Determination of the modular Jacobian varieties $J_1(M,MN)$ with the Mordell-Weil rank zero

In this paper, we determine all modular Jacobian varieties $J_1(M,MN)$ over the number field $\mathbb{Q}(\zeta_M)$ with the Mordell-Weil rank zero following the method of Derickx, Etropolski, van Hoeij, Morrow, and Zureick-Brown.


Introduction
The possible torsion groups of Mordell-Weil groups of elliptic curves over Q are completely classified by Mazur [Maz77]. By generalizing Mazur's method, the possible torsion groups over quadratic fields are also classified by Kenku and Momose [KM88] and by Kamienny [Kam92]. Very recently a corresponding theorem for cubic fields is proven [DEvH + 21], but the higher degree cases are still unknown.
These are proven by considering the corresponding modular curves. More precisely, the existence of an elliptic curve with certain torsion points is essentially equivalent to the existence of certain rational points of the modular curve. Hence this kind of problem leads us to studying the rational points of modular curves.
On the other hand, in general the set of rational points of a curve is related to its Jacobian variety. For example, if the Mordell-Weil rank of the Jacobian variety is zero, then considering the Riemann-Roch spaces of divisors, we can determine all rational points of the curve by finite steps, at least in theory.
This observation leads us to considering the problem of determining the curves X 1 (M, M N ) (for the definition see below) whose Jacobian varieties have the Mordell-Weil ranks zero. Following the method of [DEvH + 21, Theorem 3.1], in this paper we show the following (the cases of M = 1 and 2 are taken from [DEvH + 21, Theorem 3.1]): For a subgroup Γ of GL 2 (Z/N ), let X Γ denote the modular curve over Z[1/N ][ζ N ] det Γ corresponding to Γ. The space X Γ is a smooth proper scheme of relative dimension 1 with geometrically connected fibers. Let J Γ denote its Jacobian variety.
For particular subgroups Γ 0 (N ) = a b 0 d ∈ GL 2 (Z/N ) , respectively, we denote the curve X Γ by X 0 (N ) and X 1 (M, M N ) respectively, and denote the curve X 1 (1, N ) and X 1 (N, N ) by X 1 (N ) and by X(N ) respectively. The curve X 1 (M, M N ), which is defined over Z[1/M N ][ζ M ], is the modular curve parametrizing elliptic curves and their independent two rational points of orders M and N M . Moreover we denote similarly for their Jacobian varieties.
Moreover for a subgroup ∆ of (Z/N ) * /{±1}, we denote the curve X 1 (N )/∆ by X ∆ , and J ∆ its Jacobian variety. The curve X ∆ is isomorphic to X Γ∆ for the group For a normalized eigenform f of weight 2 of level Γ 1 (N ), let K f be the number field generated by the Fourier coefficients of f , and let A f denote the abelian variety associated to f . It is of dimension [K f : Q] and an order of K f acts on it ([Shi73, Theorem 1]). Then for a prime number l, since the homology group H 1 (X 1 (N )(C), Q) is free of rank 2 over T ⊗ Z Q, we have that the Tate module V l (A f ) = T l (A f ) ⊗ Z l Q l is free of rank 2 over K f ⊗ Q Q l , and by the construction, for a prime p = l satisfying that p ∤ N , the trace of a p-th arithmetic Frobenius on V l (A f ) is a p (f ). Moreover, if f is a newform, then the modular abelian variety A f is simple and End A f ⊗ Z Q = K f ([Rib80, Corollary 4.2]). Moreover, for a subgroup ∆ of (Z/N ) * /{±1}, the modular Jacobian variety J ∆ is isogenous to the product where M runs over positive divisors of N , f runs over the set of Galois conjugacy classes of newforms of level M whose characters ǫ satisfy that ǫ(a) = 1 for all a ∈ Z/M with a mod ± ∈ ∆ mod M , and m f = σ 0 (N/M ) is the number of positive divisors of N/M ([Rib80, Proposition 2.3]). For the ranks of Mordell-Weil groups of abelian varieties, Birch and Swinnerton-Dyer conjectured the following: Conjecture 2.1 (Birch-Swinnerton-Dyer conjecture). Let A be an abelian variety over Q, and assume that the L-function L(s) of A has an analytic continuation to C. Then the order of zero of L(s) at s = 1 and the rank of the Mordell-Weil group A(Q) are the same.
Kato ([Kat04, Corollary 14.3]) proves one direction of its special case: Namely, for a normalized eigenform f of weight 2 of level Γ 1 (N ), if the order of the zero of the L-function L(f, s) at s = 1 is zero, then the rank of A f (Q) is zero. In the proof of [DEvH + 21, Theorem 3.1] the authors use the converse, which is, according to a communication with the authors of the article, still an open problem, and hence the proof is incomplete. For the conditional results in this paper, we assume its converse.
For an abelian extension K/Q with the Galois group G, we identify the characters χ : G → Q * with the Dirichlet characters corresponding to it.
For a Dirichlet character χ of conductor M and for a modular form f = n a n q n of weight k of level Γ 1 (N ) with the character ϕ of conductor N ′ , we denote by f χ the twist n χ(n)a n q n of the modular form f . This is a modular form of weight k of level Γ 1 (N ′′ ) with the character χ 2 ϕ of conductor dividing lcm(M, N ′ ), where N ′′ = lcm(N, N ′ M, M 2 ). ([AL78, Proposition 3.1.]) Note that, by checking their Fourier coefficients, we have that if a modular form f is a normalized eigenform, then so is the twist f χ .
Let f be a normalized eigenform of weight 2 of level Γ 1 (N ). Then there exists a newform g of level M dividing N such that a p (f ) = a p (g) for every prime p not dividing N/M . Such g is unique by the multiplicity one theorem. We call such a newform the newform associated with f , and denote it by f new . In this case, since the number field K g is generated by a p (g) for every p not dividing N , the number field K f contains K g .
Let ∆ be the group Let N F denote the set of newforms of weight 2, level dividing M 2 N , and conductor dividing M N , and let N F be the set of Galois conjugacy classes of N F . Let D be the group of Dirichlet characters modulo M . These play a crucial role in this paper.
Lemma 2.2. The modular Jacobian variety J ∆ is isogenous to the product Proof. We show the statement by showing that the space S 2 (Γ ∆ ) of the cusp forms of level Γ ∆ is equal to the direct product ⊕ ǫ S 2 (Γ 1 (M 2 N ), ǫ), where ǫ runs over the Dirichlet characters of conductor dividing M N . The space S 2 (Γ ∆ ) is a subspace of S 2 (Γ 1 (M 2 N )), and the later space is the direct product ⊕S 2 (Γ 1 (M 2 N ), ǫ), where the sum is taken over all characters ǫ mod M 2 N . Now by the definition of ∆, for f ∈ S 2 (Γ 1 (M 2 N ), ǫ), we have that f is of level Γ ∆ if and only if the character ǫ is of conductor dividing M N .

Proof of the main theorem
In order to determine whether the Mordell-Weil rank of J 1 (M, M N ) is zero or not, we use the following corollary of the theorem of Kato: This seems to be well-known, but for the luck of reference we give a proof.
Lemma 3.1. Let L/K be a finite abelian extension of number fields with the Galois group G, F a number field containing all exp(G)-th roots of unity, and for every character χ : G → C * , let A χ be an abelian variety over K on which an order of F acts. Assume that there exists a prime l such that the Galois . Under this isomorphism, for σ ∈ G K , the automorphism σ ⊗ id on K ⊗ K L corresponds to the canonical action on K[G].
Hence we have that Thus the claim follows. Therefore since G is abelian, we have that Therefore by Faltings [Fal83,Korollar1] we have the result.
Corollary 3.2. Let f be a normalized eigenform of weight 2 of level Γ 1 (N ), and K/Q be an abelian extension with the Galois group G. Then we have where the sum is taken over all Dirichlet's characters of K/Q.
Proof. Let F be a finite extension of K f which contains all exp(G)-th roots of unity. The endomorphism ring of A up to isogeny is isomorphic to With this action, for every prime l, the Galois module V l (A and for a prime p not dividing N l, a p-th arithmetic Frobenius ϕ p has the trace χ(p)a p (f ). Therefore by the semisimplicity of the Tate modules of abelian varieties ( [Fal83,Satz3]) and by Cheb- Proof. Let L = K f . Then for a prime l and for a prime p not dividing lN , the traces of a p-th arithmetic Frobenius on the Galois modules V l (A f ) and . Therefore considering the decomposition of L ⊗ Q Q l into a product of fields and corresponding decomposition of these modules, we have, by semi-simplicity and by Chebotarev's density theorem, that these two modules are isomorphic. In particular V l (A f ) and V l (A g ) ⊕[K f :Kg] are isomorphic over Q l . Thus by Faltings' theorem we have that A f and A Let ∆ be the group Then by [JK05], we have that (X ∆ ) Q(ζM ) ≃ X 1 (M, M N ). Recall that N F is the set of newforms of weight 2, level dividing M 2 N , and conductor dividing M N , that N F is the set of Galois conjugacy classes of N F , and that D is the group of Dirichlet characters modulo M .
Using these statements we deduce the following crucial proposition.
Proposition 3.4. Consider the following statements: (2) The rank of J ∆ (Q) is zero.
(3) For every newform f ∈ N F , the special value of the L-function L(f, s) at s = 1 is nonzero.
. Moreover if the Birch-Swinnerton-Dyer conjecture is true, then these statements are equivalent.
Proof. Since (X ∆ ) Q(ζM ) ≃ X 1 (M, M N ), the condition (1) implies (2). By Lemma 2.2, the modular Jacobian variety J ∆ is isogenous to Thus by the theorem of Kato, we have that (3) implies (2), and that if the Birch-Swinnerton-Dyer conjecture is true, then these two statements are equivalent. Proof. By [DEvH + 21, Theorem 3.1]. As we remark in the introduction, the proof of [DEvH + 21, Theorem 3.1] seems to be invalid since it requires the unproved part of the Birch-Swinnerton-Dyer conjecture, however the 'if' part is still valid.
Proof of the main theorem. First of all, for the group ∆ as in the Proposition 3.4, there are morphisms Using the algorithm of [AS05, Theorem 4.5.], we can compute for a newform f whether L(f, 1) is zero or not. (For example use Magma [BCP97].) According to it we have that L(f, 1) = 0 for every newform of level dividing M 2 N with the character of conductor dividing M N , except the newforms in the Galois conjugacy class of newforms labeled "G1N144I" by Magma [BCP97], whose character has the conductor 36, and conversely for those newforms f , we have L(f, 1) = 0. Therefore by Proposition 3.4 we have the desired result.
Lastly from this theorem we deduce a corollary about the existence of certain kinds of elliptic curves.
Proof. Consider the modular curves X(N ) over Q(ζ N ) which classifies generalized elliptic curves with their full level N structures. The third statement is just Faltings' theorem. So assume N = 1, . . . , 10 or 12. For each N we consider, by the theorem, the Mordell-Weil rank of its Jacobian variety is zero. For N = 1, . . . , 4 or 5, since the curve X(N ) has genus 0 and of course has a rational point (the ∞-cusp), it is isomorphic to P 1 . Therefore the result in this case is trivial. Next assume N = 6, . . . , 10 or 12. For such an N , the curve X(N ) has genus greater than 0 and is the fine moduli scheme of the corresponding stack. Hence X(N )(Q(ζ N )) → J(N )(Q(ζ N )) is injective. By [Kat81,Appendix], for a good prime p of the curve X(N ) over Q(ζ N ) above p satisfying e(p/p) < p − 1, since the Mordell-Weil rank is zero, the reduction map J(N )(Q(ζ N )) → J(N )(F q ) at p is injective. Note that the condition e(p/p) < p−1 is almost automatic: the good primes of X(N )/Q(ζ N ) are exactly the primes which do not divide the level N , and the unramified primes of Q(ζ N ) are exactly the primes which do not divide N , except the characteristic 2. Thus in this case, by the diagram we have that the reduction map X(N )(Q(ζ N )) → X(N )(F q ) is injective, where q is the norm of p. On the other hand, by the Hasse bound, if N 2 > (1 + √ q) 2 , i.e., if q < (N − 1) 2 , then the later set consists of cusps. Since for a field k with the characteristic prime to N , the cusps of X(N )(k) correspond to the full level N structures of the Neron N -gon over k, both of X(N )(Q(ζ N )) and X(N )(F q ) contain all cusps. Since of course the number of cusps of X(N )(k) is independent on k, we have that if q < (N − 1) 2 then the reduction map Hence now what we need to show is the claim that, for our N , there is a good prime p of Q(ζ N ) whose norm over Q is less than (N − 1) 2 and whose characteristic is odd. This is easy, and we have done.

Higher rank
In the previous section we prove that the rank of J 1 (M, M N )(Q(ζ M )) is zero if and only if the rank of J ∆ (Q) is zero, where ∆ is the subgroup as in Proposition 3.4. With more careful computation it is possible to show a more general statement. Recall that N F is the set of newforms of weight 2, level dividing M 2 N , and conductor dividing M N , and N F = G Q \N F is the set of Galois conjugacy classes of N F . Also recall that D is the group of the Dirichlet characters modulo M .
Proposition 4.1. We have the following formula: where N (fχ) new is the level of (f χ ) new , and σ 0 is as in the preliminaries.
Note that, decomposing J ∆ into simple factors as in Lemma 2.2, we have the following formula: Thus using Corollary 3.2 we can compute the rank of J 1 (M, M N )(Q(ζ M )). However, in order to compute rank J 1 (M, M N )(Q(ζ M )) using this formula, we need to compute the newform g associated with the twist f χ for every χ : (Z/M ) * → C * and for every f . The importance of Proposition 4.1 is that we only need to compute the newform associated with the twist f χ only for f satisfying that rank A f (Q) is nonzero.
Before proving the proposition, we compute rank J 1 (M, M N )(Q(ζ M )) for some (M, N ) using Proposition 4.1.
For the subgroup ∆ as in Proposition 3.4 for (M, N ) = (13, 1), the modular Jacobian variety J ∆ has only one simple factor A f whose Mordell-Weil rank over Q is nonzero. The abelian variety A f has the dimension three and f has the analytic rank one. If Birch-Swinnerton-Dyer conjecture is true, then it follows that rank A f (Q) = 3, and thence we also can compute the rank of J(13)(Q(ζ 13 )). For a modular abelian variety A f if its L-function has a simple pole at s = 1, then it seems that we obtain dim K f A f (Q) ⊗ Q = 1 unconditionally. We, however, could not find a reference. Newforms of low levels have low analytic ranks. For example, the newforms of level less than 389 have analytic ranks at most 1. (See [LMFDB].) Hence with this proposition we can easily compute the Mordell-Weil ranks of many modular Jacobian varieties.

Proof of Proposition 4.1. First by Lemma 2.2 one obtains
For newforms f and g, if these are Galois conjugate to each other, then the fields K f and K g generated by their Fourier coefficients are isomorphic to each other, and also the modular abelian varieties A f and A g are isogenous to each other. Moreover for a newform f , the dimension of A f equals to [K f : Q]. Therefore we have For each f ∈ N F , by Corollary 3.2 and Lemma 3.3 we have Thus combining them together we obtain Since for every ν ∈ D, the map is well-defined and bijective by Lemma 2.3, we obtain Thus again by the same reason as in the equation (2), the result follows.
Using the same method as in [DEvH + 21, Theorem 3.1, Remark 3.4], for fixed integer r, we can get a necessary condition for that rank J 0 (N )(Q) = r, assuming the Birch-Swinnerton-Dyer conjecture. For a prime p, let g + 0 (p) be the genus of the modular curve X + 0 (p), which is the quotient of the modular curve X 0 (p) by the Atkin-Lenner involution.
Lemma 4.5. If the Birch-Swinnerton-Dyer conjecture is true, then the following inequality holds: Proof. The modular Jacobian variety J 0 (p) is isogenous to J − 0 (p)×J + 0 (p). Hence we have rank J 0 (p)(Q) ≥ rank J + 0 (p)(Q). The abelian variety J + 0 (p) is the Jacobian variety of the modular curve X + 0 (p), and is isogenous to a product ⊕ f A f , where the direct sum is taken over all Galois conjugacy classes of the newforms of level Γ 0 (p) fixed by the Atkin-Lehner involution. Thus for every simple factor of J + 0 (p), its analytic rank is odd, and in particular is nonzero. Hence if the Birch-Swinnerton-Dyer conjecture is true, then every simple factor of J + 0 (p) has the nonzero Mordell-Weil rank. Moreover, for every such newform f , since an where the sum is taken over all Galois conjugacy classes of the newforms of level Γ 0 (p) fixed by the Atkin-Lehner involution. Thus the result.
Note that Lemma 4.5 does not hold for composite numbers, for example rank J 0 (28)(Q) = 0 but g + 0 (28) = 1. We also note that we know the complete list of prime numbers p so that the genera g + 0 (p) are less than 7, see [AAB + 21, Proposition 4.5].
2. For the second statement let M i be positive integers as in the statement. Since M 1 and M 2 are relatively prime to each other, considering the simple decomposition of J 0 (N ), we obtain that J 0 (N ) contains ) up to isogeny, where f runs over the Galois conjugacy classes of the newforms of level dividing M 1 , and where g runs over those of level dividing M 2 . Thus again since M i are relatively prime to each other, we obtain σ 0 (N/N f ) = σ 0 (M 1 /N f )σ 0 (M 2 ) and σ 0 (N/N g ) = σ 0 (M 2 /N g )σ 0 (M 1 ). Since σ 0 (M 1 /N f ) (and σ 0 (M 2 /N g )) is the multiplicity of the modular abelian variety A f (and A g ) as a simple factor of J 0 (M 1 ) (and J 0 (M 2 ) respectively), the result follows.
3. For the third statement, let M be a positive proper divisor of N , and assume that the rank of J 0 (N )(Q) is nonzero. If rank J 0 (M )(Q) = 0, then there is nothing to show. Assume that rank J 0 (M )(Q) > 0. In this case there exists a simple factor A of J 0 (M ) whose Mordell-Weil rank is nonzero. By the first statement, for each simple factor A of J 0 (M ), we have that J 0 (M ) × A is contained in J 0 (N ) up to isogeny. Thus the result.
Proposition 4.7. Assume that the Birch-Swinnerton-Dyer conjecture is true. Let r be an integer. Then there exist only finitely many integers N such that rank J 0 (N )(Q) = r, i.e., the rank of J 0 (N )(Q) tends to infinity as the level N tends to infinity.
Proof. We show it by induction on r. First by [DEvH + 21, Theorem 3.1], the statement for r = 0 is true. Next assume that r > 0 and suppose that there exist only finitely many integers N such that rank J 0 (N )(Q) < r. Define We can compute that the function h(x) is monotonically increasing for sufficiently large x, for example for x > 10 4 . Since h(x) tends to infinity as x tends to infinity, we have that there exist only finitely integers N such that h(N ) < r. Therefore by Lemma 4.5 and by [AAB + 21, Proposition 4.4], there exist only finitely many primes p such that rank J 0 (p)(Q) = r. By Lemma 4.6 (3), if N is a composite number then N is of the form of M 1 M 2 for positive integers M i > 1 satisfying that rank J 0 (M i )(Q) < r. Therefore the induction hypothesis yields the result.
Moreover the proof above inductively constructs, for each integer r, a finite set S r such that if the rank of J 0 (N )(Q) is equal to r then N ∈ S r . Namely: Definition 4.8. Let S 0 be the set of positive integers N such that the rank of J 0 (N )(Q) is zero ([DEvH + 21, Theorem 3.1]). For r ≥ 1, assuming that we have defined the set S r ′ for every r ′ < r, we define S r to be the set of positive integers N satisfying that, N is either a prime such that g + 0 (N ) ≤ r, or a composite number such that every positive proper divisor M of N lies in the set ∪ r−1 i=0 S i . For example we obtain that From statements above we can obtain a rough but easy-to-understand necessary condition for that rank J 0 (N )(Q) ≤ r, in the form of a lower bound of rank J 0 (N )(Q).
Proof. We show the statement by showing that, for a nonnegative integer r, if rank J 0 (N )(Q) ≤ r, then we obtain N ≤ 180 r+1 . First we show a sharper result for prime numbers: Namely, for a prime number p and for a positive integer r, if rank J 0 (p)(Q) ≤ r then p ≤ 180 r , and if rank J 0 (p)(Q) = 0 then p ≤ 180. Let r be a nonnegative integer and p a prime. Define − √ x π (log(16x) + 2).
We show the statement for general case by induction on r. Let N be an integer so that rank J 0 (N )(Q) ≤ r. We may assume that N is a composite number. In the case r = 0, the statement follows from [DEvH + 21, Theorem 3.1]. For r = 1, we computed S 1 explicitly above, hence the result. Let r ≥ 2 and suppose that the statement holds for every r ′ < r. By the induction hypothesis, we may assume that rank J 0 (N )(Q) = r.
Next assume that N = p e for a prime p and for an integer e ≥ 2. In this case by Lemma 4.6 (1) the rank of J 0 (p)(Q) does not exceed r/e. If r/e < 1, then rank J 0 (p)(Q) = 0, and hence p ≤ 71. On the other hand by Lemma 4.6 (3) we have rank J 0 (p e−1 )(Q) ≤ r − 1, which implies p e−1 ≤ 180 r by the induction hypothesis. Thus in this case N ≤ 180 r+1 . If r/e ≥ 1 then we have shown that in this case we obtain p ≤ 180 ⌊r/e⌋ in the argument above. Hence N ≤ 180 r . This completes the proof.
Note that, although the statements in this section treat the higher rank, these, as we have remarked in preliminaries, assume only the converse of Kato's theorem [Kat04,Corollary 14.3], but not require the full strength of Birch-Swinnerton-Dyer conjecture.