has no cyclic extension of degree \(\ell \) in which \(\mathfrak {Q}\) ramifies, and hence \(\overline{M}_2\) is unramified at \(\mathfrak {q}\) over \(K(\sqrt{d})\).
Remark 5.6
Since \(73  (2^{36}  1),\) \(73  (2^9  1)\), and \(73  (2^{18}  1)\), we may not assume that there is a unique cyclic extension of \(K'\) with degree 73 in which \(\mathfrak {Q}\) is ramified, and hence the above argument does work for \(\ell = 73\). This precludes us from extending Theorem B to number fields K of degree \(\ge 6\).
Therefore, we can assume that \(\mathfrak {p}\not \mid \mathfrak {q}\cdot \mathfrak {l}\), but \(\mathfrak {p} N(E)\).
Lemma 5.7
Let \(\phi \) be an element in \({{\mathrm{Sel}}}_{\ell }(E^d,K)\). Then \( \overline{M}_1/K'\) is unramified outside of \(S_E \cup \left\{ \mathfrak {l} \right\} \) and \( \overline{M}_2/K(\sqrt{d})\) is unramified outside \(\widetilde{S}_E \cup \left\{ \mathfrak {l} \right\} \).
Proof
 1.
If \({{\mathrm{ord}}}_{\mathfrak {p}}(j_E) \ge 0\), then it follows from Néron’s list of minimal models of elliptic curves with potentially good reduction that \(\ell \) must be equal to 3 [18, p.124]. Since we only consider primes \(\ell > 3\), we can exclude this case from consideration.
 2.Now assume that \({{\mathrm{ord}}}_{\mathfrak {p}}(j_E) < 0\). We have two subcases:
 (a)If \({{\mathrm{ord}}}_{\mathfrak {p}}(j_E) \equiv 0 \mod \ell \), we have that \(\mathfrak {p}\notin S_E\) and so \(E^d\) is not a Tate curve over \(K_{\mathfrak {p}}.\) Moreover, \(K_{\mathfrak {p}}(E[\ell ])\) is unramified over \(K_{\mathfrak {p}}\) and hence \(\overline{M_1}/K'\) and \(\overline{M_2}/K(\sqrt{d})\) are unramified at all divisors of \(\mathfrak {p}\) if and only if \(M_1/L_d\) (resp. \(M_2/L_d\)) are unramified at all divisors of \(\mathfrak {p}\). We now use the triviality of the \(\phi \in {{\mathrm{Sel}}}_{\ell }(E^d,K)\) over \(K_{\mathfrak {p}}\) from Lemma 4.2. Also recall that M is the fixed field of the kernel of \(\phi \). We shall show that \(\mathfrak {Q}_M\) is unramified over \(L_d\). There is a \(\widetilde{P} \in E^d(M_{\mathfrak {P}})\) where \(\mathfrak {P}_M\mathfrak {p}\) such that for all \(\sigma \in D(\mathfrak {P}_M)\), we have \(\sigma \widetilde{P}  \widetilde{P} = \phi (\sigma )\). Henceand so \(2 P'\) is in the connected component of unity modulo \(\mathfrak {p}\) via Remark 2.6. Hence \(\widetilde{P} = \widetilde{P}_1 + P_2 \) with \(P_2 \in E^d[\ell ]\) and \(2\widetilde{P}_1\) in the component of the unity of \(E \mod \mathfrak {P}_M\), so \(\widetilde{P}_1\) corresponds to a \(\mathfrak {P}_M\)adic unity u under the Tate parametrization. Now take$$\begin{aligned} P' := \ell \cdot \widetilde{P} \in E^d(K_{\mathfrak {p}}) \end{aligned}$$where \(I(\mathfrak {P}_M)\) is the interia group of \(\mathfrak {P}_M\). Then \(2(\alpha \widetilde{P}  \widetilde{P})\) corresponds to \(\alpha u / u\) and is an \(\ell ^{\text {th}}\) root of unity. By Hilbert’s Theorem 90, we have that \(\alpha = {{\mathrm{id}}}\), and thus, \(\mathfrak {P}_M\) is unramified over \(L_d\).$$\begin{aligned} \alpha \in \langle \alpha _1,\alpha _2 \rangle \cap I(\mathfrak {P}_M) \end{aligned}$$
 (b)If \({{\mathrm{ord}}}_{\mathfrak {p}}(j_E) \not \equiv 0\mod \ell \), then the values at the Hecke characters \(\chi \) of order \(\ell \) tell us that either E is a Tate curve over \(K_{\mathfrak {p}}\) or that \(\mathfrak {p}\in S_E\). Consider the former situation. Our assumptions from Theorem B tell us that \(\mathfrak {q}\) is not completely decomposed in \(K(\sqrt{d})\) and \(K'\). Sincefor all \(\mathfrak {P}_{K'}\mathfrak {p}\), we see that for all cyclic extensions \(\overline{M}_1\) of \(K'\) and \(\overline{M}_2/K(\sqrt{d})\) of degree \(\ell \) and divisors \(\mathfrak {P}_{M_i}\mathfrak {p}\), one has that \({{\mathrm{Gal}}}(\overline{M}_{i,{\mathfrak {P}_{M_i}}}/K_{\mathfrak {q}})\) is abelian of even order. But this implies that$$\begin{aligned} K_{\mathfrak {p}}^{\times }/(K_{\mathfrak {p}}^{\times })^{\ell }\cong K_{\mathfrak {p}}(\sqrt{d})^{\times }/(K_{\mathfrak {p}}(\sqrt{d})^{\times })^{\ell } \cong K'^{\times }_{\mathfrak {P}}/(K'^{\times }_{\mathfrak {P}})^{\ell } \end{aligned}$$which is absurd. Thus \(\mathfrak {p}\in S_E\) and our lemma follows.\(\square \)$$\begin{aligned} \overline{M}_{1,\mathfrak {P}} = K_{\mathfrak {p}}' \quad \text { and }\quad \overline{M}_{2,\mathfrak {P}} = K_{\mathfrak {P}}(\sqrt{d}), \end{aligned}$$
 (a)
The next step is to describe the behavior of \(\overline{M}_i\) at divisors of \(\mathfrak {l}\).
Lemma 5.8
Assume that \({{\mathrm{ord}}}_{\mathfrak {l}}(j_E) < 0\) and \(\phi \in {{\mathrm{Sel}}}_{\ell }(E^d,K)\). Then \(\overline{M}_2/K(\sqrt{d})\) is unramified at \(\mathfrak {l}\) and \(\overline{M}_1/K'\) is little ramified at divisors of \(\mathfrak {l}\).
Proof
Finally, we look at the case where \({{\mathrm{ord}}}_{\mathfrak {l}}(j_E) \ge 0\).
Lemma 5.9
Assume that E / K has a Krational point P of order \(\ell > 3\), that \({{\mathrm{ord}}}_{\mathfrak {l}}(j_E) \ge 0\), and that P is not contained in the kernel of reduction modulo \(\mathfrak {l}\), in particular, this means that E is not supersingular modulo \(\mathfrak {l}\). Let \(\phi \) be an element in \({{\mathrm{Sel}}}_{\ell }(E^d,K)\) with corresponding fields \(\overline{M}_1\) and \(\overline{M}_2\). Then \(\overline{M}_1/K'\) is little ramified at \(\mathfrak {l}\), and \(\overline{M}_2/K(\sqrt{d})\) is unramified at \(\mathfrak {l}\).
Proof
Suppose that \({{\mathrm{ord}}}_{\mathfrak {l}}(j_E) \ge 0\), which implies that E has potentially good reduction at \(\mathfrak {l}\). Since E / K has a Krational point P of order \(\ell >3\), we know that \({{\mathrm{Gal}}}(K(E[\ell ])/K(\zeta _{\ell }))\) is a subgroup of the additive group \(\mathbf F _{\ell }^+\). We want to show that all divisors of \(\mathfrak {l}\) are not ramified in \(K(E[\ell ])/K(\zeta _{\ell })\). If E has good reduction over \(K(\zeta _{\ell })\), then we are immediately done. If E does not have good reduction over \(K(\zeta _{\ell })\), then there must exist some extension \(N/K(\zeta _{\ell })\) such that \([N{:}K(\zeta _{\ell })]  6\) and that E has good reduction at all divisors \(\mathfrak {L}_N  \mathfrak {l}\); this divisibility condition is similar to the proof of [1, Proposition VII.5.4.c]. From our assumptions, it follows that \(N_{\mathfrak {L}}\) contains \(K(E[\ell ])\) and that \(\langle Q \rangle \) is the subgroup of order \(\ell \) of the kernel of reduction modulo \(\mathfrak {L}_N\). Hence all divisors of \(\mathfrak {l}\) are not ramified in \(K(E[\ell ])/K(\zeta _{\ell })\), and we can prove the lemma by looking at the behavior of \(\mathfrak {l}\) in \(M/L_d\).
Now assume that \(I(\mathfrak {L}_M) = \langle \alpha _2 \rangle \). Then \(Q = \alpha _2\widetilde{Q}  \widetilde{Q}\) and since \(\langle \alpha _2 \rangle \) acts trivially on \(\widetilde{E}(N\cdot M_{\mathfrak {L}})/\widetilde{E}_(N\cdot M_{\mathfrak {L}})\), we may assume that \(\widetilde{Q} \in \widetilde{E}_(N\cdot M_{\mathfrak {L}})\) and hence \(\ell \cdot \widetilde{Q} \in \widetilde{E}_(N\cdot K_{\mathfrak {l}}).\) Since \(\widetilde{E}\) has ordinary reduction modulo \(\mathfrak {L}_M\), we have that \(N\cdot K_{\mathfrak {l}}(\widetilde{Q})\) is little ramified at divisors of \(\mathfrak {l}\). Thus, our lemma follows. \(\square \)
5.3 Proof of Corollary E
Lemma 5.10
\({{\mathrm{cl}}}_{\emptyset }(K')[\ell ](\chi _{\ell }) \,  \, {{\mathrm{cl}}}(K(\sqrt{d}))[\ell ]\).
Proof
6 Elliptic curves satisfying Corollary D
Let E be an elliptic curve over a number field K. In a recent work [13], Zywina has described all known, and conjecturally all, pairs \((E/\mathbf Q ,\ell )\) such that mod \(\ell \) image of Galois, \(\rho _{E,\ell }(G_\mathbf{Q })\), is nonsurjective. Using Zywina’s classification, we can find elliptic curves \(E/\mathbf Q \) that will satisfy the conditions of Corollary D. First, we present an example of this technique for the case when \(\ell = 3\). We remark that this case does not apply to Corollary D; however, it best illustrates the technique.
Proposition 6.1
Let \(E/\mathbf Q \) have mod 3 image of Galois conjugate to B(3). Then \(\mathbf Q (E[3]) = \mathbf Q (x(E[3])) \cdot K\) where K is an explicitly computable quadratic extension.
Before we prove Proposition 6.1, we prove the following lemma which tells us over which extension E obtains a 3torsion point.
Lemma 6.2
For \(E/\mathbf Q \) from Proposition 6.1, there exists some quadratic extension K such that E has a Krational 3torsion point. In particular, \(E(K)[3] = \langle P \rangle \).
Proof
Let \(E:y^2 = x^3  Ax  B\) for \(A,B\in \mathbf Q \). Via the Weilpairing, we know that \(\mathbf Q (\zeta _3) \subseteq \mathbf Q (E[3])\). It is also a well known fact that \(B(3) \cong S_3 \times \mathbf Z / 2 \mathbf Z \). Combining these results with our assumptions, we have the following diagram of Galois subfields of \(\mathbf Q (E[3])\):
Remark 6.3
From the above proof, one can easily see that \({{\mathrm{Gal}}}(\mathbf Q (x(E[3]))/\mathbf Q ) \cong S_3\). Indeed, since \(\mathbf Q (x(E[3]))\) is Galois, we showed that the Galois group of \(\psi _3(x)\) is actually the Galois group of the cubic g(x). Since \([\mathbf Q (x(E[3])) : \mathbf Q ] = 6\), we know g(x) must be an irreducible cubic with nonsquare discriminant, which immediately implies our claim.
Proof of Proposition 6.1
(Proof of Proposition 6.1) Let K denote the quadratic extension from Lemma 6.2. It is clear that \(K \subset \mathbf Q (E[3])\) and that \(K \nsubseteq \mathbf Q (x(E[3]))\), so we have \(\mathbf Q (E(3))\) is the compositum of \(\mathbf Q (x(E[3]))\) and K. \(\square \)
Let \(\ell \in \left\{ 5,13 \right\} \). Below, we provide examples of elliptic curves \(E/\mathbf Q \) that do not have a \(\mathbf Q \)rational point of order \(\ell \) but attain a Krational point P of order \(\ell \) over some extension of small degree K that satisfies the conditions of Corollary D. The final step in our verification is showing P is not contained in the kernel of reduction modulo \(\mathfrak {l}\); in particular, this means that E / K is not supersingular modulo \(\mathfrak {l}\) if \({{\mathrm{ord}}}_{\mathfrak {l}}(j_E) \ge 0\). This condition is computable via the Magma command IsSupersingular.
In order to conduct a thorough search, we consider all subgroups H which can occur as an image of Galois for a nonCM \(E/\mathbf Q \) and satisfy the above containment. In particular, we run through a large list elliptic curves \(E/\mathbf Q \) with prescribed nonsurjective mod \(\ell \) image of Galois coming from the modular curves \(X_H\) of Zywina [13]. Since this list is comprehensive, we also give examples of elliptic curves over \(\mathbf Q \) that do not satisfy and potentially satisfy Corollary D, modulo some computations.
For \(\ell = 5\), we only have one example.
Example 6.3.1
For \(\ell = 7\), we have two possibilities.
For \(\ell = 11\), there do not exist any subgroups coming from [13] that have our desired condition. For \(\ell = 13\), we find a few examples of curves satisfying Corollary D.
Example 6.3.4
Example 6.3.5
Example 6.3.6
Potential example 6.3.7 (\(\ell = 13\)) Suppose that an elliptic curve \(E/\mathbf Q \) has mod 13 image conjugate to B(13). The curve E will attain a Krational point of order 13 over an extension of degree 12. The difficultly in verifying the conditions of Corollary D is computing the class number and ramification indicies for the duodecic extension K.
Finally for \(\ell = 37\), there is only one \(E/\overline{\mathbf{Q }}\) that we need to consider.
As before, the difficultly in verifying the conditions of Corollary D is computing the class number and ramification indicies for the duodecic extension K.
We say that \(d \in \mathcal {O}_K^{\times }/(\mathcal {O}_K^{\times })^2\) is negative if the image of d under each real embedding is negative.
Declarations
Open Access
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Acknowledgements
The author wishes to thank Ken Ono for initially suggesting this project, David ZureickBrown for his guidance and patience in explaining the finer details and for his help generalizing the conditions of [2] in a series of conversations, [19] for help in Lemma 5.5, and [20] for help in Remark 2.2. The computations in this paper were performed using the Magma computer algebra system [21]. For Magma code verifying the claims in Sect. 6, we refer the reader to [22].
Authors’ Affiliations
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