SUPERCUSPIDAL RAMIFICATION OF MODULAR ENDOMORPHISM ALGEBRAS

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1 SUPERCUSPIDAL RAMIFICATION OF MODULAR ENDOMORPHISM ALGEBRAS SHALINI BHATTACHARYA AND EKNATH GHATE Abstract. The endomorhism algebra X f attached to a non-cm rimitie cus form f of weight at least two is a 2-torsion element in the Brauer grou of a number field F. We gie formulas for the ramification of X f locally at rimes lying aboe the odd suercusidal rimes of f. We show that the local Brauer class is determined by the underlying local Galois reresentation together with an auxiliary Fourier coefficient. 1. Introduction Let f = n=1 a n q n S k N, ɛ be a rimitie non-cm cus form of weight k 2 and leel N 1. Let M f denote the abelian ariety k = 2 or the Grothendieck motie k > 2 attached to f. The Q-algebra of endomorhisms of M f is denoted by X = X f := End Q M f Z Q. The Hecke field E = Qa n is either a totally real or CM number field. Let F denote the subfield of E generated by all the elements of the form a 2 nɛ 1 n with n, N = 1. Then F is a totally real number field, and may be thought of as the Hecke field of the adjoint lift of f. The algebra X has the structure of a crossed roduct algebra oer F defined in terms of the inner twists of f, as roed in [Ri80] and [Ri81] for k = 2, and generalized to higher weight forms in [BG03] and [GGQ05]. The class of X is a 2-torsion element in the Brauer grou BrF of F. Ribet has asked whether one can determine the class [X] BrF exlicitly. The Brauer class of X can be studied locally under the ma BrF BrF, where aries oer all the rimes in F. The algebra X := X F F is central simle oer F and its class [X ] 2 BrF Z/2. When this class is triial, X is a matrix algebra oer F and X is said to be unramified, and when it is non-triial, X is Brauer-equialent to a quaternion diision algebra oer F and X is said to be ramified. The Brauer class of X is determined by the Brauer classes of all the X, only finitely many of which can be non-triial. For lying aboe a rime of good reduction, a Steinberg rime, or a ramified rincial series rime, with a 0, the class [X ] is essentially determined by the arity of the sloe at of the twisted adjoint lift of f. For instance, if N, then [X ] is essentially determined by the -adic aluation of a 2 ɛ 1 F. See [BG13]. In articular, if ρ f is the l-adic Galois reresentation attached to f, then the Q l -isomorhism class of the local reresentation ρ f G at essentially determines the Brauer class [X ], for l. Thus, if the Fourier coefficient a 0, then it essentially determines [X ] Mathematics Subject Classification. Primary:11F30, Secondary:11F11, 11F80. 1

2 2 SHALINI BHATTACHARYA AND EKNATH GHATE Assume, therefore, that a = 0. For lying aboe a rime of good reduction with a = 0, the sloe of the twisted adjoint lift of f at is not finite. Moreoer, the local l-adic Galois reresentation ρ f G does not determine the class [X ], een if l =. Howeer, in [BG13, Thm. 11] it is shown that [X ] is essentially determined by a Fourier coefficient a 0 at an auxiliary rime N. More recisely, [X ] is essentially determined by the -adic aluation of a 2 ɛ 1, the Fourier coefficient of the twisted adjoint lift of f at. Assume now that lies oer a rime of bad reduction with a = 0. Since the Brauer class of [X f ] is inariant under twisting f by Dirichlet characters, we may as well assume that f is -minimal, in which case is a suercusidal rime. Almost nothing is known about the ramification of X in this case. Again the Q l - isomorhism class of ρ f G does not determine [X ]; see Examle 1 in Section 8. In this aer, we obtain formulas for [X ] in terms of a Fourier coefficient at an auxiliary rime related to, as in the good reduction case. It turns out that the Q l - isomorhism class of ρ f G, for l, together with this coefficient, determines [X ] comletely. This soles the roblem of determining the Brauer classes at the odd suercusidal rimes under an extra hyothesis in the leel 0 case. 2. Statement of results Throughout this article, will denote an odd rime. Let f S k N, ɛ be a non- CM rimitie cus form of weight k 2, leel N and nebentyus ɛ. We write N = N N, such that N, and ɛ = ɛ ɛ, where ɛ is a Dirichlet character of conductor C for some C N, and the conductor of ɛ diides N. If f is - minimal and if C < N 2, then is a suercusidal rime for f and a = 0. Let K, θ be an admissible air attached to f at, where K is a quadratic extension of Q and θ is a continuous character of K. The rime is called a ramified or unramified suercusidal rime for f, if K Q is a ramified or unramified extension. When C < N = 2, then is an unramified suercusidal rime and θ is a tamely ramified character of K. The corresonding local automorhic reresentation has leel 0, and we say is suercusidal of leel 0. When K Q is unramified, we let s denote a fixed rimitie 2 1-th root of unity in K. For any quadratic field extension K 1 K 2, and for an arbitrary x K2, let 1, if x N K1 K x, K 1 K 2 := 2 K 1, 1, otherwise. For a rime in F lying aboe, let f := f F Q be the residue degree and let e := ef Q be the ramification index. Let υ : F Z be the standard surjectie aluation. Since f is non-cm, there exist infinitely many rimes with non-zero Fourier coefficient in any congruence class modulo N. Let us choose two rimes and, both corime to N, satisfying: 1 mod N, mod N and a 0. 1 mod N, has order 1 in Z/ N Z and a 0. The Fourier coefficients of the twisted adjoint lift of f at and, namely a 2 ɛ 1 and a 2 ɛ 1, gie elements of F /F 2 which do not deend on the choice of and. Let us write 1, if X is ramified, [X ] 1, if X is unramified.

3 SUPERCUSPIDAL RAMIFICATION OF MODULAR ENDOMORPHISM ALGEBRAS 3 With notation as aboe, our main result is as follows. Theorem 2.1. Let be a rime of F lying oer a rime 2. a If is an unramified suercusidal rime, and θs + θs 0 if the - minimal twist of f is of leel 0, then [X ] 1 f υa 2 ɛ 1. b If 1 mod 4 is a ramified suercusidal rime, then [X ] 1 f υa 2 ɛ 1. c If 3 mod 4 is a ramified suercusidal rime, then [X ] 1 k a 2 ɛ 1, KF F. Remark One reason behind the similarity and simlicity of the formulas in a and b aboe is that in both cases the extension KF F turns out to be unramified, and hence the symbol a 2 ɛ 1, KF F = 1 υa 2 ɛ 1. On the other hand, we will see that in case c, the extension KF F can be ramified. We list some interesting consequences of the theorem. Corollary 2.2. Suose we are in case a or b of the theorem. If either Z 1 N = 1 i.e., N is a rime ower or if has odd order in N, or, Z 2 disce F, then X is unramified. Corollary 2.3. Suose we are in case a or c of the theorem. If K F, then X is unramified. We will show that 1 diides 2e, if is a ramified suercusidal rime. Corollary 2.4. Suose we are in case b of the theorem. If K = Q, then [X ] ɛ 1 2[F :Q ]/ Endomorhism algebra and its cocycle class Let f and E be as aboe. Then AutE contains an abelian subgrou defined by Γ := {γ AutE a Dirichlet character χ γ such that a γ = a χ γ, N}. The subfield of E fixed by Γ equals the field F mentioned in Section 1. As f is non-cm, for each γ Γ, the character χ γ is unique, and is called an inner twist of f. For a fixed g G Q, the ma γ χ γ g is a 1-cocycle. By Hilbert s theorem 90, αg E such that for all γ Γ, 1 χ γ g = αg γ 1. Thus we get a well-defined continuous character α : G Q E /F, sending each g to αg mod F. Let ρ f denote the λ-adic reresentation attached to f by Deligne, for some rime λ l of E. The following result of Ribet [Ri04, Thm. 5.5] for k = 2, holds for all weights k 2. Proosition 3.1. Let α be any lift of the character α to E. Then 1 For all g G Q, α 2 g ɛg mod F. 2 For all g G Q, αg Traceρ f g mod F, roided that the trace is non-zero.

4 4 SHALINI BHATTACHARYA AND EKNATH GHATE For any continuous lift α of α to E, the 2-cocycle c α defined by c α g, h = αgαhα 1 gh gies a 2-torsion class [c α ] in H 2 G F, F BrF. The class [c α ] H 2 G F, F corresonds to the global Brauer class [X f ] BrF. Similarly the local Brauer class [X ] is determined by the restriction [c α GF ] H 2 G F, F, for any rime of F. Let G be the decomosition grou at. If is a suercusidal rime, then the local Galois reresentation ρ f G is induced from a character of G K for some quadratic extension K of Q. This imlies that for any lift g G of the generator of GalK Q, the trace in art 2 always anishes, so one cannot use Proosition 3.1 to comute αg. In fact, the auxiliary rimes and introduced in the reious section are chosen in such a way that there is a lift g for which αg a or a mod F. 4. Galois reresentations and Langlands corresondence The air K, θ is called an admissible air, if K is a quadratic extension of Q and θ : K Q Q l is a continuous character of K satisfying 1 θ does not factor through the norm ma N K Q : K Q, 2 If K Q is ramified, then een θ U 1 does not factor through N K Q. K Here U i K denotes the i-th ste in the standard filtration of the units of K. Two airs K 1, θ 1 and K 2, θ 2 are equialent if there is a Q -isomorhism ι : K 1 K 2 with θ 2 ι = θ 1. The equialence classes of admissible airs are in bijection with Q l -isomorhism classes of irreducible 2-dimensional reresentations of G as well as with Q l -isomorhism classes of suercusidal reresentations of GL 2 Q for 2. If is a suercusidal rime for f, then ρ f G is absolutely irreducible. We call K, θ to be an admissible air attached to f at, if ρ f G Ind G G K θ. Gien a suercusidal rime 2 for f, the quadratic extension K is unique, though it is not easy to determine the admissible air K, θ attached to f at exlicitly, see [LW12]. For L Q a finite extension and an l-adic reresentation ρ of G L, let cρ be the conductor of the corresonding reresentation of the Weil grou of L. It equals the Artin conductor of a suitable unramified twist of ρ. For the character θ of G K, it equals the usual cθ = min{i : θ U i K 1}. By [CF10, Pro. 4b, 4.3, Ch. 6], we get c Ind G G K θ = υ δk Q + f K Q cθ, where υ is the normalized aluation on Q, δk Q stands for the discriminant, and f K Q is the residue degree. Alying this formula in our setting, we get 2cθ, if K = Q 2 N = 2 is unramified, 1 + cθ, if K Q is ramified. Thus if K = Q 2, then N is een. A newform f of leel N = N N is said to be -minimal, if N is the smallest among all ossible twists f χ of f by Dirichlet characters χ. If f is a -minimal form for which is a suercusidal rime, then by [AL78, Thms. 4.3, 4.3 ], C N /2 1; moreoer, if N is een, then K Q is unramified. While roing the l-adic local Langlands corresondence for GL 2 Q, the suercusidals of leel zero are always treated searately. They are exactly the suercusidal reresentations attached to some admissible airs K, θ with cθ = 1,

5 SUPERCUSPIDAL RAMIFICATION OF MODULAR ENDOMORPHISM ALGEBRAS 5 hence N = 2. By the definition of an admissible air, cθ = 1 forces K to be unramified. For the exact definition of the leel of a suercusidal reresentation we refer to [BH06, 12.6]. The next lemma will be used to inestigate the case of suercusidal rimes of ositie leel. Let σ be any lift of the generator of GalK Q to G, and let θ σ denote the conjugate character of θ by σ. Then we hae Lemma 4.1. Let be a suercusidal rime for f with N 3 and suose ɛ is tamely ramified at. Then there is an element τ I w K, the wild inertia grou of K, satisfying: 1 θτ = ζ and θ σ τ = ζ 1, where ζ is some rimitie -th root of unity. 2 ατ ζ + ζ 1 1 mod F. Thus Qζ + ζ 1 F. In articular, 1/2 diides ef Q = e, for all. Proof. By equation 2 aboe, N 3 cθ > 1, so θ U 1 K 1. So τ I w K with θτ = ζ, for some rimitie -th root of unity ζ. As ɛ is tame, ɛτ = 1, and χ 2 γτ = ɛτ γ 1 = 1, for all γ Γ. But since τ is an element of a ro- grou, and is odd, we conclude that χ γ τ = 1, for all γ Γ. In other words, ατ 1 mod F. Both θτ and θ σ τ hae some -ower order. The sum of two roots of unity of odd order cannot be zero. Hence by Proosition 3.1, ατ θ + θ σ τ 1 mod F. Since F is totally real, θτ and θ σ τ are two roots of unity whose sum is a non-zero real number. Hence they are comlex conjugates. Lemma 4.2. If 2 is a suercusidal rime of leel > 0 for a -minimal f, then F contains Qζ + ζ 1 and e is a multile of 1/2, for each rime in F aboe. Proof. Since is odd, we can twist f by a suitable character to make the nebentyus tame at, without changing the field F. As f was -minimal, N 3 must hold een after twisting. Now aly the reious lemma. We know that detρ f = χ k 1 l ɛ, where χ l is the l-adic cyclotomic character. The nebentyus character ɛ may be adelized as follows. For x Q, let [x] denote the corresonding element 1,, x,, 1 A Q. Then the idèlic character ɛ, restricted to Q is gien by the following formula. For any m Z and u Z, 3 ɛ[ m u] = ɛ m ɛ u 1. The Galois character ɛ G is also determined by this formula ia the norm residue ma of class field theory, which mas Q A Q onto a dense subset of the decomosition grou G at. The local Langlands corresondence for GL 2 is described using the theory of admissible airs. In [BH06], for any admissible air K, θ the authors construct an automorhic reresentation π θ with central character θ, as well as a 2-dimensional Galois reresentation, say ρ θ, both unique u to isomorhism. Let be a suercusidal rime for f and K, θ be an admissible air attached to f at. There exists a character θ of K, such that K, θ θ is also an admissible air, and the suercusidal reresentation π θ θ of GL 2 Q is in Langlands corresondence with ρ f G. Equating the central character with the determinant on the Galois side, we get θ θ Q = χ k 1 l ɛ Q.

6 6 SHALINI BHATTACHARYA AND EKNATH GHATE We refer to [BH06, 34], for the exlicit descrition of θ. But let us mention here that it is a quadratic character unless we are in case c of Theorem 2.1 in which case θ has order 4, and that θ Q is always the unique non-triial character factoring through N K Q K. Using the equations aboe, we get that for any u Z, ɛ 4 θu = u 1, if K = Q 2, u, K Q ɛ u 1, if K Q is ramified. 5. Local symbols We use the same notations as in the reious section. Consider the nebentyus ɛ as a Galois character, and fix a square root ɛg, for each g G Q. For all γ G F, there exists a unique quadratic of order 1 or 2 Dirichlet character ψ γ, such that g G Q, 5 χ γ g = ɛg γ 1 ψ γ g. Note that here χ γ means the inner twist corresonding to the image of γ G F in its quotient Γ = GalE F. For any γ G F, let t γ denote the fundamental discriminant corresonding to the character ψ γ. The set A = {ψ γ : γ G F } is an elementary 2-grou. Let Γ 0 G F be a fixed subset such that {ψ γ : γ Γ 0 } forms a basis for the grou A. For each γ Γ 0, choose a square-free ositie integer n γ rime to N, with a nγ 0, and such that for all γ Γ 0, 1, if γ 6 ψ γ n = γ γ = 1, otherwise. For each n γ, set z nγ := a 2 n γ ɛ 1 n γ F. Let be a rime in F aboe some odd rime. Let [c ɛ ] denote the class of the cocycle c ɛ Z 2 G F, {±1} defined by c ɛ g, h = ɛg ɛh ɛgh 1. It follows from [Qu98] that, 7 [c ɛ ] ɛ 1 = ɛ 1 [F :Q ]. For any two elements a, b F, let us write a = π υa a and b = π υb b, where π is a uniformizer in F. Then the local symbol a, b, which is indeendent of the choice of π, is gien by the following equation: b 8 a, b = 1 υaυbn 1/2 υa a υb, where is the standard quadratic residue symbol in the residue field of F. The next formula exressing the local Brauer class in terms of symbols follows from [GGQ05, Thm. 4.1]. Theorem 5.1. The local Brauer class is gien by [X ] [c ɛ ] z nγ, t γ. γ Γ 0 Note that each t γ must diide N. If diides N, := 1 may or may not diide t γ for a gien γ Γ 0. For a fixed set Γ 0 as aboe, let S := {t γ : γ Γ 0 }. Let us write S as a disjoint union of three sets: S = S S S +, where S = {t γ S : t γ }, S tγ = {t γ S \S : = 1}, S + tγ = {t γ S \S : = 1}.

7 SUPERCUSPIDAL RAMIFICATION OF MODULAR ENDOMORPHISM ALGEBRAS 7 The statement of Theorem 5.1 is true for any choice of Γ 0 G F, as long as {ψ γ : γ Γ 0 } forms a basis of the elementary 2-grou A. If S consists of more than one element, we can choose some t γ0 S and multily the quadratic characters corresonding to all other elements of S by ψ γ0, to construct a new basis of A, for which S is singleton. Similarly we can multily ψ γ0 by a character corresonding to any element of S, if necessary, and then assume that Γ 0 satisfies the following two conditions: a If S, then it is singleton; denote the unique element of S by t γ0, b If S = {t γ0 } and S too, then = 1, where t γ0 := t γ0 /. When S, we slit the symbol z nγ0, t γ0 inoled in the statement of Theorem 5.1 into a -art and a rime-to- art t υznγ0 z nγ0, t γ0 = z nγ0, γ0. Let be as in Section 2, satisfying. Then the following general lemma comutes the rime-to- art of the formula for [X ] in Theorem 5.1. Lemma 5.2. If Γ 0 satisfies the conditions a and b aboe, then we hae t υznγ0 γ0 9 z nγ, t γ = 1 f υa2 ɛ 1. t γ S \S If S =, then the first symbol on the left hand side is assumed to be 1. Proof. Note that if t γ S, then υt γ = 0, hence z nγ, t γ = t γ υznγ tγ f υz nγ =, by equation 8. First we consider the case where S =. Hence S \ S = S +, and so z nγ, t γ = f υz nγ = 1, for all tγ S \ S. Thus the left hand side of 9 reduces to just the tγ υznγ0 tγ0 tγ0 f υz nγ0 first symbol, which is =. If either S = or if S = {t γ0 } tγ0 with = 1, then we use and the condition a on Γ 0 to check that ψ γ = tγ = 1, γ Γ0. Hence ψ γ 5 χ γ γ 1 γ 1 a = = = 1, γ G F, ɛ ɛ as the characters ψ γ for γ Γ 0 generates the grou {ψ γ : γ G F }. Therefore a ɛ F and υa 2 ɛ 1 0 mod 2. Thus, both sides of equation 9 tγ0 equal 1. On the other hand, if S and = 1, then one checks that is a f υz nγ0 f candidate for the integer n γ0 defined in 6. Hence = 1 υa 2 ɛ 1, as desired. So now assume that S = {t γ1 = t 1,, t γm = t m }. By condition b on Γ 0, υznγ0 we hae the first symbol = 1 in this case. Choose distinct rimes r j, with tγ0 a r j 0, for j = 0, 1, 2,, m 1 recursiely, satisfying the following roerties: 1 r 0 =, 2 3 tγ tγ0 tγ0 r j = 1, for all t γ S + S, r j = 1 δ i j ti r j 1, for all i = 1, 2,, m and j 1. ti

8 8 SHALINI BHATTACHARYA AND EKNATH GHATE Note that our choice of r 0 is consistent with roerty 2 aboe. Indeed, if S, then by condition b, t γ0 tγ0 = = 1, and if t γ S +, then t γ = t γ = 1. r i 1 r i, if 1 i m 1, Next we define n i := r m 1, if i = m. It can be checked that each n i satisfies the criterion gien in 6 for n γi. By the telescoing argument used in the roof of [GGQ05, Thm. 4.3] or [BG03, Thm ], we get f υz tγ nγ m z nγ, t γ = = 1 f υz ni = 1 f υa 2 ɛ 1. t γ S \S t γ S S + i=1 Alying equation 7 and Lemma 5.2, we get the following simlification of Theorem 5.1: 10 [X ] ɛ 1 [F :Q ] z nγ0, 1 f υa 2 ɛ 1, where, by equation 8, the middle symbol 11 z nγ0, = 1 υz nγ 0 e f 1/2 υznγ0 when S = {t γ0 }, and is taken to be triial when S =. The formulas 10 and 11 will be the starting oint in our comutation of the local Brauer class at a suercusidal rime. Note that these formulas deend on the choice of Γ 0 satisfying the conditions a and b stated before Lemma Unramified suercusidal rimes In this section, we will roe Theorem 2.1 a in two arts, and then study some of its consequences. Theorem 6.1. Let be an odd unramified suercusidal rime with θs + θs 0. Then, we hae [X ] 1 f υa 2 ɛ 1, for. Proof. Let the set Γ 0 satisfy the two conditions before Lemma 5.2. Suose S = {t γ0 }, then by equation 10 it is enough to show that ɛ 1 [F :Q ] z nγ0, = 1. Let g s G 2 be an element which is maed to s Q under the recirocity ma. By local class field theory, we hae ψg 2 s = ψ[n Q 2 Q s], for any Dirichlet or idèlic character ψ. Note that diides the conductor of a quadratic character ψ if and only if ψ is non-triial on Z/Z. The norm N Q 2 Q s = s +1 = x, say, is a generator of the grou Z/Z inside Z. Hence for any γ Γ 0, ψ γ g s = 1 t γ γ = γ 0, where the last imlication follows from condition a on Γ 0. Looking at the definition of n γ0 gien in 6, we conclude that γ Γ 0 and hence γ G F, ψ γ g s = ψ γ n γ0. Hence it follows from 1 and 5 that αg γ 1 γ 1 s a nγ0 = ɛnγ0, for all γ G F. So we hae ɛgs z nγ0 = a2 n γ0 ɛn γ0 α2 g s ɛg s θs + θs 2 ɛ 1 mod F 2, x where the last congruence is by art 2 of Proosition 3.1, noting that the trace θs+θs is non-zero by assumtion. As θs+θs = Tr Q 2 Q θs Q F, z n γ0 e,

9 SUPERCUSPIDAL RAMIFICATION OF MODULAR ENDOMORPHISM ALGEBRAS 9 we get z nγ0 ɛ x mod F 2, and υz nγ0 0 mod 2. By equation 11, we z get z nγ0, e nγ0 = = ɛ x e = ɛ x f e = ɛ x 1/2 e f = ɛ 1 e f, therefore ɛ 1 [F :Q ] z nγ0, = ɛ 1 e f 2 = 1. If S =, then ψ γ g s = 1 for all γ G F, and a similar argument shows that θs + θs 2 ɛ 1 x mod F 2. Thus we get ɛ x F 2, which imlies that ɛ x ɛ x f = = ɛ 1 f = 1. Hence ɛ 1 [F :Q ] = 1, and the result follows by equation 10. Note that if is a suercusidal rime of leel zero, then by the regularity condition 1 on θ in the definition of an admissible air, we hae θs 1 1. The condition on θs in the hyothesis of Theorem 6.1 is equialent to θs 1 1. Howeer, for an unramified suercusidal rime of ositie leel we remoe this condition now. Theorem 6.2. Let be an odd unramified suercusidal rime of leel > 0 for a -minimal non-cm newform f. Then, we hae [X ] 1 f υa 2 ɛ 1, for. Proof. Since is an unramified suercusidal rime of leel > 0, N 4. We twist f by a suitable character at if necessary, and assume that ɛ is tame. Note that the formula to be roed is inariant under twist by a character of -ower conductor. As f was -minimal to begin with, N cannot decrease by twisting. Thus the hyothesis of Lemma 4.1 is satisfied. If θs + θs 0, we are done by Theorem 6.1. So assume θs + θs = 0, i.e., θs 1 = 1. As exlained in the roof of Theorem 6.1, if S, then 12 z nγ0 = a2 n γ0 ɛn γ0 α2 g s ɛg s mod F 2. Note that θ σ s = θs = θs. By Lemma 4.1, τ I w K, such that 13 αg s αg s τ θ + θ σ g s τ θsζ ζ 1 mod F. Using equations 3 and 4, we get 14 ɛg s = ɛ[n Q 2 Q s] = ɛ 1 s +1 = θs +1. By the equations 12, 13, and 14 and using that θs 1 = 1, we get 15 z nγ0 ζ ζ ζ ζ 1 2 mod F 2. Suose 3 mod 4, then ζ ζ 1 is a totally real element in Qζ, hence it is contained in the field Qζ + ζ 1 F, by Lemma 4.1. Thus we hae ζ ζ 1 2 F 2, hence z n γ0 16 = 2 ζ ζ 1 2 =, where we write for the rime-to-π art of to distinguish it from the auxiliary rime. From equation 15, we hae the aluation 17 υz nγ0 υ ζ ζ 1 2 = 2e / 1 e mod 2. As 3 mod 4, N 1/2 = f 1/2 f mod 2. Hence by 11 and 16, z nγ0, = 1 e e f e e = 1 e f 1 e = 1 e f 1 e f = 1.

10 10 SHALINI BHATTACHARYA AND EKNATH GHATE By 4, ɛ 1 = θ 1 = θs 2 1/2 = θs 1 +1/2 = 1 +1/2 = 1, and therefore by 10, we get [X ] 1 f υa 2 ɛ 1. Now suose 1 mod 4. By Lemma 4.1, we get that e is een, and = Q ζ + ζ 1 F, hence = 1. As all the other symbols inoled in ɛ 1 [F :Q ] z nγ0, hae an e in the exonent, we get [X ] 1 f υa 2 ɛ 1, by equation 10. If S =, then the result follows triially from the fact that ɛ 1 [F :Q ] = 1 in all cases, as shown aboe. Remark The result aboe is false in general for suercusidal rimes of leel zero. There exist -minimal newforms with C < N = 2, with θs + θs = 0, such that f is een, but X is ramified at some ; see Examle 5 in the last section. Theorems 6.1 and 6.2 imly Theorem 2.1 a. Indeed, if θs + θs 0, the first theorem alies. Otherwise, θs + θs anishes. By the hyothesis made in Theorem 2.1, we may assume that the -minimal twist of f has ositie leel, and the second theorem alies. We remark here that the anishing of θs + θs does not deend on the twist. Corollary 6.3. Let be an odd unramified suercusidal rime for f. Also assume that θs + θs 0, if the -minimal twist of f is of leel 0. a If N = 1, or if has odd order in Z/N Z, then X is a matrix algebra oer F. b If Q 2 F, then X is a matrix algebra oer F. c If X has non-triial Brauer class in BrF, then must diide disce F. Proof. We hae: a For any Dirichlet character χ, let χ denote its rime-to- comonent. If N = 1, then χ γ = χ γ = 1, for all γ Γ. Hence a F. If N 1, let n = 2m + 1 be the order of mod N, i.e., n 1 mod N. Hence we hae χ γ n = χ γ n = 1, for all γ Γ. We know that χ γ 2 = χ γ 2 = ɛ γ 1, for all γ Γ. Hence χ γ = χ γ n 2m = χ γ 2 m = ɛ γ 1 m, which imlies that a γ 1 = ɛ m γ 1, for all γ Γ. So a ɛ m mod F. Now use Theorems 6.1 and 6.2 to get the result in both cases. b Q 2 F would imly that f is een. Now aly Theorems 6.1 and 6.2. Note that here K = Q 2, so we hae roed a art of Corollary 2.3. c By Theorems 6.1 and 6.2, [X ] 1 imlies that υa 2 ɛ 1 is odd. If υ is extended to a aluation on E, then clearly υa is not an integer. Hence ramifies in E, or disce F. 7. Ramified suercusidal rimes In this section we will roe art b and c of Theorem 2.1, and derie some consequences. Let be an odd ramified suercusidal rime for f. As K Q is ramified, there are two ossible choices for K. If, K Q = 1, then K = Q, and if, K Q = 1, then K = Q ζ 1. Deending on K, we can always choose a uniformizer π = or ζ 1, and write K = Q π. For σ G, any lift of the generator of GalK Q, we hae π σ = π, and N K Q π = π 2.

11 SUPERCUSPIDAL RAMIFICATION OF MODULAR ENDOMORPHISM ALGEBRAS 11 Let and be as in Section 2, satisfying the roerties and resectiely. Theorem 7.1. Let 1 mod 4 be a ramified suercusidal rime. Then, for any rime in F, the Brauer class is gien by [X ] 1 f υa 2 ɛ 1. Proof. The formula to be roed is inariant under twist by a Dirichlet character of -ower conductor. So we twist f by a suitable character, if necessary, and then assume that f is -minimal. Since is a ramified suercusidal rime, een after alying the twist the leel will still be ositie, i.e., the hyothesis of Lemma 4.2 is satisfied. Since 1 mod 4, e is een and Qζ + ζ 1 F, by Lemma 4.2. As = is a square in F, we get = 1. Hence by 10 and 11, we get [X ] 1 f υa 2 ɛ 1. Remark Since the formula turns out to be the same as in the case of unramified suercusidal rimes, Corollary 6.3 also holds for ramified suercusidal rimes 1 mod 4, as stated in Corollary 2.2. But note that none of the results in Corollary 2.2 are true for ramified suercusidal rimes 3 mod 4; see the examles in the last section. Lemma 7.2. For any odd ramified suercusidal rime, the aluations of a and a are related by the following equation: 2e / 1 1 υa2 ɛ 1 = 1 e k 1 1 ɛ 1, K Q υa2 ɛ 1. Proof. The equation to be roed is inariant under twist by a character of -ower conductor. So without loss of generality we assume ɛ to be tame, so that we can aly Lemma 4.1. We write the ramified quadratic extension K as K = Q π, where either π =, or π = ζ 1. Let g π G K be an element whose image under the recirocity ma is π K. Taking the determinant of ρ f described in Section 4 at g π, we get θπθ σ π = θπθ π = χ k 1 l g π ɛg π 18 = θπ 2 ɛ 1 g π = θ 1 k 1. If θ 1 = 1, then αg π θ + θ σ π = 2θπ θπ mod F. So by 18, we hae α 2 g π ɛ 1 g π k 1 mod F 2. If θ 1 = 1, we use the element τ I w K gien by Lemma 4.1, to get αg π αg π τ θ + θ σ g π τ θπζ ζ 1 mod F, and by 18, we hae α 2 g π ɛ 1 g π k 1 ζ ζ 1 2 mod F 2. Therefore using υ = e and υ ζ ζ 1 2 = 2e / 1, we get υ α 2 g π ɛ 1 g π e k 1 mod 2, if θ 1 = 1, e k 1 + 2e / 1 mod 2, if θ 1 = 1. By 4, θ 1 = 1 ɛ 1, so the congruence aboe can be written as 2e / υα2 g π ɛ 1 g π = 1 e k 1 1 ɛ 1. By class field theory, g π G G Q is maed to [N K Q π] Q A Q. Each χ γ is realized as an idèlic character as in equation 3, and hence as a Galois character.

12 12 SHALINI BHATTACHARYA AND EKNATH GHATE If, K Q = 1, then N K Q π =, and for all γ Γ, we hae χ γ g π = χ γ [] = χ γ = χ γ. Hence alying 1 and 3, we get that α 2 g π ɛ 1 g π a 2 ɛ 1 mod F 2. If, K Q = 1, then N K Q π = ζ 1. So for all γ Γ, we hae χ γ g π = χ γ [ζ 1 ] = χ γχ 1 γ, = χ γ χ 1 γ. Alying 1 and 3, we get α 2 g π ɛ 1 g π a 2 ɛ 1 a 2 ɛ 1 mod F 2. Now the result follows from equation 19 aboe. Corollary 7.3. Let 1 mod 4 be a ramified suercusidal rime such that K = Q. Then, for any rime in F, we hae [X ] ɛ 1 2[F :Q ]/ 1. Proof. As exlained in the roof of Theorem 7.1, without loss of generality we may assume that f is -minimal, where is a suercusidal rime of leel > 0. By the hyothesis K = Q = Q, as 1 mod 4. Hence we hae 1 = 1 =, K Q. By Lemma 4.2, e is een. Now we aly Lemma 7.2 to the statement of Theorem 7.1 to get the result. Theorem 7.4. Let 3 mod 4 be a ramified suercusidal rime. Let be a rime in F such that e is odd. Then [X ] 1 k a 2 ɛ 1, KF F. Proof. Since e is odd, K F and KF F is a ramified roer quadratic extension. So we can and do choose a uniformizer π N KF F KF F. Note that in this case, we hae N KF F O KF = O 2 F, where O denotes units. Hence for any a = π υa a F, we hae a = a, KF F. If, K Q = 1, then =, KF F = 1. If, K Q = 1, then K = Q ζ 1 and N KF F ζ 1 ζ 1 = ζ 1, so = ζ 1, KF F = 1, which imlies that ζ 1 = = 1 f. Combining both the cases we get 20 = 1 = f 1, K Q f =, K Q f. If S = {t γ0 }, then as in the roof of Theorem 6.1, we use 6 and to choose n γ0 to be, hence z nγ0 = a 2 ɛ 1. Now using 10, 11, 20, Lemma 7.2, and the facts that e is odd, 1/2 is odd and so f 1/2 f mod 2, we get

13 SUPERCUSPIDAL RAMIFICATION OF MODULAR ENDOMORPHISM ALGEBRAS 13 [X ] 10 ɛ 1 [F :Q ] z nγ0, 1 f υa 2 ɛ 1 11 = ɛ 1 e f 1 υz nγ 0 e f 1/2 20, 7.2 υznγ0 nγ0 z 1 f υa 2 ɛ 1 = ɛ 1 f 1 f υz nγ0, K Q f υz nγ0 z nγ0 1 k 1 ɛ 1, K Q υa2 ɛ 1 f z = 1 k nγ0 1 k z nγ0 f = 1 k z, KF nγ0 F = 1 k a 2 ɛ 1, KF F. e If S =, then ψ γ = 1, γ G F, so it follows from 5 that a 2 ɛ 1 F 2 F 2. Therefore the symbol z nγ0, in equation 10 can be relaced by the triial symbol a 2 ɛ 1,, and then the same roof as aboe works. The next lemma is a basic alication of algebraic number theory and hence we state it without roof. Lemma 7.5. Let ϖ be any uniformizer in Z. If e is een, then either ϖ F or ϖζ f 1 F. Moreoer, if K Q is a ramified quadratic extension, then KF F is an unramified extension of degree 1 or 2. Theorem 7.6. Let 3 mod 4 be a ramified suercusidal rime and suose that e is een. Then, the formula [X ] 1 k a 2 ɛ 1, KF F still holds. Proof. As is a candidate for the integer n γ0, by 10, 11, Lemma 7.2 and the assumtion that e is een, we get that if S, then 21 [X ] υa 2 ɛ 1 1 f υa 2 ɛ 1 υa 2, K Q f ɛ 1. case by case. We use the fact that since 3 So we comute the symbol mod 4, the ossibilities for K are Q and Q = Q ζ 1, and these occur exactly when, K Q = 1 or 1, resectiely. Case 1: Assume K F. If, K Q = 1, then K F = 1. If, K Q = 1, then K F = 1 = 1 = 1 f. Case 2: Assume K F. By Lemma 7.5, KF F is a roer quadratic unramified extension. If, K Q = 1, then F, so by Lemma 7.5 with ϖ =, ζ f 1 ζ f 1 ζ f 1 F = 1 = = 1. If, K Q = 1, then F, so by Lemma 7.5 with ϖ =, ζ f 1 ζ f 1 ζ f 1 F = 1 = = 1 f+1.

14 14 SHALINI BHATTACHARYA AND EKNATH GHATE Alying these to equation 21, we get 1, if K F, [X ] 1 υa2 ɛ 1, otherwise. It can be checked that this formula is alid een if S =, as in the roof of Theorem 7.4. Equialently, [X ] 1 k a 2 ɛ 1, KF F, as KF F is unramified. Remark Note that the factor 1 k aboe does not hae any significance, since KF F is unramified. We kee it only to get a uniform formula for all e, een or odd. 8. Numerical examles Here are some numerical examles in suort of our results. We used the rogram Sage to comute the admissible air K, θ attached to a suercusidal rime of a newform in some cases. One may also directly check the hyothesis on θs in Theorem 6.1 in the case of leel zero suercusidals, using equation 4, together with the fact that the order of θs diides 2 1, but not 1. To determine the local Brauer class [X ], we used the rogram Endohecke for newforms with quadratic character, and the data from the tables in [GGQ05] and [Qu05] for newforms with arbitrary character. 1 f S 5 75, [1, 0], E = Q 35, F = Q; = 5 is an unramified suercusidal rime of leel zero with θs + θs 5 0. For = 5, = 101 satisfies and we comuted a = So f υ a 2 ɛ 1 = 1 υ = 5 1 mod 2. By Thm. 6.1, X 5 is ramified. There exists another newform, say g S 5 75, [1, 0], with F = Q and E = Q 14, such that = 5 is an unramified suercusidal rime of leel zero with θs + θs 5 0. By Cor. 6.3 c, X 5 has to be unramified. But, the admissible airs attached to f and g at are equialent. This shows that the Q l -isomorhism class of the local l-adic Galois reresentation at a suercusidal rime fails to redict the Brauer class [X ] aboe, een in the simlest case F = Q. 2 f S 2 72, [1, 1, 3], E = Q 2, 3, F = Q. Here = 3 is an unramified suercusidal rime of leel zero with order oθs = 8, so θs + θs 3 0. We choose = 19 and checked that a 19 = 4. Clearly υ 3 a 2 19 ɛ 1 19 = 0, so X 3 is unramified. Note that 3 diides disce F, thus the conerse of Cor. 6.3 c is false. 3 f S 2 405, [0, 2] is 3-minimal with E = Q 2, 3 and F = Q. = 3 is an unramified suercusidal rime of ositie leel, with oθs 4, so θs + θs 3 may anish. But since the leel > 0, we can still aly Thm We choose = 163 and checked that a 2 ɛ 1 6 mod Q 2. Hence X 3 ramifies. 4 f S 2 99, [3, 5], E = Q 2, 3, F = Q. = 3 is an unramified suercusidal rime of leel zero with θs + θs 3 0. The order of 3 Z/11Z is 5, hence by Cor. 6.3 a, X 3 is unramified. 5 f S 2 99, [0, 2], [E : Q] = 8 and F = Q 5. = 3 is an unramified suercusidal rime of leel zero. The order of θs is 4, so θs+θs 3 = 0, and we cannot aly Thms. 6.1 or 6.2. Note that = 3 is the unique rime

15 SUPERCUSPIDAL RAMIFICATION OF MODULAR ENDOMORPHISM ALGEBRAS 15 in F lying aboe 3, and F = Q 3 5 = Q 3 2. Thus f = 2, but still [X ] 1. This roes the necessity of the condition on θs in Thm f S 2 375, [1, 25], [E : Q] = 16 and F = Q 5. = 5 1 mod 4 is a ramified suercusidal rime for f, and K = Q 5 5. There is a unique rime in F aboe 5, and [F : Q 5 ] = 2. As ɛ 1 2[F :Q ]/ 1 = 1, X is ramified by Cor In fact, X f is ramified at the rimes aboe 5, f S 5 27, [9] with E = Q 1 and F = Q. = 3 is a ramified suercusidal rime for f. For = 3, = 53 satisfies, and we checked that a 2 53 = = So 1 k a 2 ɛ 1 = a mod N K Q K. For any ramified quadratic extension K of Q 3, 1, K Q 3 = 1. Hence by Thm. 7.4, X 3 is ramified. Note that here 3 disce F, and N = 1, but still [X 3 ] 1. So the analogue of Cor. 6.3 does not hold in this case. 8 f S 5 27, [9] with E = Q 6, and F = Q. = 3 is a ramified suercusidal rime and K = Q 3 3. We choose = 53 as before, and comute a 2 = a 2 53 = = Hence 1 k a 2 ɛ 1 = a mod N K Q3 K. But 3, K Q 3 = 1, so X 3 is unramified by Thm In fact, X f is ramified only at 2 and. Errata to [BG13] 1 Page 517, line 8: The condition N 2 > C should be relaced by N 2, N > C. 2 Page 522, line 4: In the definition of m, [F : Q ] should be relaced by [F : Q ]. 3 Page 523, line 1: The inequality should be relaced by <. 4 Page 539, Pro. 33: One should add the following condition in the hyotheses: If K/Q is unramified, then assume χg s + χg s 0, where g s G K corresonds to a rimitie 2 1-th root of unity s K. Without this assumtion, the usual argument referred to on line 16 of age 540 does not work. References [AL78] A. Atkin and W. Li. Twists of newforms and seudo-eigenalues of W-oerators. Inent. Math., 483: , [BG03] A. Brown and E. Ghate. Endomorhism algebras of moties attached to ellitic modular forms. Ann. Inst. Fourier Grenoble, 536: , [BG13] D. Banerjee and E. Ghate. Adjoint lifts and modular endomorhism algebras. Israel J. Math., 1952: , [BH06] C. J. Bushnell and G. Henniart. The local Langlands conjecture for GL2. Sringer- Verlag, Berlin, [CF10] J.W.S. Cassels and A. Fröhlich. Algebraic Number Theory. London Mathematical Society, 2nd Edition, [GGQ05] E. Ghate, E. González-Jiménez and J. Quer. On the Brauer class of modular endomorhism algebras. Int. Math. Res. Not., 12: , [LW12] D. Loeffler and J. Weinstein. On the comutations of local comonents of a newform. Math. Com., 81: , [Qu98] J. Quer. La classe de Brauer de l algèbre d endomorhismes d une ariété abélienne modulaire. C. R. Acad. Sci. Paris, 3273: , [Qu05] J. Quer. Tables of modular endomorhism algebras X f = EndM f of moties M f attached to non-cm newforms. 255 ages, [Ri80] K. Ribet. Twists of modular forms and endomorhisms of abelian arieties. Math. Ann., 2531:43-62, 1980.

16 16 SHALINI BHATTACHARYA AND EKNATH GHATE [Ri81] K. Ribet. Endomorhism algebras of abelian arieties attached to newforms of weight 2. Progr. Math., 12, Birkhaüser, Boston, Mass., , [Ri04] K. Ribet. Abelian arieties oer Q and modular forms. Modular cures and abelian arieties, Prog. Math., 224, Birkhaüser, Basel, , Tata Institute of Fundamental Research, Homi Bhabha road, Mumbai , India address: shalini@math.tifr.res.in address: eghate@math.tifr.res.in