On congruences between normalized eigenforms with different sign at a Steinberg prime

Let $f$ be a newform of weight $2$ on $\Gamma_0(N)$ with Fourier $q$-expansion $f(q)=q+\sum_{n\geq 2} a_n q^n$, where $\Gamma_0(N)$ denotes the group of invertible matrices with integer coefficients, upper triangular mod $N$. Let $p$ be a prime dividing $N$ once, $p\parallel N$, a Steinberg prime. Then, it is well known that $a_p\in\{1,-1\}$. We denote by $K_f$ the field of coefficients of $f$. Let $\lambda$ be a finite place in $K_f$ not dividing $2p$ and assume that the mod $\lambda$ Galois representation attached to $f$ is irreducible. In this paper we will give necessary and sufficient conditions for the existence of another Hecke eigenform $f'(q)=q+\sum_{n\geq 2} a'_n q^n$ $p$-new of weight $2$ on $\Gamma_0(N)$ and a finite place $\lambda'$ of $K_{f'}$ such that $a_p=-a'_p$ and the Galois representations $\bar\rho_{f,\lambda}$ and $\bar\rho_{f',\lambda'}$ are isomorphic.


Introduction
LetQ denote the algebraic closure of Q in the field C of complex numbers. Let f be a cusp Hecke eigenform of weight 2, level N and trivial nebentypus. We attach to f a sequence {a n } n≥1 of complex numbers consisting of the Fourier coefficients of f at infinity. We say that f is normalized if a 1 = 1. In this case K f = Q({a n } n ) ⊂Q and it is a number field. Let f , f ′ be normalized eigenforms of common weight 2, level N and N ′ respectively and trivial nebentypus. Consider the composite field L = K f · K f ′ inQ and let l be a prime of L, ℓZ = l ∩ Z. We will be interested in pairs of newforms f and f ′ for which a n ≡ a ′ n (mod l) for every integer n coprime to ℓNN ′ .
The Fourier coefficients of a newform are completely determined by the a p coefficients of prime subindex. It is easy to see then that (1) is equivalent to a p ≡ a ′ p (mod l) for every prime p ∤ ℓNN ′ .
Let λ = l∩O K f and λ ′ = l∩O K ′ f and assume that the residual Galois representations ρ f,λ ,ρ f ′ ,λ ′ attached to f and f ′ are irreducible. Then f , f ′ satisfy (1) if and only ifρ f,λ andρ f ′ ,λ ′ are isomorphic. In general, it is not an easy problem to find for a given newform f another eigenform f ′ satisfying (1), neither proving the existence of such an eigenform f ′ . Ribet's level raising [6] and level lowering [7] theorems are very powerful in this context. In this article we will consider a newform f of weight 2 on Γ 0 (N) together with a prime λ ∤ 2N of K f and will give necessary and sufficient conditions for the existence of another eigenform f ′ of weight 2 on Γ 0 (N) and a prime λ ′ of K f ′ such that •ρ f,λ andρ f ′ ,λ ′ are isomorphic and The first author is partially supported by MICINN grants MTM2015-66716-P. The second author is partially supported by MICINN grant MTM2013-45075-P.
• a p = −a ′ p for a prime p dividing N once and f ′ is p-new. See [6]  ) for a definition of p-new and theorem 4.4 for the exact statement of the theorem.
Acknowledgements. The authors are grateful to K. Ribet for providing so much valuable feedback. The second author wants to thank X. Guitart and S. Anni for many stimulating conversations and N. Billerey for helpful comments.

Galois representations mod λ attached to a normalized eigenform
From now on let us fix an odd prime ℓ and an immersion Q ֒−→ Q ℓ . In particular, for every number field K we have fixed a prime λ over ℓ and a completion K ℓ ⊂ Q ℓ of K with respect to λ. Let F ℓ denote the residue field of the ring of integers of Q ℓ , which is indeed an algebraic closure of F ℓ . Let f be a normalized eigenform on Γ 0 (N) with q-expansion at infinity f (z) = q + n≥2 a n q n . As in the introduction, K f denotes the number field Q({a n } n ) of coefficients of f and by O f its ring of integers. Consider the ℓ-adic Galois representation attached to f by Deligne where K f,ℓ denotes the completion of K f with respect to (the fixed prime λ above) ℓ and O f,ℓ denotes its ring of integers. Since Gal(Q | Q) is compact one can (noncanonically) embed its image in GL 2 (O f,ℓ ), Recall that O f,ℓ is a local ring whose residue field O f,ℓ /m f,ℓ is a finite extension of F ℓ contained in F ℓ . Reducing ι • ρ f,ℓ mod m f,ℓ , we obtain the mod ℓ Galois representationρ for every rational prime p ∤ ℓN.

Lowering and raising the levels
We state here Ribet's theorems, they will be the main tools needed in the proof of our theorem. See [7] and [6] (theorem 1 and remarks in section 3) for the proofs.
Theorem 3.1 (Ribet's level lowering theorem). Let f be a newform of weight 2 on Γ 0 (N), let p be a prime dividing N once. Assume that ℓ ∤ 2p and that the mod ℓ Galois representationρ f,ℓ : Gal(Q | Q) −→ GL 2 (F ℓ ) is irreducible and unramified at p. If one or both of the following conditions hold Proof. By theorem 1.1 of [7] we may assume thatρ f,ℓ | G Q(ζ ℓ ) is irreducible. We first prove the existence of a representation ρ : Gal(Q | Q) → GL 2 (Q p ) liftingρ f,ℓ satisfying enough properties such that, if modular, it arises from a cusp eigenform of weight 2 on Γ 0 (N/p). We use some results of [8] §2, §7 and we follow the notation therein. Consider the lifting problem with (i) Σ equal to the set of primes different from p at which ρ f,ℓ is ramified and ℓ, (ii) ψ trivial, (iii) type function t equal to the one attached to ρ f,ℓ , (iv) for each q ∈ Σ the inertial type τ q of ρ f,ℓ | Gq .
By proposition 2.6.1 in [8], t is definite. Since {ρ f,ℓ | Gq } q∈Σ is a local solution to our lifting problem theorem 7.2.1 in [8] says that there is a global solution ρ that is finitely ramified weight two. It is irreducible sinceρ is irreducible, and odd since its determinant is the cyclotomic character. By theorem 1.1.4 in [8] ρ is modular, so it arises from a newform g of weight two. Comparing ρ f,ℓ | Gq and ρ| Gq at every q we have that g has level N/p and trivial nebentypus (since ψ = 1).
Theorem 3.2 (Ribet's level raising theorem). Let f be a normalized eigenform of weight 2 on Γ 0 (N) such that the mod ℓ Galois representation then there are at least two such eigenforms f ′ : one for each coefficient a ′ p ∈ {±1}.

Local decomposition at Steinberg primes and proof of the theorem
In this article we consider newforms f of weight 2 on Γ 0 (N) and we work with primes p dividing N once. Recall that in this case the local type of (the automorphic form attached to) f at p is a twist of the Steinberg representation.  With the hypothesis of definition 4.1, corollary 4.6.20 in [5] says that there is a unique newform g of weight 2 on Γ 0 (N ′ ) for some divisor N ′ of N, p | N ′ , such that f is in the old-space generated by g. Theorem 4.6.17 in [5] implies that a p (g) ∈ {−1, 1}. Since f is normalized and p ∤ N/N ′ it is easy to see that a p (f ) = a p (g).
Here we state a useful lemma related to the local behavior of the mod ℓ Galois representations at a Steinberg prime.
for every prime ℓ = p, whereχ : D p → F * ℓ denotes the unramified character that maps Frob p to a p (f ) andε ℓ denotes the mod ℓ cyclotomic character.
Proof. See [4] (Loeffler, Weinstein 2012) proposition 2.8, we follow the notation therein. The newform f is p-primitive since p N. Recall that Hecke correspondence (a modification of local Langlands correspondence) attaches to π f,p ≃ St ⊗ α, for an unramified character α of Q * p , a two dimensional Weil-Deligne representation that corresponds to a Galois representation r : where we identify α with a character of Gal(Q p | Q p ) via local class field theory. It is a theorem of Carayol [1] that ρ f,ℓ | Dp and r are isomorphic. (b) There is a normalized eigenform f ′ p-new of weight 2 on Γ 0 (N) such thatρ f ′ ,ℓ is isomorphic toρ f,ℓ and a p = −a ′ p where a p (resp. a ′ p ) is the p-th Fourier coefficient of f (resp. f ′ ).

Proof. (b) implies (a)
Let us writeρ =ρ f,ℓ andρ ′ =ρ f ′ ,ℓ for simplicity. Let D p ⊆ Gal(Q | Q) be a decomposition group of p. Since {a p , a ′ p } = {1, −1}, we may assume without loss of generality thatρ,ρ ′ act locally at p as due to lemma 4.3. Sinceρ andρ ′ are isomorphic so are their local behaviors. Thus, specializing at a Frobenius map Eigenvalues must coincide and ℓ > 2 so To see thatρ is unramified at p notice that χε ℓ ≡ε ℓ .

(a) implies (b)
Ribet's lowering level theorem applies to the modular representationρ f,ℓ of level N. Thus, there exists a newform g of weight 2 on Γ 0 (M), for some M | N/p such that ρ g,ℓ ∼ρ f,ℓ . Moreover, we have that by lemma 4.3. Now we can apply Ribet's raising level theorem toρ g,ℓ and there exists an eigenform f ′ p-new on Γ 0 (N) such that By §3 page 9 of [6], when both conditions are satisfied Ribet's proof allows us to choose the a p coefficient of f ′ . We shall choose f ′ such that a ′ p = −a p and the implication holds. Remark 4.5. Let f be a newform satisfying (a). We expect that theorem 4.4 can be strengthened so that f ′ can be chosen to be a newform, not necessarily unique. This would follow from a stronger version of [6]. See section 5 for an example.

An example
In this section we are going to give an example of mod 5 Galois representation to which our theorem applies. We will use many well-known properties of elliptic curves without proof.
Let ℓ be a prime, n > 0 an integer and E an elliptic curve over Q. The ℓ n -th torsion group E[ℓ n ] of E has a natural structure of free Z/ℓ n Z-module of rank 2. The action of the Galois group G = Gal(Q | Q) is compatible with the Z/ℓ n Z-module structure of E[ℓ n ], so that E[ℓ n ] has a natural structure of (Z/ℓ n Z)[G]-module. That is, the action induces a group homomorphism where the isomorphism depends on the choice of a basis in E[ℓ n ]. The case n = 1 is of special interest and is known as the mod ℓ Galois representation attached to E. The Tate module T ℓ E = lim ← − E[ℓ n ] of E at ℓ is a free Z ℓ -module of rank 2. The morphisms {ρ E,ℓ n } induce a group morphism known as the ℓ-adic Galois representation attached to E. The mod ℓ Galois representationρ E,ℓ attached to E can be recovered from ρ E,ℓ by taking reduction mod ℓZ ℓ .
Well-known modularity theorems as Wiles, Taylor-Wiles and Breuil-Conrad-Diamond-Taylor state that such a ℓ-adic Galois representation is isomorphic to the Galois representation ρ f E ,ℓ attached to some newform f E of weight 2 on Γ 0 (N) for N equal to the conductor of E and K f = Q. Moreover the p-th Fourier coefficient of f E coincides with the c p coefficient of E (defined below) for every prime p. In this section we will apply theorem 4.4 to the mod 5 Galois representationρ E,5 ≃ρ f E ,5 attached to the following elliptic curve given by a (global minimal) Weierstrass equation: Its discriminant is ∆ = 2 5 · 19 5 · 37 For every prime p letẼ p denote the curve obtained by reducing mod p a global minimal Weierstrass model of E. As usual, we consider the value for every prime p. One can check that 2 has a node whose tangent lines are defined over F 2 , E p has a node whose tangent lines have slopes in F p 2 \ F p for p ∈ {19, 37}, E p is an elliptic curve over F p , otherwise.
• c p = a p for every prime p.
Let us see now thatρ f,5 satisfies the hypothesis of the theorem 4.4 for p = 19. Since Since the discriminant of P (X) is 2 (mod 5) and 2 is not a square in F 5 , then P (X) is irreducible over F 5 . In particularρ f,5 : Gal(Q | Q) −→ GL 2 (F 5 ) is irreducible. It is well known that such a representation is irreducible if and only if it is absolutely irreducible, thusρ f,5 is absolutely irreducible. Indeed,ρ f,5 is odd and hence its image contains a matrix E with eigenvalues {±1}. Sayρ f,5 (c) = E. Ifρ f,5 is not absolutely irreducibleρ f,5 is conjugate overF 5 to a representation of the form r = θ 1 * 0 θ 2 for some characters θ 1 , θ 2 . Take ǫ := θ 1 (c) ∈ {±1}. By changing coordinates over F 5 we may assume thatρ By comparingr andρ f,5 at c it follows that the conjugation matrix is upper triangular and hence thatρ f,5 is upper triangular over F 5 .
On the other hand, it is well known (see [2] proposition 2.12) that for a prime p = ℓ dividing once the conductor of an elliptic curve E,ρ E,ℓ is unramified at p if and only if ℓ | v p (∆). Since v 19 (∆) = 5,ρ f,5 is also unramified at 19. Hence theorem 4.4 applies toρ E,5 and there exists another eigenform f ′ of weight 2 on Γ 0 (1406) such thatρ f ′ ,ℓ is isomorphic toρ E,5 . Thus, for every p ∤ 1406 · 5. and the 19th Fourier coefficient a ′ 19 of f ′ satisfies a ′ 19 = −c 19 = 1. In this example we can find an elliptic curve E ′ of conductor 1406 such that its corresponding newform f ′ satisfies part (b) of theorem 4.4. In order to find E ′ we have assumed that f ′ can be chosen to be a newform, see remark 4.5. Assuming that f ′ is newform we can determine a ′ 2 and a ′ 37 of f ′ as follows. If a ′ 2 = −c 2 theorem 4.4 applies with ℓ = 5 and p = 2 so we conclude that 2 ≡ −1 (mod 5). Hence, a ′ 2 = −c 2 . Since a ′ 2 ∈ {±1} then a ′ 2 = c 2 = 1. Similarly one can prove that a ′ 37 = c 37 = −1. There are three newforms (up to conjugation) with this configuration of signs (a ′ 2 , a ′ 19 , a ′ 37 ) = (1, 1, −1) out of sixteen newforms of weight 2 on Γ 0 (1406). One of those (the only one satisfying a ′ 3 ≡ c 3 (mod m f ′ ,5 )) corresponds to the elliptic curve E ′ given by a global minimal Weierstrass equation One can check that its conductor is N = 1406 and that c ′ 2 = c ′ 19 = −c ′ 37 = 1. We will use Sturm's bound (see theorem 9.18 in [9] Stein's book) in order to prove that E ′ corresponds to an eigenform f ′ as in theorem 4.4. Notice that c ′ 19 = 19 + 1 − #Ẽ ′ = 1 = −c 19 . Hence, Sturm's result does not apply to the pair (f E , f E ′ ). After twisting both modular forms f E , f E ′ by the quadratic character ψ of conductor 19 we get two cusp forms f ψ E , f ψ E ′ of weight 2 on Γ 0 (1406 · 19). Sturm's bound for cusp forms of weight 2 on Γ 0 (1406 · 19) is 7218.38 and one can check computationally that for every prime p < 7219. Hence, Sturm's result applies to (f ψ E , f ψ E ′ ) and (3) is true for every p. Thus c p ≡ c ′ p (mod 5) for every prime p for which ψ(p) = 0, i.e for every prime p = 19.