Tamagawa numbers of elliptic curves with prescribed torsion subgroup or isogeny
aa r X i v : . [ m a t h . N T ] F e b TAMAGAWA NUMBERS OF ELLIPTIC CURVES WITH PRESCRIBED TORSIONSUBGROUP OR ISOGENY
ANTONELA TRBOVIĆ
Abstract.
We study Tamagawa numbers of elliptic curves with Z / Z ⊕ Z / Z torsion and of ellipticcurves with an n − isogeny, for n ∈ { , , , , , , , , , , , , } . We find that Tamagawanumbers of elliptic curves with torsion Z / Z ⊕ Z / Z are always divisible by , with factors comingfrom rational primes with split multiplicative reduction of type I k , one of which is always p = 2 . Theonly exception is the curve 1922c1, with c E = c = 14 . As for n − isogenies, Tamagawa numbers ofelliptic curves with an − isogeny must be divisible by 4, while elliptic curves with an n − isogeny forthe remaining n from the mentioned set must have Tamagawa numbers divisible by 2, except for finitesets of specified curves. Introduction
Let E be an elliptic curve over a number field K and denote by Σ the set of all finite primes of K .For each v ∈ Σ , K v will denote the completion of K at v and k v = O K v / ( π ) the residue field of v , where O K v is the ring of integers of K v and π is a uniformizer of O K v . The subgroup E ( K v ) of E ( K v ) consists of all the points that reduce modulo π to a non-singularpoint of E ( k v ) . It is known that this group has finite index in E ( K v ) so we can define the Tamagawanumber c v of E at v to be that index, i.e. c v := [ E ( K v ) : E ( K v )] . Consequently, we define the Tamagawa number of E over K to be the product c E/K := Q v ∈ Σ c v . Wewill write c E instead of c E/K wherever it does not cause confusion.It makes sense to study how the value c E depends on E ( K ) tors , since c E / E ( K ) tors appears as afactor in the leading term of the L − function of E/K in the conjecture of Birch and Swinnerton-Dyer(see, for example, [7, Conj. F.4.1.6]).Some results on Tamagawa numbers of elliptic curves with a specific torsion subgroup and on thequotient c E / E ( K ) tors are given by Lorenzini in [12, Chapter 2] for elliptic curves over the rationalsand over quadratic extensions. Krumm [9, Chapter 5] proved some further results on Tamagawa numbersof elliptic curves with prescribed torsion over number fields of degree up to 5. He also conjectured that ord ( c E ) is even for all elliptic curves defined over quadratic fields with a point of order 13 and thesame conjecture was later proved by Najman in [14].In this paper we explore this problem further and prove in Section 2 that the Tamagawa numbers ofelliptic curves defined over cubic fields with torsion subgroup Z / Z ⊕ Z / Z are always divisible by ,except in the case of the curve 1922c1 in [4], where c E = c = 14 . For each such curve we prove that at p = 2 the reduction is split multiplicative, so c = 14 k, and there always exists one more prime, distinctfrom 2, at which the reduction is also split multiplicative of type I t . The question which naturally appears next is how does the Tamagawa number of an elliptic curvedepend on the isogenies of that elliptic curve. In Section 3 we give a series of propositions which gives usfirst results about Tamagawa numbers of elliptic curves with prescribed isogeny. For elliptic curves definedover Q , we were able to prove that if an elliptic curve has an − isogeny, then its Tamagawa numberis always divisible by , and if it has an n − isogeny, for n ∈ { , , , , , , , , , , , , } ,then it has to be divisible by 2. There are finitely many exceptions for some of these results, all of whichwe list and give their Tamagawa numbers.Let E be an elliptic curve defined over K v , given by a Weierstrass equation y + a xy + a y = x + a x + a x + a . Date : February 10, 2021.The author was supported by the QuantiXLie Centre of Excellence, a project co-financed by the Croatian Governmentand European Union through the European Regional Development Fund - the Competitiveness and Cohesion OperationalProgramme (Grant KK.01.1.1.01.0004) and by the Croatian Science Foundation under the project no. IP-2018-01-1313. with the discriminant ∆ , invariants c and c , and j − invariant j E = c ∆ . It will be important for us todistinguish between different types of reductions at finite primes, especially to know when the reductionis multiplicative. For that, we will often use the following well known result.
Proposition 1.1. (see [16, Proposition VII.5.1.b])
With the above notation, the curve E in its minimalmodel has multiplicative reduction at v of type I k if and only if k := ord v (∆) > and ord v ( c ) = 0 . As most Tamagawa numbers that we will consider in this paper are coming from primes of multi-plicative reduction, it will be important to also distinguish between split and non-split multiplicativereductions and their Tamagawa numbers. One way to do that is by using the algorithm of Tate [17,Sections 7,8] which works in any characteristic of k v . Going through the algorithm with a specific ellipticcurve and a prime p , we get the reduction type at p , its Kodaira symbol and the Tamagawa number c p . It turns out that in the case of split multiplicative reduction I k we have c v = k and in the case ofnon-split multiplicative reduction I k we have c v = 1 or c v = 2 , depending on the parity of k , as indicatedin Table 1, where we can find all the Tamagawa numbers associated to different reduction types. Fordistinguishing reduction types in char ( k v ) = 2 , one can also use the tables in [16, Table 15.1] or [17,Section 6]. reduction type at v Kodaira symbol, k ≥ Tamagawa number at v good I I k k non-split multiplicative I k non-split multiplicative I k − additive II, II ∗ III, III ∗ IV, IV ∗ I ∗ , , potentially multiplicative I ∗ k , potentially multiplicative I ∗ k − , Table 1. types of reduction and their Tamagawa numbersThe computations in this paper were executed in the computer algebra system Magma [1]. The codeused in this paper can be found at https://web.math.pmf.unizg.hr/~atrbovi/magma/magma3.htm .Many of the proofs in this paper omit the information used in them, such as polynomials of very highdegree or with large coefficients, but those can be computed with the given code. For the untrustingreader, we recommend that they go through the proofs and the code simultaneously.All of the specific curves will be mentioned using their Cremona labels, with a clickable link to thecorresponding webpage in [10].2.
Tamagawa numbers of elliptic curves with torsion subgroup Z / Z ⊕ Z / Z Filip Najman and the author have examined the reduction types of primes with multiplicative re-duction of the elliptic curves with torsion Z / Z ⊕ Z / Z over a cubic field in [15, Prop. 3.1]. We willexamine those primes further, as we want to be able to say more about their Tamagawa numbers. Itwas proved in [15, Prop. 3.2] that the prime 2 always has multiplicative reduction of type I k in cubicextensions over which X (2 , has a non-cuspidal point. In this section we are going to prove that thementioned multiplicative reduction always has to be split multiplicative, giving the Tamagawa number c = 14 k , as shown in Table 1. We are also going to prove that there always exists one more prime p ,with the exception of the curve 1922c1, in which we have split multiplicative reduction of type I t and c p = 14 t , which means that the Tamagawa number of the elliptic curve contains the factor . Bruin and Najman [2] showed that every elliptic curve with torsion Z / Z ⊕ Z / Z over a cubic fieldis a base change of an elliptic curve defined over Q . Those elliptic curves are also parameterized with P ( Q ) , so we can write each such curve as E u , for some u ∈ Q . They also provided a model, which wasused for obtaining the results of [15, §3]. We used a different model here, specifically, the one given byJeon and Schweizer in [8, §2.4], since the one in [15] was dependant on 2 parameters. It did not imposea problem there, since we did not have the need to work with the coefficients of the curve. Even though
AMAGAWA NUMBERS OF ELLIPTIC CURVES WITH PRESCRIBED TORSION SUBGROUP OR ISOGENY 3
Jeon and Schweizer do not state that their family consists of all elliptic curves over cubic fields withtorsion Z / Z ⊕ Z / Z , it turns out that it is the case and the reasoning behind it can be found in theaccompanying Magma code. Briefly, we compute the isomorphism between different fields of definitionof elliptic curves with torsion Z / Z ⊕ Z / Z , those are F and L given in [2] and [8, §2.4], respectively.With that isomorphism we map every curve from the family in [2] and we see that it is isomorphic toone of the curves from the family in [8, §2.4]. Since [2] gives us all of the elliptic curves with neededproperties, we see that it suffices to only look at the family from [8, §2.4].Jeon and Schweizer provided two models for E u , one of which is y + xy = x + A ( u ) x + A ( u ) x + A ( u ) , and its short Weierstrass model y = x + A ( u ) x + B ( u ) , where we omit A ( u ) , A ( u ) , A ( u ) , A ( u ) , B ( u ) , since they are very large, but thay can be found in theaccompanying Magma code or in [8, §2.4]. We will be working with the long Weierstrass model whenconsidering the reduction at the prime 2, but generally we will be using the short Weierstrass model,since it is easier to work with.In Proposition 1.1 we mentioned a way of confirming whether the curve has multiplivcative reductionat a finite prime. As already stated, it will be very important to distinguish between split and non-split multiplicative reduction, since the associated Tamagawa numbers are different (see Table 1). Thefollowing lemma will be useful in differentiating between those, and it is taken directly from a step inTate’s algorithm. Lemma 2.1. ([17, §7. Case 2)])
Let E be an elliptic curve and let p be a prime of multiplicative reductionof type I t for E . Let ord p ( a i ) > , for i = 3 , , , and ord p ( b ) = 0 . If T + a T − a splits over k p , then E has split multiplicative reduction at p and c p = t. As a part of the proof of the following proposition we will show that the reduction at the prime 2 ismultiplicative of type I k , which is already proved in [15, Proposition 3.2]. We had to include it hereagain and could not continue from there because of the already mentioned differences in the models weused. Proposition 2.2.
Let E be an elliptic curve defined over a cubic field with torsion subgroup Z / Z ⊕ Z / Z . Then the reduction at is split multiplicative of type I k and c = 14 k. Proof.
From the long Weierstrass model of E u from [8, §2.4] we get the associated discriminant and the c − invariant: ∆( u ) = 2 ( u − ( u + 1) f ( u ) f ( u ) f ( u ) , c ( u ) = f ( u ) f ( u ) f ( u ) , where f i ( u ) are monic polynomials of degree i , which can be computed with the accompanying Magmacode. We will go through all of the possibilities of the prime 2 dividing u and see that the reduction at2 in all of those cases is split multiplicative and | c . • If ord ( u ) > , then it is obvious from the polynomials above that ord (cid:16) ∆( u )2 (cid:17) = ord ( c ( u )) = 0 and from Proposition 1.1 we conclude that the reduction at 2 is multiplicative of type I . Wecompute a = 1 and ord ( a ) > and since our model satisfies the conditions of Lemma 2.1, weget that c = 14 . • If ord ( u ) < , then we make the substitution u m so ord ( m ) > , and in the new model weget ∆( m ) = 2 ( m − m ( m + 1) g ( m ) g ( m ) g ( m ) , c ( m ) = 37 g ( m )3 g ( m ) g ( m ) , where g i ( m ) are monic polynomials of degree i , which can be computed with the accompanyingMagma code. Since ord ( m ) > , it is obvious from the polynomials that ord (cid:16) ∆( m )2 m (cid:17) = ord ( c ( m )) = 0 and as in the previous case, using Proposition 1.1 and Lemma 2.1 we get thatthe reduction at 2 is split multiplicative of type I k +1) and c = 14( k + 1) . • If ord ( u ) = 0 , then ord ( u − > . After the substitution u − m we have k := ord ( m ) > and ∆( m ) = 2 m ( m + 2) h ( m ) h ( m ) h ( m ) , c ( m ) = h ( m ) h ( m ) h ( m ) , where h i ( u ) are monic polynomials of degree i , which can be computed with the accompanyingMagma code. AMAGAWA NUMBERS OF ELLIPTIC CURVES WITH PRESCRIBED TORSION SUBGROUP OR ISOGENY 4
We can divide both numerator and the denominator of ∆( m ) with and we get ord (∆( m )) =14( k − and if we divide the numerator and the denominator of c ( m ) with and we get ord ( c ( m )) = 0 . So if k > , by Proposition 1.1 we have that the reduction at 2 is multiplicativeof type I k − . We compute a = 1 and ord ( a ) > (after dividing both numerator and thedenominator with ) and since our model satisfies the conditions of Lemma 2.1, we get that c = 14( k − . Obviously we have to look at the case k = 1 separately. This means that u = 2 n + 1 , where ord ( n ) = 0 . After the substitution u n + 1 we have ∆( n ) = n ( n + 1) p ( n ) p ( n ) p ( n ) , c ( n ) = p ( n ) p ( n ) p ( n ) , where p i ( n ) are monic polynomials of degree i , which can be computed with the accompanyingMagma code. Since ord ( n ) = 0 , we have t := ord ( n + 1) > and ord ( p i ( n )) = 0 , for each i ,so ord (∆( n )) = 14 t and ord ( c ( n )) = 0 . By Proposition 1.1 we see that the reduction at 2 ismultiplicative of type I t and similarly as in previous cases, Lemma 2.1 gives that the reductionis split multiplicative with c = 14 t. (cid:3) In the following proposition we will deal with primes distinct from 2, for which we have a simplerway of determining split multiplicative reduction than going through Tate’s algorithm as we did in theprevious proposition.
Lemma 2.3. ([3, Lemma 2.2])
Let p = 2 and let E be an elliptic curve defined over Q p with multiplicativereduction at p . The reduction is split multiplicative if and only if − c is a square in F × p . Proposition 2.4.
Let E be an elliptic curve defined over a cubic field with torsion subgroup Z / Z ⊕ Z / Z . Then there exist at least 2 rational primes with split multiplicative reduction of type I k , one ofwhich is always the prime , so | c E , except for the curve c , where c E = c = 14 . Proof.
In Proposition 2.2 we have already seen that the reduction at 2 is split multiplicative of type I k and therefore c = 14 k. It remains to prove that there exists one more prime with the same property foreach of those curves.From the short Weierstrass model of E u from [8, §2.4] we get the associated discriminant and the c − invariant: ∆( u ) = 2 ( u − ( u + 1) f ( u ) f ( u ) ,c ( u ) = f ( u ) f ( u ) f ( u ) , where f i ( u ) are polynomials of degree i , which can be computed with the accompanying Magma code. • Assume that there exists a prime p such that k := ord p ( u − > . Let res ( f, g ) denote theresultant of the polynomials f and g . We compute res (cid:18) u − , ∆( u )( u − (cid:19) = 2 ,res ( u − , c ( u )) = 2 . For p = 2 this means that p k | ∆( u ) and p ∤ c ( u ) and from Proposition 1.1 we find that thereduction of E at p is multiplicative of type I k . We want to see that c p = 14 k, i.e. that thereduction at p is split multiplicative. According to Lemma 2.3, it will suffice to check the valueof − c modulo p . Having in mind that u ≡ p ) , we get that − c ≡ (mod p ) , which isa square mod p . • Assume now that there exists a prime p such that k := ord p ( u − < . We put m := u − so ord p ( m ) = k > and we get an elliptic curve with ∆( m ) = 12 m ( m + 1 / g ( m ) g ( m ) ,c ( m ) = g ( m ) g ( m ) g ( m ) , where g i ( m ) are polynomials of degree i , which can be computed with the accompanying Magma code. res (cid:18) m, ∆( m ) m (cid:19) = 2 − ,res ( m, c ( m )) = 2 − . AMAGAWA NUMBERS OF ELLIPTIC CURVES WITH PRESCRIBED TORSION SUBGROUP OR ISOGENY 5
For p = 2 this means that p k | ∆( m ) and p ∤ c ( m ) and from Proposition 1.1 we find that thereduction of E at p is multiplicative of type I k . Having in mind that m ≡ p ) , we getthat − c ≡ − (mod p ) , which is a square mod p , so by Lemma 2.3 we have c p = 14 k. So far we have proved that if we have a prime p = 2 and k := ord p ( u − = 0 , then we have splitmultiplicative reduction at p with c p = 14 | k | . We have several possibilities when ord p ( u −
1) = 0 andthose are u − or u − ± k , k ∈ Z . When u − ± k , k = 0 , , then ord p ( u +1) > , for some prime p = 2 . In the cases u − ± k , k = 0 , ,or u − we get that u ∈ { , ± , } . For u = ± we get a singular curve and for u ∈ { , } we get thesame curve, c , with c E = c = 14 . Therefore, if we have a curve distinct from c , it certainly has a prime p such that ord p ( u − = 0 or ord p ( u + 1) > . It remains to see what happens in the case ord p ( u + 1) > . • Assume that there exists a prime p such that k := ord p ( u + 1) > . We compute res (cid:18) u + 1 , ∆( u )( u + 1) (cid:19) = 2 ,res ( u + 1 , c ( u )) = 2 . For p = 2 this means that p k | ∆( u ) and p ∤ c ( u ) and from Proposition 1.1 we find that thereduction of E at p is multiplicative of type I k . Having in mind that u ≡ − p ) , we getthat − c ≡ (mod p ) , which is a square mod p , so by Lemma 2.3 we have c p = 14 k. (cid:3) Tamagawa numbers of elliptic curves with prescribed isogeny
In [13, Table 3] we can find the j − invariants of elliptic curves parameterized by points on modularcurves X ( n ) defined over Q , for X ( n ) of genus , and in [13, Table 4] there are j − invariants of ellipticcurves parameterized by points on modular curves X ( n ) defined over Q , with genus of X ( n ) largerthan . In this section we will examine the properties of Tamagawa numbers of elliptic curves definedover Q with an n − isogeny, i.e. the properties of Tamagawa numbers of elliptic curves obtained from thementioned j − invariants.In Section 2 we worked with a specific model for the curve X (2 , . However, the points on X ( n ) give us j − invariants of curves with an n − isogeny, which give us elliptic curves up to a twist, so now, asopposed to the situation in Section 2, we also have to take into consideration the twists of the curveswe get from those j − invariants. Therefore, we will be interested in how the reduction types at primes p ∈ Q change under the twisting of the curve.Let E be an elliptic curve, which will always be defined over Q in this section. Denote by E d itsquadratic twist by d , where d is a squarefree integer. When p = 2 , the reduction type change is quitestraightforward, and is presented in Table 2. In essence, if p ∤ d, the reduction type does not change, andwhen p | d, reduction types change as indicated in the third column.reduction typeof E at p reduction typeof E d at p ∤ d reduction typeof E d at p | dI I I ∗ I m I m I ∗ m II II IV ∗ III III III ∗ IV IV II ∗ I ∗ I ∗ I I ∗ m I ∗ m I m IV ∗ IV ∗ IIIII ∗ III ∗ IIIII ∗ II ∗ IV Table 2. change of reduction types at p = 2 under twisting [5, Prop.1]When p = 2 , the situation gets more complicated. As most of the relevant Tamagawa numbers we willhave in the following proofs come from primes of multiplicative reduction, we give a lemma that will be AMAGAWA NUMBERS OF ELLIPTIC CURVES WITH PRESCRIBED TORSION SUBGROUP OR ISOGENY 6 especially useful for dealing with quadratic twists of a large family of elliptic curves with multiplicativereduction at p = 2 . Lemma 3.1. ([6, Thm.A.5], [11, Thm.2.8])
Let E be an elliptic curve with multiplicative reduction oftype I n at p = 2 . Denote by E d the twist of E by d, where d is a squarefree integer.(a) If d ≡ , , then the reduction of E d at p is of type I ∗ n . (b) If d ≡ , then the reduction of E d at p is of type I n . For other types of reduction, some results can also be found in [5, Section 2]. Since we will deal herewith only finitely many explicitly known elliptic curves with non-multiplicative reduction at p = 2 , forthose curves we can simply check all of the possibilities for reduction type at p = 2 of quadratic twists,since Q × / (cid:0) Q × (cid:1) = h− , , i . Proposition 3.2.
Let E be an elliptic curve over Q with an − isogeny. Then | c E , except for thecurves a , a , a , a , where c E = 2 .Proof. From [13, Table 3] we take the parameterization of the j − invariants of the curves that are non-cuspidal points on X (18) , j ( h ) = ( h − ( h − h − h − h ( h − h + 1) , h ∈ Q . From it we can acquire the discriminant and the c − invariant of the minimal model up to a twist, ∆( h ) = ( h − h ( h + 1) ( h − h + 1) ( h + 2 h + 4) f ( h ) f ( h ) f ( h ) f ( h ) ,c ( h ) = f ( h ) f ( h ) f ( h ) f ( h ) , where f i ( h ) are polynomials of degree i , which can be computed with the accompanying Magma code.Assume that there exists a prime p such that k := ord p ( h + 1) > . We compute res (cid:18) h + 1 , ∆( h )( h + 1) (cid:19) = 3 ,res ( h + 1 , c ( h )) = 3 . If p = 3 , this means that p k | ∆( h ) and p ∤ c ( h ) , so from Proposition 1.1 we find that the reduction of E at p is multiplicative of type I k , and therefore c p is even (see Table 1). If there exists a second prime p ′ distinct from p and 3 with k ′ := ord p ′ ( h + 1) > , then we have another prime with multiplicativereduction of type I k ′ and therefore with even c p ′ . Assume now that there exists a prime p such that k := ord p ( h + 1) < . We put m := h +1 and afterthe substitution x x · − , y y · − we get an elliptic curve with ∆( m ) = ( m − (3 m − m (3 m − m + 1) (3 m + 1) g ( h ) g ( h ) g ( h ) g ( h ) , up to a twist, and c ( m ) = g ( h ) g ( h ) g ( h ) g ( h ) , up to a twist, where g i ( h ) are polynomials of degree i , which can be computed with the accompanyingMagma code. We compute res (cid:18) m, ∆( m ) m (cid:19) = 1 ,res ( m, c ( m )) = 1 . We see from Proposition 1.1 that the reduction at p is multiplicative of type I k , with even c p (seeTable 1). If we have another prime p ′ = p with k ′ := ord p ′ ( h + 1) < , then the reduction at p ′ is alsomultiplicative of type I k ′ with even c p ′ . Assume now that there exists only one prime p such that k := ord p ( h + 1) = 0 . That means thateither h + 1 ∈ Z or m = h +1 ∈ Z . We consider the following cases:(1) If h + 1 ∈ Z and h + 1 = ± p k , p = 3 , we have ord p ( h − h + 1) = 0 , since res ( h + 1 , h − h + 1) = 3 ,where h − h + 1 is one of the factors in the discriminant. Then there exists p ′ = p, such that ord p ′ ( h − h + 1) > . Otherwise, we have h − h + 1 ∈ {± , ± } , i.e. h ∈ { , ± , } . For h = 1 we get a twist of the curve 14a4 which has c E = 2 , while for h = 0 , − , we do not get an ellipticcurve (look at the j-invariant). AMAGAWA NUMBERS OF ELLIPTIC CURVES WITH PRESCRIBED TORSION SUBGROUP OR ISOGENY 7 (2) If h + 1 ∈ Z and h + 1 = ± k , for k = 0 we have h + 1 = ± , i.e. h ∈ { , − } . We alreadyknow that h cannot be 0, but for h = − we get a twist of the curve 14a6, for which we have c E = 2 . When k = 1 we have h + 1 = ± , i.e. h ∈ {− , } . For h = − we get a twist of14a5, with c E = 2 , and h = 2 cannot happen. Assume now that h + 1 = ± k , k > . Countingthe multiplicities of 3 in ∆( h ) and c ( h ) we get that the factor − k appears in the j-invariant.Furthermore, if we write ± k − instead of h in the equation for E and make the substitution x x · , y y · , we get a model where ord ( c ) = 0 , and it follows from Proposition 1.1that for k > we have multiplicative reduction I k − in 3, with c being even (see Table 1).Note that in any case we also have a prime p = 3 dividing h − h + 1 in ∆( h ) with multiplicativereduction I n , which makes c E divisible by 4.(3) If m ∈ Z , then m = ± p k , for some prime p, and clearly ord p (3 m − m +1) = 0 , since res ( m, m − m + 1) = 1 . Then there exists p ′ = p such that ord p ′ (3 m − m + 1) > . Otherwise, we have m − m + 1 ∈ {± } , i.e. m ∈ { , } which only makes sense for h = 0 but, as we noted earlier, h cannot be 0.The only thing left to consider is when we have only 2 primes with ord p ( h + 1) = 0 , one of which is3 and divides the numerator; in other words the cases h + 1 = ± k p l and h + 1 = ± k p l , p = 3 , k, l > . From the reasoning in (2) above, it is clear that if k > , we have multiplicative reduction in 3 and fromthe part of the proof where we had ord p ( h + 1) < we see that the reduction is multiplicative in p aswell, which gives us c E that is divisible by 4. For k = 1 , we have h + 1 = ± p l or h + 1 = ± p l . • If h + 1 = ± p l , we have another prime p ′ = p, dividing h − h + 1 in the discriminant (similarlyas in (1)) with multiplicative reduction. • If h + 1 = m = ± p l , we also have another prime p ′ = p dividing the numerator of m − m + 1 inthe discriminant (as in (3)) with multiplicative reduction, except possibly when m − m + 1 = a n , a ∈ Z , n > (this situation couldn’t have happened in (3), because we had m ∈ Z ). Byputting ± p l instead of m , we get ± p l ∓ p l + 1 = 1 a n , which only has solutions for a = 3 , n = 1 , p = 2 , l = 1 , i.e. if h ∈ (cid:8) − , (cid:9) . For h = we get atwist of the elliptic curve 14a3, with c E = 2 , and for h = − we get a curve that already has 2primes of reduction type I k , namely 2 and 13.To conclude the proof of this proposition, it remains to see how these reduction types and Tamagawanumbers would change under the twisting of the curves. All even Tamagawa numbers mentioned inthe proof above come from multiplicative reductions I n at primes p, so by using Table 1, Table 2 andLemma 3.1, we conclude that all reduction types of twists at p are either I n or I ∗ n , so the Tamagawanumbers stay even.As for the curves 14a3, 14a4, 14a5 and 14a6, they have c E = c = 2 . By using the fact that Q × p / (cid:0) Q × p (cid:1) = h− , , i , we explicitly compute all possible reduction types of quadratic twists at p = 2 and conclude that for every twist of those curves | c E . (cid:3) Proposition 3.3.
Let E be an elliptic curve over Q with a − isogeny. Then | c E . Proof.
From [13, Table 3] we take the parameterization of the j − invariants of the curves that are non-cuspidal points on X (10) , j ( h ) = ( h − h + 16 h + 16) ( h + 1) ( h − h , h ∈ Q . From it we can acquire the discriminant and the c − invariant up to a twist, ∆( h ) = ( h − h ( h + 1) ( h − h − ( h − h + 2) f ( h ) f ( h ) f ( h ) ,c ( h ) = ( h − h − ( h − h + 2) f ( h ) f ( h ) f ( h ) , where f i ( h ) are polynomials of degree i , which can be computed with the accompanying Magma code.Assume that there exists a prime p such that k := ord p ( h + 1) > . We compute res (cid:18) h + 1 , ∆( h )( h + 1) (cid:19) = 5 ,res ( h + 1 , c ( h )) = 5 . AMAGAWA NUMBERS OF ELLIPTIC CURVES WITH PRESCRIBED TORSION SUBGROUP OR ISOGENY 8 If p = 5 , this means that p k | ∆( h ) and p ∤ c ( h ) , and we find from Proposition 1.1 that the reductionof E at p is multiplicative of type I k , and therefore c p is even (see Table 1).For the case h + 1 = ± k , after the change of variables x x · , y y · , counting the multiplicitiesof 5 in ∆( h ) and c ( h ) we get that the factor − k appears in the j − invariant, with ∤ c ( h ) . Therefore,when k > , by Proposition 1.1 we have multiplicative reduction at 5 of type I k − with even c p (seeTable 1). For k ∈ { , } we have h ∈ {− , − , , } . For the values h ∈ { , } we do not have an ellipticcurve, and for the values h ∈ {− , − } we get twists of curves 768d3 and 768d1, which have c E = 2 , both with bad prime 2 with reduction type III, so c = 2 . Assume now that there exists a prime p such that k := ord p ( h + 1) < . We put m := h +1 and afterthe substitution x x · − , y y · − we get an elliptic curve with ∆( m ) = ( m − (5 m − m (5 m − m + 1) (5 m − m + 1) g ( h ) g ( h ) g ( h ) up to a twist, and c ( m ) = (5 m − m + 1) (5 m − m + 1) g ( h ) g ( h ) g ( h ) up to a twist, where g i ( h ) are polynomials of degree i , which can be computed with the accompanyingMagma code. We compute res (cid:18) m, ∆( m ) m (cid:19) = 1 ,res ( m, c ( m )) = 1 . We see by Proposition 1.1 that the reduction at p is multiplicative of type I k , with even c p (see Table 1).To conclude the proof of this proposition, it remains to see how these reduction types and Tamagawanumbers would change under the twisting of the curves. All even Tamagawa numbers mentioned inthe proof above come from multiplicative reductions I n at primes p, so by using Table 1, Table 2 andLemma 3.1, we conclude that all reduction types of twists at p are either I n or I ∗ n , so the Tamagawanumbers stay even.As for the curves 768d3 and 768d1, they have c E = c = 2 . By using the fact that Q × p / (cid:0) Q × p (cid:1) = h− , , i , we explicitly compute all possible reduction types of quadratic twists at p = 2 and concludethat for every twist of those curves | c E . (cid:3) Proposition 3.4.
Let E be an elliptic curve over Q with an − isogeny. Then | c E , except for the curves a , a , a , where c E = 1 . Proof.
From [13, Table 3] we take the parameterization of the j − invariants of the curves that are non-cuspidal points on X (8) , j ( h ) = ( h − h + 16) ( h − h , h ∈ Q . From it we can acquire the discriminant and the c − invariant up to a twist, ∆( h ) = ( h − h ( h + 4) f ( h ) f ( h ) f ( h ) ,c ( h ) = f ( h ) f ( h ) f ( h ) , where f i ( h ) are polynomials of degree i , which can be computed with the accompanying Magma code.Assume that there exists a prime p such that k := ord p ( h ) > . We compute res (cid:18) h, ∆( h ) h (cid:19) = 2 ,res ( h, c ( h )) = 2 . If p = 2 , then this means that p k | ∆( h ) and p ∤ c ( h ) , and from Proposition 1.1 we find that thereduction of E at p is multiplicative of type I k , and therefore c p is even (see Table 1).For the case h = ± k , after the change of variables x x · , y y · , counting the multiplicitiesof 2 in ∆( h ) and c ( h ) we get that the factor − k appears in the j − invariant, with ∤ c ( h ) . Therefore,when k > , we have multiplicative reduction at 2 of type I k − with even c p , by Proposition 1.1 andTable 1. For k = 0 we have h = ± and for the both values we get a twist of the curve 15a8 with c E = 1 . For k = 1 we have h = ± , i.e. a curve 48a4 up to a twist, with c E = 1 . When k = 2 we do not geta curve and, for k = 3 and h = ± we have a twist of 24a3, where c E = 2 , and finally for k = 4 and h = ± we have a twist of 15a7, where c E = 1 . AMAGAWA NUMBERS OF ELLIPTIC CURVES WITH PRESCRIBED TORSION SUBGROUP OR ISOGENY 9
Assume now that there exists a prime p such that k := ord p ( h ) < . We put m := h and after thesubstitution x x · − , y y · − we get an elliptic curve with ∆( m ) = − (4 m − m (4 m + 1) g ( h ) g ( h ) g ( h ) up to a twist, and c ( m ) = g ( h ) g ( h ) g ( h ) up to a twist, where g i ( h ) are polynomials of degree i , which can be computed with the accompanyingMagma code. We compute res (cid:18) m, ∆( m ) m (cid:19) = 1 ,res ( m, c ( m )) = 1 . We see from Proposition 1.1 and Table 1 that the reduction at p is multiplicative of type I k , with even c p . To conclude the proof of this proposition, it remains to see how these reduction types and Tamagawanumbers would change under the twisting of the curves. All even Tamagawa numbers mentioned inthe proof above come from multiplicative reductions I n at primes p, so by using Table 1, Table 2 andLemma 3.1, we conclude that all reduction types of twists at p are either I n or I ∗ n , so the Tamagawanumbers stay even.As for the curves 15a7, 15a8 and 48a4, they have c E = 1 . By using the fact that Q × p / (cid:0) Q × p (cid:1) = h− , , i , we explicitly compute all possible reduction types of quadratic twists at p = 2 and concludethat for every twist of those curves | c E . Lastly, for every twist of the curve 24a3 we have | c E . (cid:3) Proposition 3.5.
Let E be an elliptic curve over Q with a − isogeny. Then | c E , except for the curves a , b , where c E = 1 , and also the curves a , b and possibly their twists.Proof. From [13, Table 3] we take the parameterization of the j − invariants of the curves that are non-cuspidal points on X (6) , j ( h ) = ( h + 6) ( h + 18 h + 84 h + 24) h ( h + 8) ( h + 9) , h ∈ Q . From it we can acquire the discriminant and the c − invariant up to a twist, ∆( h ) = h ( h + 6) ( h + 8) ( h + 9) f ( h ) f ( h ) f ( h ) ,c ( h ) = ( h + 6) f ( h ) f ( h ) f ( h ) , where f i ( h ) are polynomials of degree i , which can be computed with the accompanying Magma code.Assume that there exists a prime p such that k := ord p ( h + 9) > . We compute res (cid:18) h + 9 , ∆( h )( h + 9) (cid:19) = 3 ,res ( h + 9 , c ( h )) = 3 . If p = 3 , this means that p k | ∆( h ) and p ∤ c ( h ) , and from Proposition 1.1 we find that the reductionof E at p is multiplicative of type I k , and therefore c p is even (see Table 1).For the case h + 9 = ± k , after the change of variables x x · , y y · , counting the multiplicitiesof 3 in ∆( h ) and c ( h ) we get that the factor − k appears in the j − invariant, with ∤ c ( h ) . Therefore,when k > , we have multiplicative reduction at 3 of type I k − with even c p , by Proposition 1.1 andTable 1. For k = 0 we have h ∈ {− , − } . With h = − we have a twist of the elliptic curve 20a2which has c E = 3 , coming from the reduction at 2 of type IV, and h = − does not give us an ellipticcurve. For k = 1 we have h ∈ {− , − } . For h = − we get the curve 36a2 with c E = 6 and for h = − we have 27a3, where c E = 1 . Lastly, if k = 2 , then h ∈ {− , } . For h = 0 we do not get an ellipticcurve, but for h = − we get a twist of the curve 80b4, with c E = 1 . Assume now that there exists a prime p such that k := ord p ( h + 9) < . We put m := h +9 and afterthe substitution x x · − , y y · − we get an elliptic curve with ∆( m ) = ( m − (3 m − (9 m − m g ( h ) g ( h ) g ( h ) up to a twist, and c ( m ) = (3 m − g ( h ) g ( h ) g ( h ) , AMAGAWA NUMBERS OF ELLIPTIC CURVES WITH PRESCRIBED TORSION SUBGROUP OR ISOGENY 10 up to a twist, where g i ( h ) are polynomials of degree i , which can be computed with the accompanyingMagma code. We compute res (cid:18) m, ∆( m ) m (cid:19) = 1 ,res ( m, c ( m )) = 1 . We see that the reduction at p is multiplicative of type I k , with even c p , by Proposition 1.1 and Table 1.To conclude the proof of this proposition, it remains to see how these reduction types and Tamagawanumbers would change under the twisting of the curves. All even Tamagawa numbers mentioned inthe proof above come from multiplicative reductions I n at primes p, so by using Table 1, Table 2 andLemma 3.1, we conclude that all reduction types of twists at p are either I n or I ∗ n , so the Tamagawanumbers stay even.As for the curve 20a2, using the fact that Q × p / (cid:0) Q × p (cid:1) = h− , , i , we explicitly compute all possiblereduction types of quadratic twists at p = 2 and conclude that for every twist of that curve | c E . Thecurve 36a2 already has c = 2 , which will stay the same under every twist. For curves 27a3 and 80b4 wewere unable to conclude what will happen with all of the twists, since some of the twists have c E = 1 . (cid:3) Proposition 3.6.
Let E be an elliptic curve over Q with an n − isogeny, n ∈ { , , , , , , } .Then | c E . Proof.
For each value of n , from [13, Table 4] we took all the possible j − invariants. They can be foundin Table 3 in the second column. In the third column we have a Cremona label of one of the curves inthe class of twists represented by each j − invariant. For each of those curves in the fourth column wehave a prime of bad reduction of type III.
That reduction can only change to
III ∗ , and vice versa, aftertwisting, as we see in Table 2. Table 1 tells us that the Tamagawa number at primes of reduction type III and
III ∗ is always 2, so the claim follows. (cid:3) n j − invariant Cremona label bad prime withreduction type III − · · · − · − · − · − · − · · − · · − · · · − · · · · Table 3. j − invariants of the curves X ( n ) ; their Cremona labels are representatives inthe class of twists of least conductor with reduction type III at some prime
Acknowledgments.
The author would like to thank Filip Najman for comments and helpful discussions.
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Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb,Croatia
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