aa r X i v : . [ m a t h . N T ] M a y RAISING THE LEVEL AT YOUR FAVORITE PRIME
LUIS DIEULEFAIT AND EDUARDO SOTO
Abstract.
In this paper we prove a level raising theorem for some weight 2trivial character newforms at almost every prime p . This is done by ignoring theresidue characteristic at which the level raising appears. Introduction
For a newform h and a prime l in ¯ Z consider the semisimple 2 dimensional con-tinuous Galois representation ¯ ρ h, l with coefficients in F l = ¯ Z / l attached to h and let { a p ( h ) } p ⊂ ¯ Z be the sequence of prime index Fourier coefficients of h . Let f and g be newforms of weight 2 and trivial character. We say that f and g are Galois-congruent if there is some prime l in ¯ Z such that ¯ ρ f, l , ¯ ρ g, l are isomorphic. This isequivalent to a p ( f ) ≡ a p ( g ) (mod l )for all but finitely many p . In 1990 Ribet proved the following Theorem (K. Ribet) . Let f be a newform in S (Γ ( N )) such that the mod l Galoisrepresentation ¯ ρ f, l : Gal( Q | Q ) −→ GL ( F l ) is absolutely irreducible. Let p ∤ N be a prime satisfying a p ( f ) ≡ ε ( p + 1) (mod l ) for some ε ∈ {± } . Then there exists a newform g in S (Γ ( pM )) , for some divisor M of N such that ¯ ρ f, l is isomorphic to ¯ ρ g, l . If / ∈ l then g can be chosen with a p ( g ) = ε . Hence under some conditions one can raise the level of f at p . That is, there is anewform g Galois-congruent to f with level divisible by p once. When consideringlevel-raisings of f at p we will tacitly assume that p is not in the level of f . In thispaper we do level raising at every p > l .More precisely, we prove the following. Theorem 1.
Let f be a newform in S ( N ) and let p be a prime not dividing N .Assume that ( AbsIrr ) ¯ ρ f, l is absolutely irreducible for every l . ( a ) If p = 2 assume that a ( f ) = 8 .Then there exists some M dividing N and some newform g in S ( M p ) such that f and g are Galois-congruent. The first author is partially supported by MICINN grant MTM2015-66716-P.The second author is partially supported by MICINN grant MTM2016-78623-P.
We proof Theorem 1 and a variant of it in section 3. A theorem of B. Mazur impliesthat Theorem 1 applies to most of the rational elliptic curves. We exhibit an infinitefamily of modular forms satisfying ( AbsIrr ) with coefficient fields constant equalto Q , see section 4 Remark.
It is worth remarking the existence of infinite families with coefficientfields of unbounded degree satisfying all of the condition ( AbsIrr ) , see [9] . Lemma 2.3 together with Ribet’s theorem imply that we can choose the sign of a p ( g ) when the congruence of f and g is in odd characteristic. An obstructionappears in characteristic 2 since Ribet’s methods identify 1 , − f arising fromelliptic curves using 2-adic modularity theorems of [1] for the ordinary case andquaternion algebras for the supersingular case. In this paper we treat the ordinarycase. Theorem 2.
Let f be a newform in S ( N ) , let p be a prime not dividing N andchoose a sing ε ∈ {± } . Assume that f satisfies ( AbsIrr ) and for every l ∋ assume that ( DiehReal ) ¯ ρ f, l has dihedral image induced from a real quadratic extension, (2 Ord ) ¯ ρ f, l | G ≃ (cid:18) ∗ (cid:19) .Then there exists some M dividing N and some newform g in S ( M p ) such that f and g are Galois-congruent and a p ( g ) = ε . We deal with this obstruction in section 3.2 following techniques in [12].Let E/ Q be an elliptic curve. Modularity theorems attach to E a newform f ( E )such that ¯ ρ f, l ≃ E [ ℓ ] ⊗ F ℓ F l modulo semisimplification, for every l . We obtain anapplication to elliptic curves. Theorem 3.
Let E/ Q be an elliptic curve such that • E has no rational q -isogeny for every q prime, • Q ( E [2]) has degree over Q .Let p be a prime of good reduction. Then there exists some divisor M of the conduc-tor of E and some newform g in S ( M p ) such that f ( E ) and g are Galois-congruent.Let ε ∈ {± } . Assume further that • p ≥ , • E has good or multiplicative reduction at and • E has positive discriminantthen g can be chosen with a p ( g ) = ε . Notation.
Let ¯ Q denote the algebraic closure of Q in C . Let ¯ Z be the ring ofalgebraic integers contained in ¯ Q . We use p, q, ℓ to denote rational primes and l , l ′ to denote primes of ¯ Z . We use prime of ¯ Z to refer to maximal ideals of ¯ Z , i.e. non-zero prime ideals. We denote by F l the residue field of l and ℓ its characteristic. Weconsider modular forms as power series with complex coefficients and for a newform f we define by Q f its field of coefficients, that is the number field Q f = Q ( { a p } p ).We denote by G Q the absolute Galois group Gal( ¯ Q | Q ) and by G p a decompositiongroup of p contained in G Q . AISING THE LEVEL AT YOUR FAVORITE PRIME 3 Newforms
Let Γ ( N ) be the subgroup of SL ( Z ) corresponding to upper triangular matricesmod N . The space S ( N ) := S (Γ ( N )) of weight 2 level N trivial Nebentypus cuspforms is a finite dimensional vector space over C . For every M dividing N , S ( M )contributes in S ( N ) under the so-called degeneracy maps S ( M ) ֒ −→ S ( N ). Let S ( N ) old be the subspace of S ( N ) spanned by the images of the degeneracy mapsfor every M | N . Let S ( N ) new be the orthogonal space of S ( N ) old with respect tothe Peterson inner product. A theorem of Atkin-Lehner says that S ( N ) new admitsa basis of Hecke eigenforms called newforms, this basis is unique.1.1. Galois representation.
Let f be a newform of level N and let l be a prime ofresidue characteristic ℓ . A construction of Shimura (see [7] section 1.7) attaches to f an abelian variety A f over Q of dimension n = [ Q f : Q ]. A f has good reductionat primes not dividing N . Let Q f, l denote the completion of Q f with respect to l .Working with the Tate module V ℓ ( A f ) = lim ←− n A f [ ℓ n ] ⊗ Z ℓ Q ℓ one can attach to A f acontinuous Galois representation ρ f, l : G Q −→ GL ( Q f, l )such that detρ f, l is the ℓ -adic cyclotomic character and trρ f, l ( F rob p ) = a p ( f ) forevery p ∤ N ℓ . Indeed, V ℓ ( A f ) and ρ f, l are unramified at p ∤ N ℓ by Neron-Ogg-Shafarevich criterion. Then ¯ ρ f, l is obtained as the semisimple reduction of ρ f, l mod l tensor F l .Next definition is central in this paper. Definition 1.1.
Let f , g be newforms of weight two, level N , N ′ respectively andtrivial character. • We say that f and g are Galois-congruent if there is some prime l in ¯ Z suchthat ¯ ρ f, l ≃ ¯ ρ g, l . • We say that g is a level-raising of f at p if f and g are Galois-congruent and p k N ′ but p ∤ N . • We say that g is a strong level-raising of f at p over l if g is a level raisingof f at p with N ′ = N p . Remark 1.2.
From Brauer-Nesbitt theorem ([6] theorem 30.16) we have that asemisimple Galois representation ¯ ρ : Gal( ¯ Q | Q ) → GL (¯ F ℓ ) is uniquely determinedby the characteristic polynomial function. Hence ¯ ρ is determined by tr and det .Since all Galois representations we consider have cyclotomic determinant Galois-congruence is equivalent to congruence on traces of unramified Frobenius (cf. Cheb-otarev density theorem, [17] Corollary 2). Remark 1.3.
Let R be a ring at which 2 is invertible and A a free R -moduleof rank 2. For an endomorphism f of A we have that tr ( f ) = tr ( f ) − det ( A )and hence det is determined by tr . This is [7] Proposition 2.6 b) for d = 2 andgives necessary conditions on existence of Galois-Congruency for general weightsand levels (cf. [16]). LUIS DIEULEFAIT AND EDUARDO SOTO
Remark 1.4.
With our definition Le Hung and Li [12] do strong level raising at aset of primes with some extra requirements always in characteristic 2.Let ω ℓ denote the mod ℓ cyclotomic character and for α ∈ ¯ Z let λ α be the uniqueunramified character λ α : G p → F × l sending the arithmetic Frobenius F rob p to α mod l . We collect in the following theorem work of Deligne, Serre, Fontaine,Edixhoven, Carayol and Langlands. It gives some necessary conditions for level-raising existence. Theorem 1.5.
Let g be a newform in S ( M p ) with p ∤ M and fix a prime l . Then ¯ ρ g, l | G p ≃ (cid:18) ω ℓ ∗ (cid:19) ⊗ λ a p ( g ) . Let f be a newform S ( N ) , p ∤ N and fix a prime l containing p . Then either ¯ ρ f, l | G p is irreducible or ¯ ρ f, l | G p ≃ (cid:18) ω p λ a p ( f ) − ∗ λ a p ( f ) (cid:19) with ∗ ‘peu ramifi´e’.Proof. Case p ∈ l is Theorem 6.7 in [3] for k = 2. Because a p ( g ) ∈ {± } , we havethat g is ordinary at p . Case p / ∈ l follows from Carayol’s theorem in [5].The statement for f is Corollary 4.3.2.1 in [4]. (cid:3) Following corollaries were inspired by Proposition 6 in [2].
Corollary 1.6.
Let f ∈ S ( N ) , g ∈ S ( M p ) be Galois-congruent newforms over l , p ∤ N M . Then a p ( f ) ≡ a p ( g )( p + 1) (mod l ) . Proof. Case p / ∈ l . We have that tr ¯ ρ g, l ( F rob p ) = a p ( g )( p + 1). On the other hand¯ ρ f, l is unramified at p and has trace a p ( f ) at F rob p . Case p ∈ l . The isomorphism ¯ ρ f, l | G p ≃ ¯ ρ g, l | G p implies that ¯ ρ f, l | G p reduces and wehave equality of characters { ω p λ a p ( g ) , λ a p ( g ) } = { ω p λ a p ( f ) − , λ a p ( f ) } The mod p cyclotomic character is ramified since p = 2. In particular a p ( g ) ≡ a p ( f )(mod l ) . (cid:3) As a consequence we obtain a result on congruent modular forms with level-raising.
Corollary 1.7.
Let f ∈ S ( N ) , g ∈ S ( M p ) be newforms, p ∤ N M . If f and g are congruent , that is a n ( f ) ≡ a n ( g ) (mod l ) for every n ,then ℓ = p and a p ( f ) ≡ a p ( g ) = ± l ) . Proof.
Congruency implis Galois-congruency. We have that a p ( g ) ≡ a p ( f ) ≡ a p ( g )( p + 1) (mod l ) . The corollary follows since a p ( g ) ∈ {± } . (cid:3) AISING THE LEVEL AT YOUR FAVORITE PRIME 5
Ribet’s level raising.
Ribet’s theorem says that the necessary condition ofCorollary 1.6 for level-raising turns out to be enough modulo some irreducibilitycondition.
Theorem 1.8 (Ribet’s level raising theorem) . Let f be a newform in S ( N ) suchthat the mod l Galois representation ¯ ρ f, l : Gal( Q | Q ) −→ GL ( F l ) is absolutely irreducible. Let p ∤ N be a prime satisfying a p ( f ) ≡ ε ( p + 1) (mod l ) for some ε ∈ {± } . Then there exists a newform g in S ( pM ) , for some divisor M of N such that ¯ ρ f, l is isomorphic to ¯ ρ g, l . If / ∈ l g can be chosen with a p ( g ) = ε . Remarks 1.9. • Ribet’s original approach deals with modular Galois repre-sentations ¯ ρ so that in particular there is some newform f such that ¯ ρ ≃ ¯ ρ f, l .His approach deals with traces of Frobenii, this forces him to deal with un-ramified primes only, hence the hypothesis p = ℓ . As he explains later thetheorem can be stated in terms of Hecke operators and hence in terms ofFourier coefficients even if p = ℓ , when f is p -new. • Every normalized Hecke eigenform f ′ has attached a unique newform f sothat the l -adic Galois representations attached to f ′ are the ones attached to f . Furthermore, the level of f divides the level of f ′ . Hence p -new in Ribet’sarticle means new of level pM for some M | N .We introduce a definition in order to deal with the irreducibility condition. Definition 1.10.
Let f ∈ S ( N ) be a newform, let l be a prime of Z and let¯ ρ f, l : Gal( Q | Q ) → GL ( F l ) denote the semisimple mod l Galois representationattached to f by Shimura. We say that f satisfies condition ( AbsIrr ) if ¯ ρ f, l isabsolutely irreducible for every prime l .In section 5 we provide explicit examples with Q f = Q . See [9] Theorem 6.2 for aconstruction of families { f n } for which the set of degrees { dim Q Q f n } n is unbounded.2. Bounds and arithmetics of Fourier coefficients
In light of Ribet’s theorem and the following well-known properties of Fouriercoefficients we study some arithmetic properties of p + 1 ± a p . Theorem 2.1.
Let f be a newform in S ( N ) and let a p be the p -th Fourier coefficientof f , p prime. Then(i) a p ∈ Z ,(ii) a p is totally real. That is, its minimal polynomial splits over R ,(iii) (Hasse-Weil Theorem) | σ ( a p ) | ≤ √ p for every embedding σ : Q ( a p ) ֒ −→ R . We say that a p ∈ Q is a p -th Fourier coefficient if a p satisfies conditions ( i ) − ( iii ). LUIS DIEULEFAIT AND EDUARDO SOTO
Arithmetic lemmas.
Let K be a number field with ring of integers O . Let S be the set of complex embeddings σ : K → C of K . For every α ∈ O we considerthe norm N K ( α ) = Y σ ∈ S σ ( α )and the characteristic polynomial P α ( X ) = Y σ ∈ S X − σ ( α ) . One has that P α (0) = ( − | S | N K ( α ). Following well-known lemma is basic algebraicnumber theory. Lemma 2.2.
Let K be a number field and α an algebraic integer of K . Then P α ( X ) ∈ Z [ X ] and N K ( α ) ∈ Z . A rational prime ℓ divides N K ( α ) if and only ifthere is some prime l in ¯ Z of residue characteristic ℓ such that α ≡ l ) . Inparticular α is a unit of O if and only if N K ( α ) = ± .Proof. Let n = | S | = dim Q K . Consider the embedding ι : K ֒ −→ End Q ( K ),where ι ( α ) is the multiplication-by- α morphism. A choice of integral basis of K induces an embedding ι : K ֒ −→ M n × n ( Q ) with ι ( O ) ⊂ M n × n ( Z ). Then P α ( X ) isthe characteristic polynomial of ι ( α ) by Cayley-Hamilton theorem. For a nonzero α ∈ O one has that | N K ( α ) | = |O /α O| = Q i |O / p e i i | where α O = Q i p e i i is thefactorization in prime ideals of the ideal generated by α . Thus ℓ divides N K ( α )if and only if some prime p of O containing ℓ divides α O . Moreover, the map M axSpec ( ¯ Z ) → M axSpec ( O ) given by l l ∩ O is surjective and the lemmafollows. (cid:3) Lemma 2.3.
Let a p be a p -th Fourier coefficient.(a) ( p + 1 + a p )( p + 1 − a p ) is unit in ¯ Z if and only if p = 2 and a = 8 .(b) If p > then p + 1 ± a p , is not unit in ¯ Z .Proof. Let K = Q ( a p ) and S = { σ ∈ : Q ( a p ) ֒ −→ R } be the set of embeddings. Itscardinality equals n the degree of Q ( a p ) | Q .(a) We have σ (( p + 1 + a p )( p + 1 − a p )) = ( p + 1) − σ ( a p ) ≥ ( p + 1) − p = ( p − ≥ . Hence N K (( p + 1 + a p )( p + 1 − a p )) ≥ ( p − n ≥
1. Equalities hold if and onlyif p = 2 and 9 − a = 1.(b) Since σ ( p + 1 ± a p ) ≥ p + 1 − √ p = ( √ p − then N K ( p + 1 ± a p ) ≥ ( √ p − n > p > (cid:3) AISING THE LEVEL AT YOUR FAVORITE PRIME 7
Lemma 2.4 (Avoiding p ) . Fix a positive odd integer n . There exists an integer C n such that if a p is a p − th coefficient of degree n with p > C n then there is a prime l not over p such that l | ( a p + p + 1)( a p − p − . Proof.
Let K = Q ( a p ) and assume that ( p + 1 + a p )( p + 1 − a p ) factors as productof primes over p in the ring of integers of K . Then N K ( p + 1 − a p ), N K ( p + 1 + a p )are powers of p in the closed interval I = [( √ p − n , ( √ p + 1) n ]. We can take p great enough so that p n is the unique power of p in I . Thus N K ( p + 1 − a p ) = N K ( p + 1 + a p ) = p n N K ( − p − − a p ) = ( − n N K ( p + 1 + a p ) = − p n In particular 0 ≡ P a p ( p + 1) − P a p ( − p −
1) = 2 p n (mod p + 1). (cid:3) We can describe the bound C n : conditions p n +1 , p n − / ∈ I are equivalent to p > (cid:18) pp − √ p + 1 (cid:19) n =: α ( p, n ) ,p > (cid:18) p + 2 √ p + 1 p (cid:19) n =: β ( p, n )Notice that β < α and that p satisfies p > α ( p, n ) if and only if x n > x n − + 1 where x n = p . Since n is odd we can take θ the greatest real root of X n − X n − − C n := θ n . Notice that C n /n has finite limit. Lemma 2.5.
The best bound for n = 1 is C = 2 .Proof. Notice that ( p + 1) − a p = 1 if p = 2, a p = ±
1. Following the notation abovewe have that θ = 2 for n = 1 and C = 4 works. Thus it is enough to check that(4 − a )(4 + a ) is not ± a power of 3. Both factors are positive by Hasse’s bound.Thus 4 + a = 3 a , 4 − a = 3 b and 3 a + 3 b = 8. (cid:3) Proofs
Proof of main result and variant.
Proof of Theorem 1.
Let f ∈ S ( N ) new and p ∤ N . We need to check that Ribet’stheorem applies for some l . By Lemmas 2.1 and 2.3, ( p + 1 + a p )( p + 1 − a p ) is notinvertible in ¯ Z . Hence it is contained in a maximal ideal l . That is, either a p ≡ p + 1(mod l ) or a p ≡ − p − l ). (cid:3) Following variant allows us to do level-raising at p over characteristic ℓ = p . Thistogether with Corollary 1.7 ensures that the predicted Galois-congruency is not acogruence of all Fourier coefficients, at least when the level-raising is at p = 2. Theorem 3.1.
Let f be a newform in S ( N ) such that n := dim Q K f is odd.Assume that ( AbsIrr ) ¯ ρ f, l is absolutely irreducible for every l .There exists a constat C > such that for every prime p > C f has a level-raising g at p over a prime l of residue characteristic different from p . C depens only on n . LUIS DIEULEFAIT AND EDUARDO SOTO
Proof.
Let f ∈ S ( N ) new. Due to ( AbsIrr ) it is enough to find a maximal ideal l not over p . This is done in Lemma 2.4. (cid:3) Choice of sign mod . In this section we adapt some ideas of [12] to ourcase. The strategy is to solve a finitely ramified deformation problem. This kind ofdeformation problem consists on specifying the ramification behavior at all but onechosen prime q . If such a deformation problem has solution and some modularitytheorem applies this provides newforms with specified weight, character and prime-to- q part level. If one chooses an auxiliary prime q , a twist argument kills theramification at q so that one recovers a newform with the specified weight, characterand level.Fix a prime ideal l ∋ Z . Let ρ be a Galois representation G Q → SL ( F l ) with dihedral image D and let E = ¯ Q D be the number field fixed by ker ρ . The order ofan element in SL ( F l ) is either 2 or odd. This forces D to have order 2 r , 2 ∤ r . Inparticular E | Q has a unique quadratic subextension K | Q and ρ is induced froma character χ : Gal( ¯ Q | K ) → F × l of order r .We say that q is an auxiliary prime for ρ if • q ≡ • ρ is unramified at q and ρ ( F rob q ) is non-trivial of odd order. Proposition 3.2.
Let g be a newform in S ( M q α ) , q ∤ M , such that ¯ ρ g, l is unramifiedat an auxiliary prime q . Then either g or g ⊗ (cid:0) · q (cid:1) has level M .Proof. ¯ ρ g, l ( F rob q ) has different eigenvalues by the order condition, thus ρ g, l | I q factorsthrough a quadratic character η of I q (Lemma 3.4 in [12]). By the structure of tameinertia at q = 2 there is a unique open subgroup in I q of index 2 and η : I q ։ Gal( Q urq ( √ q ) | Q urq ) ≃ {± } . If η is trivial then α = 0 and we are done. Otherwise, η extends locally to G q ։ Gal( Q q ( √− q ) | Q q ) and globally to the Legendre symbol (cid:18) · q (cid:19) : G Q ։ Gal( Q ( √− q ) | Q ) ≃ {± } . Legendre symbol over q is only ramified at q and the proposition follows. (cid:3) Auxiliary primes are inert at Q ( i ) and split at K by a parity argument. Inparticular, ρ has auxiliary primes only if K = Q ( i ). Lemma 3.3.
Let ρ : G Q → SL ( F l ) be a Galois representation as above. Assumethat ρ is not ramified at p and that K = Q ( i ) . Then the set of auxiliary primes for ρ splitting at Q ( √ p ) has positive density in the set of all primes.Proof. As in Lemma 3.2 of [12] E and Q ( i, √ p ) are linearly disjoint since E isunramified at p and K = Q ( i ). Chebotarev density theorem implies the lemma. (cid:3) Theorem 3.4.
Let f be a newform in S ( N ) , p be a prime not dividing N and ε ∈ {± } a sign. Assume that a p ≡ p mod l for some prime l containing .Assume that(1) ¯ ρ f, l has dihedral image induced from a real quadratic extension, and(2) (2 Ord ) ¯ ρ f, l | G ≃ (cid:18) ∗ (cid:19) . AISING THE LEVEL AT YOUR FAVORITE PRIME 9 then there exists some M dividing N and some newform g in S ( M p ) such that f and g are Galois-congruent and a p ( g ) = ε .Proof. Let q be an auxiliary prime for ¯ ρ f, l | G splitting at Q ( √ p ). By Theorem 4.2of [12] there is some newform g in S ( N pq α ) with a p ( g ) = ε . Let g ′ be the newformin S ( N p ) obtained by Proposition 3.2. Then a p ( g ′ ) = a p ( g ) since (cid:0) pq (cid:1) = 1. (cid:3) Proof ot Theorem 2.
By Lemma 2.3 there are some maximal ideals l + , l − such that a p ( f ) ≡ p + 1 (mod l + ) and a p ( f ) ≡ − p − l − ). If 2 / ∈ l + , l − then Ribet’stheorem applies and we are done. Otherwise apply previous theorem. (cid:3) Case n = 1 . Elliptic curves and Q -isogenies Let E/ Q be an elliptic curve and let E p / F p be the mod p reduction of (the N´eronmodel of) E for a prime p . Consider the integer c p = p + 1 − E p . Then thereis a unique newform f of weight 2 such that a p ( f ) = c p for every prime p . Thisis a consequence of modularity of elliptic curves over Q . In particular, ¯ ρ f, l and E [ ℓ ] ⊗ F l are isomorphic up to semisimplification for every prime l . In this sectionwe characterize elliptic curves whose corresponding newform f satisfies ( AbsIrr ) .Let E/ Q be an elliptic curve, ℓ an odd prime and c ∈ Gal( C | R ) ⊂ Gal( ¯ Q | Q ) bethe complex conjugation. Then c acts on E [ ℓ ] with characteristic polynomial X − E [ ℓ ] is irreducible ifand only if E [ ℓ ] ⊗ F l is irreducible. We say that E satisfies ( Irr ) if E [ ℓ ] is irreduciblefor every ℓ . From a particular study of the case ℓ = 2 one obtains the Lemma 4.1.
Let E/ Q be an elliptic curve. Then E satisfies ( AbsIrr ) if and only E satisfies ( Irr ) and Q ( E [2]) has degree 6 over Q . Isogenies.
In practice one can deal with ( Irr ) by studying the graph ofisogeny classes. LMFDB project has computed in [13] a huge amount of ellipticcurves and isogenies. We recall some well known results on this topic.Let E, E ′ be elliptic curves defined over Q . An isogeny is a nonconstant morphism E −→ E ′ of abelian varieties over Q . The map { E → E ′ isogeny } / ∼ = −→ { finite Z [Gal( ¯ Q | Q )]-submodules of E } ϕ Ker ϕ defines a bijection. Hence, the torsion group E [ n ] corresponds to the multiplication-by-n map E [ n ] → E under the bijection. Lemma 4.2.
Let E/ Q be an elliptic curve. The following are equivalent(1) E satisfies ( Irr ) .(2) the graph of isogeny classes of E is trivial.(3) every finite Z [Gal( ¯ Q | Q )] -submodule of E is of the form E [ n ] for some n .Proof. Let G Q denote the absolute Galois group Gal( ¯ Q | Q ). We will prove that(1) ⇒ (2) ⇒ (3) ⇒ (1). For the first implication let E ϕ → E ′ be an isogeny. Thenthere exists a maximal n such that ϕ factors as E [ n ] −→ E ψ −→ E ′ for some isogeny ψ : E → E ′ . It can be checked that n is the biggest integersatisfying E [ n ] ⊂ Ker ϕ . Let r = Ker ψ . If p | n then E [ p ] ∩ Ker ψ is a nontrivialsubrepresentation of E [ p ]. If E satisfies ( Irr ) then r = 1 and ψ is an isomorphism.For the second let H be a finite Z [ G Q ]-submodule of E . It corresponds to someisogeny E ϕ → E ′ with kernel equal to H . By hypothesis E and E ′ are isomorphic say E ′ h → E . Thus h ◦ ϕ is an endomorphism of E defined over Q and hence h ◦ ϕ = [ n ]for some n . The last implication is trivial. (cid:3) If the isograph is unkown one can still do something. In 1978 Barry Mazur proved(see [14]) the
Theorem 4.3 (B. Mazur) . Let E/ Q be an elliptic curve and let ℓ be a prime suchthat E [ ℓ ] is reducible. Then ℓ ∈ T := { , , , , , , , , , , , } . Hence, there is a complete list of possible irreducible submodules of E [ ℓ ]. We willuse Mazur’s theorem later in order to exhibit a family of elliptic curves satisfying ( Irr ) .4.2. Twists.
Condition ( AbsIrr ) is invariant under ¯ Q -isomorphism. This followsfrom the fact that Galois representations attached to ¯ Q -isomorphic rational ellip-tic curves differ by finite character. The useful invariant in this context is the j -invariant. More precisely, the map j : Ell := { E/ Q elliptic curve } / ∼ = ¯ Q −→ Q E j ( E )is a bijection, hence ( AbsIrr ) is codified in the j -invariant. Definition 4.4.
Let a/b ∈ Q , with a, b coprime integers. The Weil height of a/b is h ( a/b ) := max {| a | , | b |} . Let S be a subset of Ell . We say that S has Weil density d iflim n →∞ { E ∈ S : h ( j ( E )) ≤ n } { x ∈ Q : h ( x ) ≤ n } = d. Proposition 4.5.
Let S be the set elliptic curves satisfying ( AbsIrr ) modulo iso-morphism. Then S has Weil density .Proof. j -invariant morphism extends to an isomorphism X (1) Q → P Q of rationalalgebraic curves. Here X (1) Q denotes a rational model of the trivial-level modularcurve. Hence Ell is the set Y (1)( Q ) ⊆ X (1)( Q ) of rational non-cuspidal points of X (1). Let p ≥ X ( p ) Q a model over Q of the modular curveof level Γ ( p ). We have the forgetful map X ( p ) → X (1) which is a morphism ofalgebraic curves of degree p + 1. Elliptic curves not satisfying ( Irr ) correspondto non-cuspidal points in the image of f p : X ( p )( Q ) → X (1)( Q ), for p ≤
163 byMazur. Either X ( p ) has genus 0, in which case p ∈ { , , , , } , or X ( N ) haspositive genus, in which case p ∈ { , , , , , , } and X ( p )( Q ) is finite.Image of f p has 0 Weil density in X (1) for every p ≤ B. . ( Irr ) havedensity 1. One can deal similarly with the condition dim Q Q ( E [2]) = 6. (cid:3) AISING THE LEVEL AT YOUR FAVORITE PRIME 11 Examples
A family of elliptic curves.
In this section we give a family of elliptic curvesover Q satisfying ( AbsIrr ) . First we find a family of elliptic curves with irre-ducible 2-torsion as ¯ F [ G Q ]-module. This is done by exhibiting a family of rationalcubic polynomials with symmetric Galois group. Second we take a subfamily withirreducible ℓ -torsion as F ℓ [ G Q ]-module, for every ℓ ∈ T . Lemma 5.1.
Let n = ± be integer such that n is not square. The polynomial P n ( X ) = X − n + 1) X + 2( n + 1) has Galois group isomorphic to S .Proof. Let us see that P n is irreducible over Q when n = 0 , ±
1. Consider a factor-ization P n ( X ) = ( X − a )( X + bX + c ) over the integers. By equating coefficientswe have that a = b a + 3 ac − c = 0 − ac = 2( n + 1)The conic 0 = 18(2 X + 3 XY − Y ) = (6 X + 9 Y + 4)(6 X −
4) + 16 has finitelymany integer points, namely( a, b ) ∈ { (0 , , ( − , , (1 , − , (2 , − } . In particular P n is irreducible if and only if n / ∈ {− , , } . In this case either P n has Galois group of order three or P n has Galois group isomorphic to S , thelatter corresponds to the nonsquare discriminant case. The discriminant of P n is∆ n = 3 n · n + 1) and the lemma follows. (cid:3) Lemma 5.2.
Consider the elliptic curve defined over F given by the equation ¯ E : Y = X + 3 · X − · . Then ¯ E [ ℓ ] is irreducible over F ℓ for every ℓ ∈ T .Proof. It can be checked that E = 1424. Let ϕ denote the Frobenius over 1427,then ϕ satisfies ϕ − ϕ + 1427 = 0as an endomorphism of ¯ E . The polynomial X − X + 1427 is irreducible over F ℓ for every ℓ ∈ T and hence ¯ E [ ℓ ] is irreducible. (cid:3) Theorem 5.3.
Let n be an integer such that k ≡ −
11 (mod 1427) . Then the elliptic curve given by the equation E k : Y = X − kX + 2 k satisfyes ( AbsIrr ) . In particular it is Galois-congruent to infinitely many new-forms.Proof. Since −
12 is not a square in F Lemma 5.1 applies and since E k [ ℓ ] isunramified over 1427 for every ℓ ∈ T the theorem follows. (cid:3) Remark 5.4.
Notice that Theorem 3.1 together with Lemma 2.5 say that everylevel-raising of E k at p > p . This together with Corollary1.7 implies that odd level-raisings of E k can be chosen not congruent. Control of M . Let f be a newform of level N and let l be a prime. If ¯ N = N ( ¯ ρ f, l ) denotes the prime-to- ℓ conductor of ¯ ρ f, l then ¯ N | N . With this in mind wemanage in next theorem to take M = N . Theorem 5.5.
Let E/ Q be an elliptic curve such that(i) E has trivial graph of isogeny classes,(ii) Q ( E [2]) has degree over Q ,(iii) E is semistable with good reduction at ,(iv) ∆( E ) is square-free.Let N denote the conductor of E and let p ∤ N be a prime. Then there exists somenewform g ∈ S ( N p ) Galois-congruent to f ( E ) .Proof. Let l be a prime and g a newform in S ( M p ) such that g is a level raising of E over l . Let us prove that M = N . Since ∆( E ) is squre-free then E [ ℓ ] is ramifiedat every prime p | N , p = ℓ , and the prime-to- ℓ conductor N ℓ of E is the prime-to- ℓ conductor of E [ ℓ ] (cf. Proposition 2.12 in [7]). In particular M ∈ { N, N/ℓ } . Assume that M = N , then N = M ℓ and ℓ = 2 since E has good reduction at2. Theorem 1.5 (or Tate’s p -adic uniformization) says that E [ ℓ ] | G ℓ is reducible. Inparticular E [ ℓ ] | G ℓ ≃ ¯ ρ f, l | G ℓ ≃ (cid:18) ω ℓ λ a ℓ ( f ) − ∗ λ a ℓ ( f ) (cid:19) with ∗ ‘peu ramifi´e’. This together with Proposition 8.2 of [10] and Proposition 2.12of [7] leads to a contradiction. (cid:3) Remarks 5.6. • Condition ( iii ) is equivalent to N being odd and square-free. • The rational elliptic curve of conductor 43 satisfies coditions ( i ) − ( iv ).6. An application: safe chains
When considering safe chains as in [8] (Steinberg) level-raising at an appropriate(small) prime is a useful tool. In particular, this combined with a standard modularcongruence gives an alternative way of introducing a “MGD” prime to the level.Having a MGD prime in the level is one of the key ingredients in a safe chain.Therefore, one could expect to use generalizations of Theorem 1 to build safe chainsin more general settings. In the process of doing so one can rely on tools as in [9]to ensure that the condition ( AbsIrr ) holds when required. Acknowledgements
We would like to thank Samuele Anni, Sara Arias-de-Reyna, Roberto Gualdi,Xavier Guitart and Xavier Xarles for helpful conversation and comments.
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L. Dieulefait: Departament de Matem`atiques i Inform`atica, Universitat de Barcelona,Gran Via de les Corts Catalanes, 585, 08007, Barcelona, Spain
E-mail address : [email protected] E. Soto: Departament de Matem`atiques i Inform`atica, Universitat de Barcelona,Gran Via de les Corts Catalanes, 585, 08007, Barcelona, Spain
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