On elliptic curves with $p$-isogenies over quadratic fields

Let $K$ be a number field. For which primes $p$ does there exist an elliptic curve $E / K$ admitting a $K$-rational $p$-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that $K$ is a quadratic field, subject to the assumption that $E$ is semistable at the primes of $K$ above $p$. We prove results both for families of quadratic fields and for specific quadratic fields.


Introduction
One of the most important aspects of the study of elliptic curves is the investigation of the maps between them, and in particular their isogenies. Isogenies of prime degree are perhaps the most intriguing: a complete understanding would provide much insight into the arithmetic of elliptic curves, yet we still cannot answer some of the most basic questions about them. In this paper, we will investigate isogenies of prime degree over quadratic fields.
Given an elliptic curve E defined over a number field K, and a prime p, we say that E admits a K-rational p-isogeny if it admits an isogeny, ϕ, of degree p, satisfying ϕ σ = ϕ for any σ ∈ Gal(K/K). Equivalent formulations are that E has a K-rational subgroup of order p, or that the mod p Galois representation of E is reducible. We simply call an isogeny rational if it is Q-rational. The key question we would like to answer is the following: given a number field K, for which primes p does there exist an elliptic curve E/K admitting a K-rational p-isogeny? Thanks to the work of Mazur, we have a complete answer to this question over the rationals.
MSC2020 : 11F80, 11G05, 11G18. The author is supported by an EPSRC studentship and has previously used the name Philippe Michaud-Rodgers.
Theorem (Mazur,[15,Theorem 1]). Let p be a prime such that there exists an elliptic curve E/Q that admits a rational p-isogeny. Then p ∈ {2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67, 163}. Although this theorem was proven over 40 years ago, it has not been possible to obtain an analogous result for even a single other number field. Perhaps the most likely candidate for a similar result is a quadratic field of small discriminant. Recent work [2, p. 5] has shown that this is possible assuming the generalised Riemann hypothesis, although removing this assumption seems to be out of reach at this time.
Apart from the intrinsic interest of studying isogenies of elliptic curves, perhaps one of the most spectacular consequences of Mazur's theorem is the role it plays in the proof of Fermat's Last Theorem. More generally, in the 'modular approach' to studying Diophantine equations, one associates a Frey elliptic curve to a putative solution of a Diophantine equation, and applies Ribet's level-lowering theorem [20,Theorem 1.1] to relate this Frey curve to a modular form. A key hypothesis in applying Ribet's theorem at a given prime p is the non-existence of a rational p-isogeny.
More recently, the modular approach has been applied over various number fields, most commonly over real quadratic fields. See [13], [1], and [4] for a sample of papers that do this. In these examples, the Frey curve one constructs is defined over a number field, K, and in order to apply an analogue of Ribet's level-lowering theorem [11,Theorem 7], it is again necessary, for a given prime p, to rule out the existence of a K-rational p-isogeny. Since there is no analogue of Mazur's theorem over number fields, various methods have been used to achieve this. A further assumption in this analogue of Ribet's theorem is that the elliptic curve one is working with should be semistable at all primes of K above p, which one may view as a natural condition in its own right. With the assumption of semistability at the primes of K above p, it is possible to obtain results akin to Mazur's theorem, both for families of quadratic fields and for specific quadratic fields. Our main result is the following. Theorem 1. Let K be a real quadratic field and let ǫ be a fundamental unit of K. Let n be the exponent of the class group of K and assume n ≤ 3. Let p be a prime such that there exists an elliptic curve E/K which admits a K-rational p-isogeny and is semistable at all primes of K above p. Then either • p ramifies in K; or 3, 5, 7, 11, 13, 17, 19, 37}; or • p splits in K and p | Norm K/Q (ǫ 12 − 1).
Although this theorem only considers the case n ≤ 3, where n is the exponent of the class group of K, we may obtain similar results for larger values of n. For example, in Section 5 we consider the case n = 100. We also note that the list of primes appearing in this theorem is the smallest possible: for each p ∈ {2, 3,5,7,11,13,17,19, 37} and n ≤ 3 there exists a real quadratic field K with class group exponent n and an elliptic curve E/K which has a K-rational p-isogeny and is semistable at all primes of K above p (see Remark 5.2).
If we work over a fixed quadratic field, which is not imaginary of class number 1, then we can obtain more precise results. The following theorem considers certain 'small' quadratic fields, both real and imaginary. We highlight the fact that this is an 'if and only if' statement. It is also possible to produce similar results for quadratic fields with large class group exponent. As an example, in Section 5.3 we consider a quadratic field with class group Z/122Z.
We now outline the rest of the paper. In Section 2, we analyse the situation over the rationals, and prove a result analogous to Theorem 2. This result is a corollary to Mazur's theorem stated above. In Section 3, we study the mod p Galois representation of an elliptic curve with a p-isogeny, and we introduce the notions of isogeny characters and isogeny signatures. Next, in Section 4, by studying the ramification of these isogeny characters and by investigating certain properties of the modular curve X 0 (p), we see how the existence of an elliptic curve with a p-isogeny places stringent conditions on the prime p. This provides us with a method for ruling out the existence of such primes. In Section 5, we apply this method, combined with a study of quadratic points on modular curves, to prove Theorems 1 and 2. We also consider further examples.
The Magma [5] code used to support the computations in this paper can be found at: https://warwick.ac.uk/fac/sci/maths/people/staff/michaud/c/ I am grateful to my supervisors, Samir Siksek and Damiano Testa, for all their help and support. I would also like to thank Barinder Banwait for very helpful correspondence. Finally, I am grateful to the anonymous referee for many helpful suggestions that have improved the exposition of the paper.

Elliptic curves with rational p-isogenies
We start with a short analysis of the situation over the rationals. Let E/Q be an elliptic curve and let p be prime for which E admits a rational pisogeny. We will denote the kernel of this isogeny by V p , which is a rational cyclic subgroup of order p. The pair (E, V p ) then gives rise to a non-cuspidal point x ∈ X 0 (p)(Q). The study of the modular curve X 0 (p), and in particular the Eisenstein quotient of its Jacobian, allowed Mazur to prove his celebrated result [15,Theorem 1] (stated in the introduction), which classifies the primes p for which X 0 (p) has non-cuspidal rational points. This result allows us to obtain an analogue of Theorem 2 quite easily. Proof. Suppose first that E/Q is an elliptic curve which admits a rational p-isogeny and is semistable at p. By Theorem 1, it will suffice to rule out the primes p ∈ {11, 17, 19, 43, 67, 163}.
For each of these values of p, the modular curve X 0 (p) has only finitely many non-cuspidal rational points and we let x ∈ X 0 (p)(Q) denote one of these. Using Magma's small modular curve package, we can write down an elliptic curve F/Q with a rational subgroup W p of order p such that the pair (F, W p ) gives rise to the point x. In each case, the curve F (we have chosen) has additive potentially good reduction at p (so F is not semistable at p) and its j-invariant is not equal to 0 or 1728. We compute that 0 < v p (∆(F )) < 6 in each case. In particular, F is minimal at p. However, this alone is not enough to rule out the prime p. The pair (F, W p ) is one representative for the point x ∈ X 0 (p)(Q), and it is possible that a different representative is semistable at p. Suppose (F ,Ŵ p ) also represents the point x ∈ X 0 (p)(Q) for an elliptic curveF /Q with a rational subgroup of order p. We note that j(F ) = j(F ), soF also has potentially good reduction at p. The curves F andF are isomorphic (over Q), and since j(F ) = j(F ) / ∈ {0, 1728}, the curves are quadratic twists of each other (up to isomorphism over Q) by some squarefree d ∈ Z, and so we may replaceF by F d , where F d denotes the quadratic twist of F by d. Since It follows that 0 < v p (∆(F d )) < 12, so F d is minimal at p and F d does not have good reduction at p. So F d must have additive reduction at p.
For the converse, it suffices to find elliptic curves which have a rational p-isogeny and are semistable at p for p ∈ {2, 3, 5, 7, 13, 37}. Table 2 gives an example of such an elliptic curve in each case. We have chosen an elliptic curve of minimal conductor in each case.

Isogeny characters and isogeny signatures
We will now shift our attention to quadratic fields. In this section we will introduce two key concepts: isogeny characters and isogeny signatures. We will define these concepts in relation to our set-up, but they can be defined more generally for elliptic curves with p-isogenies over arbitrary number fields (see [3]). For the remainder of the paper, we will let p be a prime and let E/K be an elliptic curve over a quadratic field K such that: • E admits a K-rational p-isogeny; and • E is semistable at all primes p | p. In this section we will assume that: • p is unramified in K. We denote by ϕ this K-rational p-isogeny, and we write V p for its kernel throughout. The group V p is a K-rational cyclic subgroup of E[p] of order p. Write G K = Gal(K/K). The mod p Galois representation of E, ρ E,p : G K → GL 2 (F p ), is reducible, and we have that where χ p denotes the mod p cyclotomic character. We call λ : G K → F × p the isogeny character of (E, V p ). This character gives the action of G K on V p , and we can choose R ∈ V p such that for all σ ∈ G K , Throughout, we will let τ denote the generator of Gal(K/Q). By choosing an automorphism of K extending τ , we may also view τ as an element of Gal(K/Q). The following lemma describes the isogeny characters of (E τ , V τ p ) and (E/V p , E[p]/V p ). Lemma 3.1. Let λ be the isogeny character of (E, V p ).
Proof. These statements are well-known. We provide some short proofs here as we were unable to find any in the literature. For (i), let R be a generator of Applying τ , we see that So the isogeny character of (E τ , V τ p ) maps σ to λ(τ στ −1 ) as required. For (ii), let ϕ be the K-rational p-isogeny with kernel V p . This means that (E/V p , E[p]/V p ) = (ϕ(E), ϕ(E[p]). We fix a basis (R 1 , R 2 ) of E[p] so that R σ 1 = λ(σ)(R 1 ) for any σ ∈ G K . Then ϕ(R 2 ) generates ϕ(E[p]), and for any σ ∈ G K we have where b σ is the upper-right entry of the matrix ρ E,p (σ) (with respect to the basis (R 1 , R 2 )).
We will be particularly interested in studying the ramification of the character λ 12 . For a prime p of K above p, we will denote by I p the inertia subgroup of G K corresponding to p. . Let λ be the isogeny character of (E, V p ). Then λ 12 is unramified at the infinite primes of K and at all finite primes of K coprime to p. If p | p is a prime of K, then there exists a unique integer s p ∈ {0, 12} such that If s p = 0 then we see that λ 12 is unramified at p. We now fix, once and for all, a prime p 0 | p of K. We define the isogeny signature of (E, V p ) to be (s p 0 , s τ (p0) ). We will also refer to this as the isogeny signature of the associated character λ. This isogeny signature is one of (0, 0), (12,12), (12, 0), or (0, 12).
We will refer to (0, 0) and (12,12) as constant isogeny signatures, and we will refer to (12, 0) and (0, 12) as non-constant isogeny signatures. We note that if the prime p is inert in K, then the isogeny signature of (E, V p ) is constant, since τ (p 0 ) = p 0 . Primes p for which the isogeny signature of (E, V p ) is constant are referred to as Type 1 primes in [17] and [2]. If the isogeny signature of λ is (0, 0), then λ 12 is everywhere unramified. Remark 3.3. Our assumption that E is semistable at the primes of K above p forces the integer s p appearing in Proposition 3.2 to be 0 or 12. Without assuming this semistability condition, s p can also take the values 4, 6, and 8 (see [9, pp. 7-9]). This gives rise to many more possible isogeny signatures. In particular, one of the isogeny signatures which must be considered is (6,6). This is the isogeny signature which is the most difficult to rule out and it is the reason that the generalised Riemann hypothesis is assumed in [2] and [3]. In the case that K = Q, the case analogous to isogeny signature (6, 6) is considered by Mazur in [15, pp. 154-155], and is ruled out using some relatively elementary algebraic number theory to conclude that the class number of Q( √ −p) must be 1.
(iii) LetÊ/K be an elliptic curve with a K-rational subgroupV p of order p.
Proof. Parts (i) and (ii) follow from Lemma 3.1 and the definition of isogeny signature. For (iii), the curveÊ will be a twist of the curve E by a character, θ, whose order divides 2, 4, or 6. In particular, the order of θ divides 12 and it follows that the twelfth powers of the isogeny characters of (E, V p ) and (Ê,V p ) agree, and so the isogeny signatures must also agree. We refer to [8, pp. 6-9] for more on how ρ E,p is affected by twisting. We note that (iii) is stated in [17, p. 330].
A pair (E, V p ) gives rise to a K-rational point on X 0 (p). Part (iii) of Lemma 3.4 allows us to extend the definition of isogeny signature to noncuspidal points y ∈ X 0 (p)(K). We define the isogeny signature of a non-cuspidal point y ∈ X 0 (p)(K) to be the isogeny signature of any pair (Ê,V p ) representing y, forÊ an elliptic curve defined over K andV p a Krational subgroup of order p. If the isogeny signature of y is (a, b), then Parts (i) and (ii) of Lemma 3.4 show that: Here, w p denotes the Atkin-Lehner involution on X 0 (p).

Eliminating primes
Throughout this section we will again assume that p ≥ 17 and that p is unramified in K. We continue to denote by p 0 a fixed prime of K above p. We write O K for the ring of integers of K. We write λ for the isogeny character of (E, V p ), and (a, b) for the isogeny signature of (E, V p ).
For the remainder of the paper, we will write q for a prime of K above a rational prime q. We write n q for the norm of the ideal q, and we will denote by σ q ∈ G K a Frobenius element at q.
Our aim in this section is to see how to reduce the number of possibilities for p to a (small) finite set. Our strategy for bounding, and subsequently eliminating, p is based on the following key result.
Proposition 4.1. Let λ be the isogeny character of (E, V p ) with isogeny signature (a, b). Let q ∤ p be a prime of K, let r be the order of the class of q in the class group of K, and write q r = α · O K . Then Proof. This is a direct consequence of [17,Lemma 1]. We refer also to [12, Proposition 2.2] and [9, Proposition 2.6] for statements that use notation closer to ours. Indeed, following [12, Proposition 2.2], the quantity N s (α) is α a · (α τ ) b , and the prime q is the unique prime in the support of α.
This proposition can then be used to prove the following result, which will be crucial in our proof of Theorem 1.
Then the isogeny signature of (E, V p ) is constant.
Proof. Suppose the isogeny signature of (E, V p ) is (12,0) or (0, 12) (i.e. nonconstant). We will show that p | Norm K/Q (ǫ 12 − 1). In the notation of [12], N s (ǫ) = ǫ 12 or (ǫ τ ) 12 (according to whether the isogeny signature is (12,0) or (0, 12)), and applying [12,Corollary 3.2] gives that Taking norms, we have that This result will often allow us to focus on the case of a constant isogeny signature. We will need different methods to deal with the non-constant isogeny signatures if K is imaginary quadratic or if the prime factors of Norm K/Q (ǫ 12 − 1) are large.
We can now also obtain a bound on p. The following result is similar to [12,Theorem 2]. The key difference is that we have removed a factor of 2 from the exponent of 3 in the bound. Theorem 4.3. Let K be a real quadratic field and let ǫ be a fundamental unit of K. Write n for the exponent of the class group of K. Let p be a prime such that there exists an elliptic curve E/K which admits a K-rational p-isogeny and is semistable at all primes of K above p. Then either • p ramifies in K; or Moreover, if n = 1 then p ≡ 1 (mod 12) or p ≤ 19.
Proof. We assume p ≥ 17 with p unramified in K. We let V p denote the kernel of the K-rational p-isogeny of E, and write λ for the isogeny character of (E, V p ). We assume that if p splits in K then p ∤ Norm K/Q (ǫ 12 − 1) which ensures that the isogeny signature of E is constant (by Corollary 4.2). By if necessary, we may assume that the isogeny signature of (E, V p ) is (0, 0). So λ 12 is everywhere unramified and it follows that λ 12n = 1.
Let M denote the field cut out by λ 2 (the fixed field of the kernel of λ 2 ), which will be an extension of K of degree dividing 6n, and therefore have absolute degree dividing 12n. Then θ := λ| G M will either be trivial or a quadratic character. If θ = 1 then E has a point of order P defined over M . Otherwise, we twist E (viewed as a curve over M ) by the quadratic character θ, to obtain an elliptic curve with a point of order p defined over M . We then apply Oesterlé's torsion bound [10, Theorem 6.1], to obtain If n = 1, then we can obtain improved results. We have λ 12 = 1 and also λ p−1 = 1. So λ gcd(12,p−1) = 1. Therefore, if p ≡ 1 (mod 12) then λ 4 = 1 or λ 6 = 1. Applying the same argument as above, we conclude that there exists an elliptic curve with a point of order p over a field of absolute degree dividing 4 or 6. Applying the torsion bounds of [10, Theorem 1.2], we conclude that p ≤ 19 or p = 37, and since p ≡ 1 (mod 12), we must have p ≤ 19.
Remark 4.4. The idea used in this proof of applying a quadratic twist to reduce the degree of the field extension being considered is also used in [13, p. 888].
Although it is hidden within its proof, Corollary 4.2 (and consequently Theorem 4.3) relies on the fact that an elliptic curve will have a prime q of potentially good reduction. In what follows, we will want to choose specific primes q, and we will not know whether they are of potentially good or of potentially multiplicative reduction for E. This leads us to separate our analysis into two cases.

Primes of potentially good reduction
Let q be a prime of potentially good reduction for E. We will write q for the rational prime below q. Let r be the order of the class of q in the class group of K, and write q r = α · O K (like in Proposition 4.1). We start by recalling some facts about the Frobenius element σ q and its action on the p-adic Tate module of E, following [9, pp. [10][11]. The characteristic polynomial of Frobenius for E at q is given by P q (X) = X 2 − a q (E)X + n q .
Let β 1 , β 2 denote the roots of P q (X). Each root has absolute value √ n q .
The two roots are complex conjugates, and we write L = Q(β 1 ) for the field they generate. The field L is either Q or an imaginary quadratic field. Let P denote a prime of L above p. Then β i (mod P) ∈ F × p for i ∈ {1, 2}, and moreover The following result is a direct consequence of Proposition 4.1. We write Res for the resultant of two polynomials.  isogeny signature (a, b). Let q ∤ p be a prime of potentially good reduction for E, let r be the order of the class of q in the class group of K, and write q r = α · O K . Then If (a, b) = (0, 0), then p | Res(P q (X), X 12r − 1).
The problem with applying this proposition is that the trace of Frobenius, a q (E), is unknown. However, we know that a q (E) ≤ 2 √ n q . We Then a q (E) ∈ A q . The set A q only depends on n q and is therefore independent of the choice of prime q | q. We will therefore also write A q for this set.
Remark 4.6. Instead of using the set A q defined in (1), it is possible to run through all elliptic curves defined over the residue field of q to compute a set of possible values for a q (E). This is possible because E acquires good reduction at a totally ramified extension of the completion of K at q. This idea is used, for example, in (parts of) [2]. However, this slows down the computations we will perform in Section 5, and we did not find it led to improved results in any of the cases we considered.
In each case, R q is an integer. Also, R q is independent of the choice of prime q above q, and so we will also write R q for R q .
Corollary 4.7. Let λ be the isogeny character of (E, V p ) with isogeny signature (a, b). Let q be a prime of potentially good reduction for E. Then Proof. If q ∤ p, then the main statement follows directly from Proposition 4.5, and if q | p, then p = q and so p | R q too, which is why we have included a factor of q in the definition of R q . Finally, if (a, b) = (0, 0) and R q = 0, then for some a ∈ A q , the roots of X 2 − aX + n q (which are complex conjugate since a ∈ A q ) would both be roots of unity, and therefore their product, n q , would also be a root of unity, a contradiction.
The integer R q is independent of p. If q 1 , . . . , q t are several primes of potentially good reduction for E, then p | gcd(R q 1 , . . . , R qt ).
As we will see in Section 5, this idea will allow us to obtain a good bound on p. However, we have not yet used all the information at our disposal. We will now work with a fixed prime p that we would like to eliminate and we suppose that q ∤ p. We first note that we can cut down the possibilities for a q (E). The roots of the characteristic polynomial of Frobenius reduce to elements of F × p , and so we have that a q (E) 2 − 4 · n q is a square mod p.
We define We have that a q (E) ∈ A is independent of the choice of prime q | q and so we will also write A (p) q for this set. Now, by Proposition 4.1, we have that, and this is then used to conclude that p 0 | Res(P q (X), X 12r − α a (α τ ) b ), which gives Proposition 4.5. However, recalling that χ p (σ q ) ≡ n q (mod p), we can also see that Then n 12r An alternative way of seeing this is by swapping (E, V p ) with (E/V p , E[p]/V p ) and using Proposition 4.1 along with Lemma 3.1. For a ∈ A (p) q , let {γ a,1 , γ a,2 } ⊂ F × p denote the reductions of the roots of signature (a, b). Let q ∤ p be a prime of K of potentially good reduction for E, let r be the order of the class of q in the class group of K, and write q r = α · O K . Then for some a ∈ A (p) q , (at least) one of the following two conditions holds: a,1 = α a (α τ ) b (mod p 0 ) and γ 12r a,2 = α 12−a (α τ ) 12−b (mod p 0 ); or (ii) γ 12r a,2 = α a (α τ ) b (mod p 0 ) and γ 12r a,1 = α 12−a (α τ ) 12−b (mod p 0 ).
This gives us a strategy for eliminating a given prime p. Indeed, if for all a ∈ A  where h is the class number of K. This restores a certain symmetry and is sufficient to bound p. However, combining the two conditions places fewer conditions on p and reduces the chances of eliminating the prime p.

Primes of potentially multiplicative reduction
Let q be a prime of potentially multiplicative reduction for E. As before, we will write q for the rational prime below q, we let r be the order of the class of q in the class group of K, and write q r = α · O K . We would like to obtain results analogous to Proposition 4.5 and Corollary 4.7 for primes of potentially multiplicative reduction. If the isogeny signature of (E, V p ) is (a, b), then we start by defining The integer M q is independent of the choice of prime q | q, and so we will also write is as M q . Proof. Let λ be the isogeny character of (E, V p ), with isogeny signature (a, b). If q | p then the statement holds, so we will assume that q ∤ p. Let r be the order of the class of q in the class group of K, and write q r = α · O K . We then have that (see [9, Proposition 1.4] for example) λ 2 (σ q ) ≡ 1 or n 2 q (mod p).
Then, applying Proposition 4.1, we see that Taking norms, we see that and we conclude that p | M q . Finally, it is clear that M q = 0 if the isogeny signature of (E, V p ) is non-constant.
Unfortunately, M q = 0 for all primes q if the isogeny signature of (E, V p ) is constant, and so this result will not help us eliminate primes p in the case of a constant isogeny signature. We will use a different approach for this case.
As in Section 3 we will write x ∈ X 0 (p)(K) for the non-cuspidal point that the pair (E, V p ) gives rise to, and we recall that we extended the notion of isogeny signature to non-cuspidal points y ∈ X 0 (p)(K). We will denote the two cusps of X 0 (p) (which are defined over Q) by ∞ and 0. We write k q for the residue field of q. If y ∈ X 0 (p)(K), we denote by y kq the reduction of y mod q. Since q is a prime of potentially multiplicative reduction for E, we have that We will now assume that E has potentially multiplicative reduction at all primes q | q. Instead of working with the point x ∈ X 0 (p)(K), we will instead focus on the pair (x, x τ ), which we view as a rational point on the symmetric square of X 0 (p), which we write as We will denote by (x, x τ ) Fq the reduction of this rational point mod q. From (5), and our assumption that all primes above q are of potentially multiplicative reduction for E, we have that If the prime q does not split in K then there is a unique prime above q and it follows that (x, x τ ) Fq = (∞, ∞) Fq or (0, 0) Fq . We start by stating the following result (for which we do not need to assume that p is unramified in K). Let y ∈ X 0 (p)(K) and suppose (y, y τ ) Fq = (∞, ∞) Fq or (0, 0) Fq . Then y = ∞ or y = 0.
Proof. By applying the Atkin-Lehner involution w p (which swaps the cusps) to (y, y τ ) if necessary, we may assume that (y, y τ ) Fq = (∞, ∞) Fq , and we aim to prove that (y, y τ ) = (∞, ∞). We introduce the following notation. We write S = Spec(Z[1/p]), we write J 0 (p) for the Jacobian of X 0 (p), and we write J e (p) for the winding quotient of J 0 (p) (defined in as in [10, Definition 2.1]). We then consider the map which is the composition of the Abel-Jacobi map with basepoint 2∞ and the projection map to the winding quotient. This is the same map as the one considered in [14, p. 223], except that we project to the winding quotient instead of the Eisenstein quotient.
In order to prove that (y, y τ ) = (∞, ∞), following the argument of [14, p. 225], it suffices to verify that the map f p is a formal immersion along (∞, ∞) in characteristic q. This is precisely what is done in [3, p. 29-33] (in particular we refer the reader to the d = 2 row in Table 7 of this paper when q > 2 and the associated data file for the case when q = 2). As noted in [3, p. 32], this computation is really an extension of [10,Lemma 5.4] to the case of quadratic fields.
We will describe any pair of primes (q, p) satisfying the conditions of Proposition 4.11 as an admissible pair. The upper bound of 2357 on p in the case q = 2 is simply taken from [3, p. 32]. We expect this to hold for all p > 2357 and this bound could most likely be increased if desired. Being able to work with the prime q = 2 will provide useful information for the computations we carry out in the next section. Proposition 4.11 already tells us that if (q, p) is an admissible pair with q a prime that does not split in K, and the unique prime of K above q is of potentially multiplicative reduction for E, then E cannot have a K-rational p-isogeny for p ≥ 23 with p = 37. However, it does not consider the case (y, y τ ) Fq = (∞, 0) Fq , which is certainly possible if q splits in K. Our next result, from the author's own work, addresses this case (for which we do not need to assume that p is unramified in K). If the isogeny signature of x ∈ X 0 (p)(K) is constant, then the isogeny signatures of the points w p (x) and x τ differ (by Lemma 3.4), so w p (x) = x τ . This observation combined with Proposition 4.11 and Lemma 4.12 gives the following result.
Proposition 4.13. Let (E, V p ) be an elliptic curve over K with a K-rational subgroup of order p. Let q be a prime for which all primes of K above q are of potentially multiplicative reduction for E. Suppose that (q, p) is an admissible pair, and that if q = 2 then q is inert or ramified in K. Then the isogeny signature of (E, V p ) is non-constant.
In order to deal with the case in which q = 2 and 2 splits in K, we will use the following result.
Lemma 4.14. Suppose that the isogeny signature of (E, V p ) is constant. Suppose that (2, p) is an admissible pair such that 2 splits in K and both primes of K above 2 are of potentially multiplicative reduction for E. Let q 2 denote a prime of K above 2 and let r be the order of the class of q 2 in the class group of K. Then Proof. This may be viewed as a special case of part of [2, Proposition 4.1]. We may assume, by replacing (E, V p ) with (E/V p , E[p]/V p ) if necessary, that the isogeny signature of (E, V p ) is (0, 0). By Proposition 4.11, the point (x, x τ ) ∈ X 0 (p) (2) (Q) that (E, V p ) gives rise to must satisfy (x, x τ ) F 2 = (∞, 0) F 2 . It follows that either x kq 2 = 0 kq 2 or x τ kq 2 = 0 kq 2 . By replacing (E, V p ) by (E τ , V τ p ) if necessary, we may assume that x kq 2 = 0 kq 2 . We note that the isogeny signature remains (0, 0).
Since x kq 2 = 0 kq 2 , applying the argument of [2, p. 19] we must be in the second case: λ 2 (σ q2 ) ≡ n 2 q 2 (mod p). Since the isogeny signature of (E, V p ) is (0, 0), using Proposition 4.1 we obtain where we have used the fact that n q 2 = 2 in the final step.
Remark 4.15. In [17, p. 338], it is claimed that if the isogeny signature of (E, V p ) is constant, if q | q is of potentially multiplicative reduction for E, and if (q, p) is an admissible pair, then p − 1 | 12h, where h is the class number of K. The argument leading to this seems to be incorrect. A correct condition is p | n 12r q − 1, where r is the order of the class of q in the class group of K, which is part of [2, Proposition 4.1] (referred to in the proof of Lemma 4.14 above).
Remark 4.16. It is perhaps worth noting at this point that the results of this section could be suitably extended to number fields of larger degree (we refer to [3] for a selection of such results). There are two main reasons we have chosen to focus on the case of quadratic fields. Firstly, we can produce strong results in the case of quadratic fields. This is due to the fact that the methods we use turn out to be very effective in the case of quadratic fields. It is also due to the extensive work done on studying quadratic points on the modular curves X 0 (p) of small genus. Secondly, as discussed in the introduction, applications of the modular approach (for solving Diophantine equations) over totally real fields are most common over (real) quadratic fields, and so we hope that our results will be directly useful in this setting.

Computations
In this section, we apply the results of Section 4 to certain specific quadratic fields and families of quadratic fields. We start by outlining the basic strategy.
As in Sections 3 and 4, we let p be a prime and let (E, V p ) be an elliptic curve over a quadratic field K with a K-rational subgroup of order p, and we assume that E is semistable at the primes of K above p. For the moment, we do not make any further assumptions on p (in particular, p could ramify in K, or p could be < 17). Our strategy consists of three main steps. In each step we try and eliminate the prime p as a possible K-rational p-isogeny prime for E.
Step 1. Assume that p ≥ 23, p = 37, that p is unramified in K, and that the isogeny signature of (E, V p ) is constant.
Requiring p ≥ 23 and p = 37 means that (q, p) will be an admissible pair for any q = 2, p, and for q = 2 when p ≤ 2357 and p = 41. We may assume, by replacing (E, V p ) by (E/V p , E[p]/V p ) if necessary, that the isogeny signature of (E, V p ) is (0, 0).
We now choose auxiliary primes q 1 , . . . , q t , with q i ≥ 3 for all i. By Proposition 4.13, unless q i = p, it is not possible for both primes of K above q i to be of potentially multiplicative reduction for E, and so there is a prime q i | q i of potentially good reduction for E. We compute the integers R q i (which we recall are independent of the prime chosen above q i ), and applying Corollary 4.7 we have that p | gcd(R q 1 , . . . , R qt ).
If q i = p for some i, then p | R q i and so this still holds. This leaves us with a finite set of primes p which we are unable to eliminate (which we hope is fairly small).
For each remaining prime, we may then perform a finer analysis to try and eliminate it. We use Lemma 4.8 to try and achieve a contradiction with each prime q i . Further, if (2, p) is an admissible pair then we may also use q = 2 to try and eliminate p. Indeed, if 2 is inert or ramified in K, then by Proposition 4.13, the unique prime of K above 2 must be of potentially good reduction for E, and we may apply Lemma 4.8. Otherwise, we may apply Lemma 4.8 in combination with Lemma 4.14.
Step 2. Assume that p ≥ 17, that p is unramified in K, and that the isogeny signature of (E, V p ) is non-constant.
In this case, p must split in K. Also, by Cororally 4.2, if K is a real quadratic field with fundamental unit ǫ, then p | Norm K/Q (ǫ 12 − 1).
We now use auxiliary primes q 1 , . . . , q t ≥ 2. For each auxiliary prime q, either there is a prime of K above q which is of potentially good reduction for E, in which case p | R q by Corollary 4.7, or there is a prime of K above q which is of potentially multiplicative reduction for E, in which case p | M q by Proposition 4.10. In both cases, we have p | R q M q , and R q M q is independent of p for each q.
Step 3. We consider the primes p = 11, 17, and 19, the primes that ramify in K, and any primes that we were unable to eliminate in Step 1 and Step 2.
For each of these primes we study X 0 (p)(K) directly to try and eliminate it.
We note that it is not possible to eliminate the primes 2, 3, 5, 7, 13, and 37. To see this, it is enough to consider the base change to K of the elliptic curves appearing in Table 2.
The following lemma will be useful in Step 1 and Step 3.
Proof. Suppose p is one of the primes in (7). For p ≤ 73 the papers [7] and [6] compute all quadratic points x ∈ X 0 (p)(K) that satisfy w p (x) = x τ ; such points are called exceptional. The recent paper [18] does the same for p ≥ 79, and we will use this result for p = 101 in our example in Section 5.3.
In each paper, we simply consult the tables and read off the possible fields over which these exceptional quadratic points are defined.

Families of quadratic fields
We start by proving Theorem 1.
Proof of Theorem 1. We assume that there there exists an elliptic curve E/K which admits a K-rational p-isogeny and is semistable at the primes of K above p. We apply the strategy described at the start of this section. Even though the quadratic field K is not fixed, knowing the exponent, n, of the class group of K will be enough to do this. We will assume that p ≥ 23, p = 37, that p is unramified in K, and that if p splits in K then p ∤ Norm K/Q (ǫ 12 − 1). These assumptions on the prime p mean that the isogeny signature of (E, V p ) is constant, and we need only focus on Step 1 of the strategy outlined above. We are aiming to achieve a contradiction. We will choose several auxiliary primes q. For each auxiliary prime q we choose, and a prime q | q, we do not know two things: • whether q splits, is inert, or ramifies in K; and • the order, r, of the class of q in the class group of K.
If q is inert in K, then r = 1 and n q = q 2 . Otherwise, q is split or ramified in K, n q = q, and r | n. This means that for each q, we have 1 + d(n) possibilities for the pair (n q , r), where d(n) denotes the number of positive divisors of n. Computing the integer R q defined in (2) only requires this pair as input, and so we simply run through all 1 + d(n) possibilities, obtain an integer R q for each of these, and take their lowest common multiple at the end. We may then apply Lemma 4.8 to try and eliminate even more primes, again running through all 1 + d(n) possibilities. We use the auxiliary prime q = 2 if (2, p) is an admissible pair. Using the auxiliary primes 3 ≤ q ≤ 19, followed by q = 2, we were able to eliminate all possibilities for the prime p, other than p = 73 in the case n = 3, which we rule out by applying Lemma 5.1.
We note that it took under a tenth of a second to perform the necessary computations to obtain this result.
We have considered n ≤ 3 here, but we can also prove similar results for larger values of n, with the caveat that the size of the set of constant isogeny signature primes which we are unable to eliminate sometimes increases. For example, the constant isogeny signature primes we are unable to eliminate in the case n = 100 are p ∈ {97, 151, 241, 401, 601, 1201, 1801}.
We expect that we should be able to eliminate these extra primes, but the method we use does not achieve this.
Remark 5.2. In contrast to the case n = 100 above, as noted in the introduction, it is not possible to eliminate the primes p ≤ 19 or p = 37 appearing in Theorem 1, and we may construct the appropriate elliptic curves to verify this. Indeed, for a given field K, if p ∈ {2, 3, 5, 7, 13, 37} then we may use the base change to K of the corresponding elliptic curve appearing in Table  2. For p ∈ {11, 17, 19} we may search for a quadratic field K for which the elliptic curve X 0 (p) has positive rank over K and follow the strategy of the proof of Theorem 2 below. For n = 1, 2, and 3, it turns out we can use the fields K = Q( √ 29), Q( √ 10), and Q( √ 79) respectively. Alternatively, for n = 1 we could use Theorem 2 and use the fields K = Q( √ 2) and Q( √ 3). These computations are presented in the accompanying Magma files.
The following result demonstrates that knowing the splitting behaviour of certain primes can produce strong results. Theorem 5.3. Let K be a real quadratic field in which the primes 2 and 3 are inert, and let ǫ be a fundamental unit of K. Let p be a prime such that there exists an elliptic curve E/K which admits a K-rational p-isogeny and is semistable at all primes of K above p. Then either • p ramifies in K; or 3, 5, 7, 11, 13, 17, 19, 37}; or • p splits in K and p | Norm K/Q (ǫ 12 − 1).
Proof. We proceed as in the proof of Theorem 1 and need only consider Step 1 of the strategy outlined at the start of this section. We assume that p ≥ 23, p = 37, p is unramified in K, and that the isogeny signature of (E, V p ) is constant. We aim to obtain a contradiction. We start by using the auxiliary prime q = 3. Since 3 is inert in K, we know that the unique prime above 3 is a principal ideal and has norm 3 2 . By Proposition 4.13, the unique prime above 3 must be of potentially good reduction for E and so p | R 3 . The largest prime factor of R 3 is 1489 and 41 ∤ R 3 , so we may also now use the auxiliary prime q = 2, since (2, p) will be admissible for all remaining primes p. We again apply Proposition 4.13 to conclude that the unique prime above 2 must be of potentially good reduction for E, and so p | R 2 . So p | gcd(R 3 , R 2 ) = 754471972800 = 2 6 · 3 4 · 5 2 · 7 2 · 13 2 · 19 · 37, giving the required contradiction.
Similarly to Theorem 1 above, we cannot eliminate the primes p ≤ 19 or p = 37 from the statement of this theorem. This can be seen by considering the field K = Q( √ 29) again (as in Remark 5.2) in which the primes 2 and 3 are inert.

Small real quadratic fields
We will consider the real quadratic fields K = Q( √ d) for d ∈ {2, 3, 5, 6, 7}. Each of these fields has class number 1.
its Mordell-Weil group. We may then write down, using Magma's 'small modular curve package', an elliptic curve E/K with a K-rational p-isogeny representing the point mQ for (small) integers m, and test its reduction type at each prime above p. In each case, we found a suitable elliptic curve using m = 1 or 2.

An example with large class group
We consider the quadratic field K = Q( √ d), with d = 47 · 67 · 101. The class group of K is Z/122Z.
There exists an elliptic curve E/K which admits a K-rational p-isogeny and is semistable at all primes of K above p if and only if p ∈ {2, 3, 5, 7, 11, 13, 19, 37} Although the class group of K is large, our quadratic field is now fixed, and we know the splitting behaviour of each auxiliary prime q in K, as well as the order of any q | q in the class group of K.
Proof. We start by following Step 1 of the strategy described at the start of this section with the auxiliary primes 3 ≤ q ≤ 20, followed by q = 2. We were able to show that if p ≥ 23 with p = 37, then the isogeny signature of (E, V p ) cannot be constant.
We then proceed with Step 2 and compute Norm(ǫ 12 − 1), for the fundamental unit ǫ = 13535 + 24 √ d. Although this has several large prime factors, it in fact has no prime factors ≥ 23 that split in K.
Next, we continue with Step 3. We start by eliminating the prime p = 17 like in the proof of Lemma 5.4. It remains to eliminate the primes that ramify in K, namely p = 47, 67, and 101. We work directly with the corresponding modular curves X 0 (p). By Lemma 5.1, each gives rise to a point x ∈ X 0 (p)(K) satisfying w p (x) = x τ . By [6, p. 337], the modular curve X 0 (67) has a single non-cuspidal rational point, and no points defined over real quadratic fields. Applying the arguments of Corollary 2.1 (see also the proof of Lemma 5.4), the pair (E, V p ) will not give rise to the non-cuspidal rational point, and so we may eliminate p = 67. Next, we consider p = 47. The curve X 0 (47) is hyperelliptic and the Atkin-Lehner involution coincides with the hyperelliptic involution. We therefore obtain a rational point on the quadratic twist of X 0 (47) by d. However, this twisted curve has no points over Q 101 , and so we obtain a contradiction. Finally, if p = 101, then we would obtain a rational point on the twisted modular curve X (d) 0 (101) (see [19, pp. 323-324]). In this case we may apply [19, Theorem 1.1 Part (5)] to obtain a contradiction. To see this, we start by writing M = Q( √ −101). The prime 67 ramifies in K, is unramified in M , and each prime of M above 67 is not principal (and therefore not totally split in the Hilbert class field of M ). This proves that X For the converse, it suffices to write down suitable elliptic curves for p = 11 and p = 19. We argue exactly as in the proof of Theorem 2 (for d > 0) above. The only difference is that we were unable to compute the Mordell-Weil group of X 0 (p)(K) directly using Magma. Instead, we find suitable points by first working with the quadratic twist of X 0 (p) by d.

An imaginary quadratic field
We consider the imaginary quadratic field K = Q( √ −5) which has class number 2. We note that we cannot obtain a finite list of possible primes in the case that K is imaginary quadratic of class number 1. This is because, in this case, if the isogeny signature of (E, V p ) is non-constant, then R q = 0 for all primes q. This is unsurprising, since if K is any number field that contains the Hilbert class field of an imaginary quadratic field then there are infinitely many primes for which there exist curves which admit K-rational p-isogenies (see [2, p. 2] for more details on this).
Proof of Theorem 2 for d = −5. If the isogeny signature of (E, V p ) is constant then we may use the proof of Theorem 1 to eliminate all primes p ≥ 23 with p = 37. We will therefore focus on the case that the isogeny signature of (E, V p ) is non-constant and proceed with Step 2. As usual, by interchanging (E, V p ) with (E/V p , E[p]/V p ) if necessary, we may assume that the isogeny signature of (E, V p ) is (12,0). We use the auxiliary primes 3 and 7. We have that for p ≥ 17, p | gcd(R 3 M 3 , R 7 M 7 ), and this tells us that p ∈ {17, 43, 71}. The prime 71 does not split in K, so we may eliminate it. Next, we proceed with Step 3 and eliminate the primes 11, 17, and 19 as in the proof of Lemma 5.4. For the converse, we must exhibit an appropriate elliptic curve when p = 43. The curve X 0 (43) is of genus 3. We start by searching for rational points on the elliptic curve X + 0 (43), and pull them back to try and find a point x ∈ X 0 (43)(K)\X 0 (43)(Q). We were able to do and then using Magma we wrote down a representative elliptic curve E defined over K. We found that this curve was not semistable at p 0 | 43. However, the quadratic twist of E by a certain element of valuation 1 at p 0 has good reduction at both primes of K above 43, and this twisted elliptic curve will still have a K-rational 43-isogeny.