ON THE BRAUER CLASS OF MODULAR ENDOMORPHISM ALGEBRAS

EKNATH GHATE, ENRIQUE GONZÁLEZ-JIMÉNEZ, AND JORDI QUER

Abstract. We investigate the Brauer class of the endomorphism algebra of the motive attached to a non-CM form. The ramification of the algebra is shown in many cases to be controlled by the normalized slopes of the form.

1. Introduction

In this paper we study the Brauer class of the endomorphism algebra X_{f} of the motive attached to a primitive elliptic modular cusp form f without complex multiplication. Our study includes the case of forms of weight 2, where the associated motive is an abelian variety.

It is a fundamental fact that X_{f} has a natural crossed product structure. This was proved by Ribet [Rib80] and Momose [Mom81] in the case of weight 2, and extended to forms of higher weight in [BG03] subject to an injectivity constraint, which we remove here. It follows that X_{f} is a central simple algebra over a subfield F of the Hecke field of f. Moreover X_{f} is 2-torsion when considered as an element of the Brauer group of F. Thus X_{f} is isomorphic to a matrix algebra over F, or a matrix algebra over a quaternion division algebra over F. Ribet has remarked in [Rib80] that it seems difficult to distinguish these cases by pure thought. His remark pertains to the case of weight 2, but is equally relevant in higher weight. The chief motivation of this paper (and to a large extent [BG03]) is to give as complete a picture as possible of the Brauer class of X_{f}.

In recent years the notion of slope has played an important role in the theory of elliptic modular forms. For instance this notion is fundamental in parameterizing families of elliptic modular cusp forms, as in the work of Hida (slope 0) and Gouvêa, Mazur, Coleman (finite slope). Remarkably, the notion of slope turns out to be useful in studying the Brauer class of modular endomorphism algebras as well. In fact our main result is that at a finite place of F not dividing the level of f the ramification of X_{f} is essentially completely determined by the parity of the (normalized) slope of f when this slope is finite.

At places of F dividing the level our knowledge of the ramification of X_{f} is less complete. However we show that it is still governed to some extent by the slopes of f, at least at certain places where the underlying local representation is in the principal series. On the other hand at places for which the local representation is of Steinberg type one knows quite a bit about the ramification of X_{f} (see [BG03]). Predicting the ramification of X_{f} at the remaining bad places (the supercuspidal places) is still an open problem.

We end this paper with tables of the Brauer class of X_{f} for all forms f of small weight and level with F = ℚ.

Date: February 5, 2005.

2. Statement of results

We now give more precise statements of our results. Let f = ∑ a_{n}q^{n} be a primitive cusp form of weight k ≥ 2, level N ≥ 1 and nebentypus ϵ. Here primitive means that f is a normalized newform that is a common eigenform of all the Hecke operators. Let M_{f} denote the abelian variety associated to f as constructed by Shimura when k = 2, and let M_{f} denote the Grothendieck motive attached to f constructed by Scholl in [Sch90] when k > 2. Let End(M_{f}) be the ring of endomorphisms of M_{f} defined over \overline{ℚ}. When k > 2 we work modulo cohomological equivalence. Set X_{f} := End(M_{f}) ⊗ ℚ.

The first result is that X_{f} has a natural structure of a crossed product algebra. To state this result more precisely we need some notation. Let E = ℚ(a_{n}) denote the Hecke field of f. Then E is either a totally real or a CM number field. Assume from now on that f does not have complex multiplication. A pair (γ, χ_{γ}) where γ ∈ Aut(E) and χ_{γ} is an E-valued Dirichlet character is said to be an extra twist for f if a_{p}^{γ} = a_{p} · χ_{γ}(p) for all but finitely many primes p. Let Γ denote the set of γ ∈ Aut(E) such that f has a twist by (γ, χ_{γ}) for some E-valued Dirichlet character χ_{γ}. In turns out that Γ is an abelian subgroup of Aut(E). For γ, δ ∈ Γ set

c(γ, δ) = \frac{G(χ_{γ}^{−1})G(χ_{δ}^{−γ})}{G(χ_{γδ}^{−1})} (2.0.1)

where G(χ) is the Gauss sum of the primitive Dirichlet character associated to χ. Then c ∈ Z^{2}(Γ, E^{×}) is a 2-cocycle which turns out to be E^{×}-valued. Let X denote the crossed product algebra associated to c. For the reader’s convenience we recall the definition of X. For each γ ∈ Γ let x_{γ} denote a formal symbol. Then as an E-vector space X is finite dimensional with basis given by the symbols x_{γ}:

X = ⨁_{γ∈Γ} E x_{γ},

and as an algebra X has structure given by the relations

x_{γ} · e = γ(e) x_{γ}
x_{γ} · x_{δ} = c(γ, δ) x_{γδ},

where e ∈ E and γ, δ ∈ Γ.

If k = 2 then it is a result of Ribet [Rib80, Theorem 5.1] and Momose [Mom81, Theorem 4.1] that X_{f} is isomorphic to X. On the other hand if k > 2 then X_{f} contains a sub-algebra isomorphic to X (see [Mom81] and [BG03, Theorem 1.0.1]). Here, building on the above results, we prove the following.

Theorem 2.0.2. Let k ≥ 2. Then X_{f} is isomorphic to X.

Let F be the number field contained in E which is the fixed field of Γ. Then X is isomorphic to a central simple algebra over F which is easily seen to be 2-torsion when considered as an element of the Brauer group of F. As a result X is either a matrix algebra over F or a matrix algebra over a quaternion division algebra over F. We wish to distinguish these cases.

Recall that by global class field theory there is an injection

Br(F) ↪ ⊕_{v}Br(F_{v})

where v runs through the places of F and F_{v} is the completion of F at v. Thus to study the Brauer class of X it suffices to study its image X_{v} = X ⊗_{F} F_{v} for each place v of F under the above map. Since X is 2-torsion in the Brauer group of F, the algebra X_{v} is a fortiori either a matrix algebra over F_{v} or a matrix algebra over a quaternion division algebra over F_{v}.

As far as the infinite places are concerned the field F is easily seen to be totally real, and it follows from a result of Momose [Mom81, Theorem 3.1 ii)] (see also [BG03, Theorem 3.1.1]) that X is totally indefinite if k is even or totally definite if k is odd.

Now suppose that v is a finite place of F of residue characteristic p with p coprime to N. Then a_{p}^{2}ϵ(p)^{−1} ∈ F. Set

m_{v} := [F_{v} : ℚ_{p}] · v(a_{p}^{2}ϵ(p)^{−1}) ∈ ℤ ∪ {∞}

where v is normalized so that v(p) = 1. We shall show that the structure of X_{v} is essentially determined by the parity of m_{v} when it is finite. If w is a place of E lying over v then w(a_{p}) = \frac{1}{2}v(a_{p}^{2}ϵ(p)^{−1}) is called the slope of f at p (with respect to w). Thus we show that there is a close connection between the ramification of X_{v} away from the level and the parity of (normalized) slopes. More precisely we have the following theorem.

Theorem 2.0.3. Let p be a prime such that • p does not divide N, and,

• p ≠ 2 if F ≠ ℚ. Also assume a_{p} ≠ 0. Let v be a place of F lying over p. Then

X_{v} is a matrix algebra over F_{v} if and only if m_{v} ∈ ℤ is even,

except possibly in the exceptional case that p splits in all the quadratic fields cut out by the extra twists of f in which case X_{v} is necessarily a matrix algebra over F_{v}.

A result of this kind was proved in [BG03, Theorem 1.0.4] for cusp forms having quadratic extra twists, or equivalently, real nebentypus character. That such a result might be true in general became clear after extensive numerical computations were made by the second author. The proof of the general case combines ideas from [Que98] and [BG03].

If f is ordinary at v (that is v(a_{p}^{2}ϵ(p)^{−1}) = 0) then m_{v} = 0, and it follows from the theorem that X_{v} is a matrix algebra over F_{v}. This was already known even if p = 2 (see [Rib81, Theorem 6] for k = 2 and [BG03, Theorem 3.3.1] for k > 2). Thus the theorem above may be considered as a generalization of these results.

For a more detailed explanation of the exceptional case mentioned in the statement of the theorem the reader is referred to Theorem 4.0.18 below. It is also possible to deal with the case a_{p} = 0 (and p still coprime to N). In this case m_{v} blows up but it may easily be substituted for by a closely related (finite) integer (see Proposition 4.0.25). We point out here that the above mentioned results imply that X can only be ramified at the primes dividing

2 · N · disc(E) · ∞

where disc(E) is the discriminant of E and ∞ is the unique infinite place of ℚ (Corollary 4.0.27).

The proof of the theorem above is based on an explicit computation of symbols appearing in a formula for the Brauer class of X. A more conceptual approach, based on a study of the filtered (ϕ, N)-modules of Fontaine attached to the local Galois representations associated to f, is available. This approach has so far been more successful for studying the ramification of X only when the slope is small compared to the weight (see for instance [BG03, Theorem 1.0.3]). However, it can be applied to study the ramification of X at some bad places v|N (see [BG03, Theorems 1.0.5 and 1.0.6]). Here we push this approach further and prove the following result.

Theorem 2.0.4. Suppose that p|N and that the power of p dividing N is the same as the power of p dividing the conductor of ϵ. Let v be a place of F lying over p. Let α ∈ ℚ be such that 0 ≤ α < (k − 1)/2 and α has odd denominator. If for each place w of E lying over v either

w(a_{p}) = α or w̄(a_{p}) = α

then X_{v} is a matrix algebra over F_{v}.

If the power of p dividing N is larger than the power of p dividing the conductor of ϵ, our knowledge of X_{v} is less complete, except in the very special case when p||N and the conductor of ϵ is prime to p (the Steinberg case). In this last case it turns out that the ramification of X_{v} is related to the parity of the weight k of f. For more precise statements see the end of section 3 of [BG03].

3. Crossed Product Structure

In this section we prove that X_{f} is isomorphic to the crossed product algebra X (Theorem 2.0.2). In view of the work of Ribet and Momose we shall assume in this section that k > 2.

Let ℓ be a prime and let M_{ℓ} denote the ℓ-adic realization of M_{f}. Recall that M_{ℓ} is a ℚ_{ℓ}-vector space with an action of Gal(\overline{ℚ}/ℚ). For a subgroup H of Gal(\overline{ℚ}/ℚ) let End_{H}(M_{ℓ}) denote the endomorphisms of M_{ℓ} which commute with H. An endomorphism of M_{f} gives rise to an endomorphism of each of its realizations. One therefore obtains a map

α : End(M_{f}) ⊗ ℚ_{ℓ} → End_{H}(M_{ℓ}). (3.0.5)

where H is a sufficiently deep finite index subgroup of Gal(\overline{ℚ}/ℚ). In [BG03] it was shown that X_{f} contains a sub-algebra generated by certain twisting operators which is isomorphic to the crossed product algebra X. Moreover it was shown that if α is injective then X_{f} is isomorphic to X because of dimension considerations. Indeed, by [Rib80, Theorem 4.4] (whose proof carries over to the case k > 2), the ℚ_{ℓ}-dimension of End_{H}(M_{ℓ}) is [E : F][E : ℚ], which is also the ℚ_{ℓ}-dimension of X ⊗ ℚ_{ℓ}.

So it suffices to prove that α is injective. To do this we work slightly more generally. Let X be a smooth irreducible projective variety over ℚ of dimension d. Let Z(X × X) be the rational vector space generated by the irreducible subvarieties of X × X over \overline{ℚ} of codimension d. Fix an embedding \overline{ℚ} ↪ ℂ. Let H_{B}^{2d}(X × X_{/ℂ})(d) denote Betti cohomology (with coefficients (2πi)^{d}ℚ) and let c_{B} : Z(X × X) → H_{B}^{2d}(X × X_{/ℂ})(d) be the cycle class map. Let Z_{h}(X × X) be the quotient

Z_{h}(X × X) := Z(X × X)/ ker(c_{B}) = Z(X × X)/ ∼

where ∼ is the cohomological equivalence relation. Thus for Z ∈ Z(X ×X) one has Z ∼ 0 if and only if the image of Z in H_{B}^{2d}(X × X_{/ℂ})(d) under c_{B} is zero. Recall that Z_{h}(X ×X) has a natural ring structure where multiplication is induced by the composition product of correspondences. Let p ∈ Z_{h}(X × X) be a projector. Let M = (X, p) be a motive, where X and p are as above. Recall that by definition

End(M) := \frac{\{Z ∈ Z_{h}(X × X) : Z ∘ p = p ∘ Z\}}{\{Z ∈ Z_{h}(X × X) : Z ∘ p = p ∘ Z = 0\}}.

We show that in this setting the natural map

α : End(M) ⊗ ℚ_{ℓ} → End(M_{ℓ}) (3.0.6)

is injective. Note that α is equivariant for the action of Gal(\overline{ℚ}/ℚ) on both sides.

Consider the cycle class map c_{ℓ} : Z(X × X) → H_{ℓ}^{2d}(X × X)(d) to ℓ-adic cohomology. There is a comparison isomorphism

I_{ℓ} : H_{B}^{2d}(X × X_{/ℂ})(d) ⊗ ℚ_{ℓ} ≅ H_{ℓ}^{2d}(X × X_{/ℂ})(d) ≅ H_{ℓ}^{2d}(X × X)(d)

where the first isomorphism is (a twist of) the canonical comparison isomorphism between Betti and ℓ-adic cohomology for smooth projective varieties over ℂ (see [Mil80, Theorem 3.12]), and the second isomorphism is (again a twist of the one) induced by the embedding \overline{ℚ} ↪ ℂ via the proper base change theorem. The two cycle class maps c_{B} and c_{ℓ} are related via I_{ℓ}, that is c_{ℓ} = I_{ℓ} ∘ (c_{B} ⊗ 1) (see page 21 of [Del82] or page 58 of [Jan90]). It follows that the ℓ-adic cycle class map factors through the cycles (Betti-) cohomologically equivalent to zero, inducing a map Z_{h}(X × X) ↪ H_{ℓ}^{2d}(X × X)(d). (It also follows that Z_{h}(X × X) and hence End(M) is defined independently of the embedding \overline{ℚ} ↪ ℂ fixed above). Since Betti cohomology gives a rational structure on ℓ-adic cohomology, the induced map

Z_{h}(X × X) ⊗ ℚ_{ℓ} ↪ H_{ℓ}^{2d}(X × X)(d) (3.0.7)

continues to be injective. This is the key observation that was missed in [BG03]. To deduce from this that the map α in (3.0.6) is also injective is purely formal. We recall the argument here for the sake of completeness. By the Künneth formula

H_{ℓ}^{2d}(X × X)(d) = ⨁_{q=0}^{2d} H_{ℓ}^{q}(X) ⊗ H_{ℓ}^{2d−q}(X)(d) = ⨁_{q=0}^{2d} End(H_{ℓ}^{q}(X))

where the last equality follows since H_{ℓ}^{2d−q}(X)(d) is dual to H_{ℓ}^{q}(X). Let

H_{ℓ}^{q}(X)(p) = Im (p : H_{ℓ}^{q}(X) → H_{ℓ}^{q}(X)) .

By definition End(M_{ℓ}) = ⊕_{q=0}^{2d}End(H_{ℓ}^{q}(X)(p)). Now, the map in (3.0.7) induces a map

{Z ∈ Z_{h}(X × X) : Z ∘ p = p ∘ Z} ⊗ ℚ_{ℓ} → End(M_{l}) (3.0.8)

Let Z = Z ⊗ 1 ∈ Z_{h}(X × X) ⊗ ℚ_{ℓ} belong to the kernel of (3.0.8). Thus we have Zp = pZ and Z = 0 in End(M_{ℓ}). Write Z = Zp + Z(1 − p) and note that Z(1 − p) ∘ p = 0, so Z(1 − p) acts as 0 on each H_{ℓ}^{q}(X)(p). Thus Zp = Z − Z(1 − p) acts as 0 on each H_{ℓ}^{q}(X)(p). Since Zp clearly acts as 0 on the spaces

ker(p : H_{ℓ}^{q}(X) → H_{ℓ}^{q}(X))

we see that Zp = 0 in H_{ℓ}^{2d}(X × X)(d). By the injectivity of (3.0.7) we see that Zp = pZ = 0 in Z_{h}(X × X). It follows that the kernel of the map (3.0.8) is {Z ∈ Z_{h}(X × X) : Z ∘ p = p ∘ Z = 0} ⊗ ℚ_{ℓ}. Thus (3.0.8) induces an injective map End(M) ⊗ ℚ_{ℓ} ↪ End(M_{ℓ}), which is precisely the map α in (3.0.6). This proves Theorem 2.0.2.

4. Ramification and Slopes

Let X ≅ X_{f} denote the endomorphism algebra of M_{f}. We now study the relation between the ramification of X and the slopes of f.

Let G = Gal(\overline{ℚ}/ℚ) and let G_{F} = Gal(\overline{ℚ}/F). We will sometimes consider Dirichlet characters as characters of G. In particular ϵ is a character of G. For each g ∈ G let \sqrt{ϵ(g)} be a fixed square-root of ϵ(g). Now, for every extra twist (γ, χ_{γ}) of f, and every pre-image of γ in G_{F}, which we again denote by γ, there is a unique primitive Dirichlet character ψ_{γ} of order 1 or 2 such that χ_{γ}(g) = ψ_{γ}(g) · \sqrt{ϵ(g)}^{γ−1} for all g ∈ G (cf. [Mom81, Lemma 1.5] and [Que98, Lemme 2]). Let ℚ(\sqrt{t_{γ}}) be the quadratic field corresponding to ψ_{γ} (in the case that it has order 2). We assume that t_{γ} is also the discriminant of this field. The characters {ψ_{γ} | γ ∈ G_{F} } form an elementary 2-group. Fix once and for all a subset Γ_{0} ⊂ G_{F} such that {ψ_{γ} | γ ∈ Γ_{0}} is a basis for this group. For each γ ∈ Γ_{0} choose square-free positive integers n_{γ} prime to N such that a_{n_{γ}} ≠ 0, and such that for all γ′ ∈ Γ_{0},

ψ_{γ′}(n_{γ}) = \begin{cases}−1 & if γ′ = γ, \\ 1 & if not. \end{cases} (4.0.9)

Also, for a square free integer n which is prime to N, set z_{n} = a_{n}^{2}ϵ(n)^{−1} ∈ F.

Let G_{F} = Gal(\overline{ℚ}/F). Let [c_{ϵ}] denote the class of the cocycle c_{ϵ} ∈ Z^{2}(G_{F} , ±1) defined by

c_{ϵ}(g, h) = \sqrt{ϵ(g)}\sqrt{ϵ(h)}\sqrt{ϵ(gh)}^{−1},

for g, h ∈ G_{F} (see [Que98, Section 2]).

The following result expressing the Brauer class of X in terms of symbols was proved in [Que98, Théorème 3] in the case k = 2.

Theorem 4.0.10. Let k ≥ 2. Then

X = [c_{ϵ}] ⊗ ⨂_{γ∈Γ_{0}} (z_{n_{γ}} , t_{γ})

up to Brauer equivalence.

Proof. We make some brief remarks which show that the proof given in [Que98] for the case k = 2 continues to hold if k > 2.

Each g ∈ G acts naturally as an automorphism on X and this automorphism fixes E since the elements of E are basically Hecke operators and so are defined over ℚ. By the Skolem-Noether theorem the action of g on X must be given by inner conjugation by some element e ∈ X, which is well defined modulo F^{×}. Since E is its own commutant in X we see that e ∈ E. The association g ↦ e defines a continuous character α : G → E^{×}/F^{×} where the target has the discrete topology. Write α̃ : G → E^{×} for any lift of α.

The first point to note is that the result [Rib92, Theorem 5.6], which says that the class of X in Br(F) is cut out by the 2-cocycle

(g, h) ↦ \frac{α̃(g)α̃(h)}{α̃(gh)} (4.0.11)

for g, h ∈ G_{F}, continues to hold if k > 2. Indeed, the class of X in Br(F) is the same as the class of the cocycle c(g, h) ∈ Z^{2}(G_{F} , F̄^{×}), naturally obtained from the Jacobi sum cocycle c(γ, δ) ∈ Z^{2}(Γ, E^{×}) in (2.0.1) by inflation. On the other hand by [Rib81, Proposition 1], which is easily seen to hold for k > 2, this last cocycle cuts out the same class in Br(F) as the cocycle

(g, h) ↦ χ_{g}(h) (4.0.12)

where χ_{g} = χ_{γ} for the image γ ∈ Γ of g ∈ G_{F}, and χ_{g} is thought of as a character of G. Now using [Rib81, Theorem 4], which says that α̃(h)^{g−1} = χ_{g}(h), we see that the cocycle (4.0.12) defines the same class in Br(F) as the cocycle (g, h) ↦ α̃(h)^{g−1}. The proof of [Rib81, Theorem 4] uses the Tate conjecture for the abelian variety attached to f. This is proved in [BG03, Corollary 1.0.2] for the motive M_{f} when k > 2 subject to an injectivity hypothesis which we have removed in Section 3. Finally, this last cocycle differs from the cocycle (4.0.11) by the map

(g, h) ↦ \frac{α̃(h)^{g}α̃(g)}{α̃(gh)}

which is a coboundary.

The second point to note is that the cocycle (4.0.11) is equal to c_{ϵ} · c_{d} up to a coboundary, where c_{d} is the cocycle defined by

c_{d}(g, h) = (α̃(h)/\sqrt{ϵ(h)})^{1−g},

for g, h ∈ G_{F} (see the beginning of Section 2 of [Que98]). The rest of the proof of the theorem, which involves writing the class of c_{d} as a product of symbols, proceeds exactly as in the case k = 2 without any change. This proves the theorem. □

Remark 4.0.13. A formula similar to that appearing in the theorem above was proved in [BG03, Theorem 4.1.3] in the case that all the χ_{γ} are quadratic characters. This formula was proved by directly computing quadratic Gauss sums.

We wish to evaluate the symbols that appear in the expression for X in Theorem 4.0.10. To do this let us recall some general facts about symbols from [Ser79]. Let F be an arbitrary number field. Let v denote a place of F which is either finite or infinite. Let F_{v} denote the completion of F at v. It is well known that

(4.0.14)  Br(F_{v}) ≅ ℚ/ℤ

if v is finite (and is ℤ/2 if v is infinite and real and is trivial if v is infinite and complex). Now let a and b be non-zero elements of F. Then the symbol (a, b) determines an element in Br(F)[2]. For each finite place v of F, let (a, b)_{v} denote the induced element of Br(F_{v})[2]. By (4.0.14) the symbol (a, b)_{v} is completely specified by a sign +1 or −1. This sign can be computed in terms of the v-adic valuations of a and b. There are two cases, the tame case: v | 2̸, and the wild case: v|2.

First assume that v is prime to 2. Fix a uniformizer π_{v} of the ring of integers of F_{v}. Write

a = π_{v}^{v(a)} · a′
b = π_{v}^{v(b)} · b′

where we consider v here to be normalized such that v(π_{v}) = 1. In this section v will refer to a valuation which is normalized in this way unless explicitly stated otherwise. Then one has

(4.0.15)  (a, b)_{v} = (−1)^{\frac{Nv−1}{2} v(a)v(b)} · (\frac{b′}{v} )^{v(a)} · (\frac{a′}{v} )^{v(b)} .

Here the symbol (\frac{c}{v} ) takes the values ±1 and is 1 exactly when the image of c is a square in the residue field at v.

Now assume that v|2. We shall only treat the case F = ℚ so that v = 2. For a unit u ∈ ℚ_{2}^{×} let ε(u) denote the residue of (u − 1)/2 in ℤ/2 and let ω(u) denote the residue of (u^{2} − 1)/8 in ℤ/2. Then for units u, v in ℚ_{2}^{×} we have

(4.0.16)  (u, v)_{2} = (−1)^{ε(u)ε(v)},
(4.0.17)  (2, u)_{2} = (−1)^{ω(u)}.

Note that these formulas completely determine (a, b)_{2} for a, b ∈ ℚ_{2}^{×}.

Now let us return to our situation. Thus F is the center of X and contains a_{p}^{2}ϵ(p)^{−1} for p prime to N. The usual local-global exact sequence for the Brauer group of F shows that the Brauer class of X is completely determined by the Brauer classes of the X_{v}, which are in turn completely determined by specifying a sign, one for each v. For notational convenience we write X_{v} ∼ a for an integer a if the sign of the Brauer class of X_{v} is the same as (−1)^{a}.

Recall that by a result of Momose [Mom81, Theorem 3.1] one knows X_{v} ∼ k if v is infinite. On the other hand if v|p is a finite place of F with p | N̸ we have X_{v} ∼ 0 if v is ordinary for f, that is if v(a_{p}^{2}ϵ(p)^{−1}) = 0 (see [Rib81, Theorem 6] for the case k = 2 and [BG03, Theorem 3.3.1] for k > 2). The following theorem generalizes this result. To state it we introduce a positive integer m_{v} for each place v of F of residue characteristic p | N̸ with a_{p} ≠ 0:

m_{v} := [F_{v} : ℚ_{p}] · v(a_{p}^{2}ϵ(p)^{−1}).

In the definition of m_{v} we take the valuation v which is normalized such that v(p) = 1. Then we have the following theorem (it is a more precise version of Theorem 2.0.3):

Theorem 4.0.18. Let p be a prime such that p | N̸ and a_{p} ≠ 0. Let v be a place of F lying over p. If p ≠ 2 we have

X_{v} ∼ \begin{cases}0 & if ψ_{γ}(p) = 1 for all γ ∈ Γ_{0}, \\ m_{v} & otherwise. \end{cases}

If p = 2 then the same conclusion holds if F = ℚ.

Proof. The proof is similar to the proof of [BG03, Theorem 4.1.11] the main difference being that we use Theorem 4.0.10 instead of [BG03, Theorem 4.1.3] to compute X locally.

Note [c_{ϵ}]_{v} = 1 if and only if the local component ϵ_{p} of ϵ is even. Since p is prime to N, ϵ_{p} is in fact trivial, so that [c_{ϵ}]_{v} = 1. It follows from Theorem 4.0.10 that

(4.0.19)  X_{v} = ⨂_{γ∈Γ_{0}} (z_{n_{γ}} , t_{γ})_{v}.

Since v is prime to N we have v(t_{γ}) = 0. First assume that p ≠ 2. Then v is

prime to 2 so that by (4.0.15) we have (z_{n_{γ}} , t_{γ})_{v} = (\frac{t_{γ}}{v} )^{v(z_{nγ})}. But (\frac{t_{γ}}{v} ) = (\frac{t_{γ}}{p} )^{f_{v}} since every element of 𝔽_{p} has a square root over a quadratic extension of 𝔽_{p}. We conclude that

(4.0.20)  (z_{n_{γ}} , t_{γ})_{v} = (\frac{t_{γ}}{p} )^{f_{v}·v(z_{nγ})} .

Thus if ψ_{γ}(p) = (\frac{t_{γ}}{p} ) = 1 for all γ ∈ Γ_{0} then X_{v} = 1 as desired.

Suppose on the other hand that the subset S^{−} of the set {t_{γ} | γ ∈ Γ_{0}} consisting

of those t_{γ} for which (\frac{t_{γ}}{p} ) = −1 is non-empty. Write the elements of S^{−} as t_{1}, t_{2}, …, t_{m} with m ≥ 1. Define distinct primes r_{j} for j = 0, 1, …, r_{m−1} as follows: set r_{0} = p and define r_{j} for j = 1, …, m − 1 recursively by

(4.0.21)  (\frac{t_{i}}{r_{j}} ) = (−1)^{δ_{ij}} · (\frac{t_{i}}{r_{j−1}} ) for all i = 1, …, m, and,
(4.0.22)  (\frac{t_{γ}}{r_{j}} ) = 1 if t_{γ} ∉ S^{−}.

We may and do assume that each a_{r_{j}} ≠ 0. This can be done for j = 0 since a_{p} ≠ 0 by hypothesis. For the other r_{j}’s we simply note that if a_{r_{j}} = 0 for all r_{j} defined by the congruence conditions (4.0.21) and (4.0.22) then the set of primes p for which a_{p} = 0 would have a positive density contradicting Serre [Ser81, Theorem 15].

Corresponding to t_{i} ∈ S^{−} set

n_{i} = \begin{cases}r_{i−1} · r_{i} & if 1 ≤ i ≤ m − 1, \\ r_{m−1} & if i = m. \end{cases} (4.0.23)

Clearly the n_{i} are square free positive integers prime to the level satisfying a_{n_{i}} ≠ 0 since Fourier coefficients are multiplicative on distinct primes and the r_{j} were chosen so that each a_{r_{j}} ≠ 0. Furthermore the n_{i} satisfy the congruence conditions (4.0.9). Indeed suppose that t_{i} corresponds to γ ∈ Γ_{0}. Assume first that i < m. Then

ψ_{γ}(n_{i}) = ψ_{γ}(r_{i−1})ψ_{γ}(r_{i}) = −1

since, by (4.0.21), ψ_{γ}(r_{i−1}) and ψ_{γ}(r_{i}) differ by a sign. Similarly if γ′ corresponds to t_{j} for j ≠ i then ψ_{γ′}(r_{i−1}) = ψ_{γ′}(r_{i}) by (4.0.21) again so that ψ_{γ′}(n_{i}) = 1. Finally if γ′ corresponds to some t_{γ′} ∉ S^{−} then by (4.0.22) ψ_{γ′}(n_{i}) = 1. Now assume that i = m. Then for any γ′ ∈ Γ_{0}

ψ_{γ′}(n_{m}) = ψ_{γ′}(r_{m−1}) = (\frac{t_{γ′}}{r_{m−1}} ) .

But (4.0.21) shows that

(\frac{t_{i}}{r_{m−1}} ) = \begin{cases}− (\frac{t_{i}}{p} ) = 1 & if i ≤ m − 1, \\ (\frac{t_{i}}{p} ) = −1 & if i = m. \end{cases}

So this along with (4.0.22) shows that ψ_{γ′}(n_{m}) = −1 if and only if γ′ = γ as desired.

We are now ready to begin computing our symbols. Only those γ for which t_{i} ∈ S^{−} contribute to the sign of X_{v} in (4.0.19) since if t_{γ} ∉ S^{−} then (\frac{t_{γ}}{v} ) = (\frac{t_{γ}}{p} ) = 1 and the corresponding local symbol is trivial by (4.0.20). Now for t_{i} ∈ S^{−} we have z_{n_{i}} = a_{n_{i}}^{2}ϵ(n_{i})^{−1} and (\frac{t_{i}}{p} ) = −1 so that by (4.0.20) we have

(z_{n_{i}}, t_{i})_{v} ∼ f_{v} · v(a_{n_{i}}^{2}ϵ(n_{i})^{−1}).

Substituting for n_{i} from (4.0.23) above and multiplying over all i in {1, …, m}, there is a mod 2 telescoping effect, the result of which is

X_{v} ∼ f_{v} · v(a_{p}^{2}ϵ(p)^{−1}).

If we take the v(p) = 1 normalization for v then the right hand side becomes m_{v}, proving the theorem in the case p ≠ 2.

Now assume that p = 2 and that F = ℚ. Write v_{2}(z_{n_{γ}}) for the power of 2 that divides z_{n_{γ}} and define z′_{n_{γ}} by z_{n_{γ}} = 2^{v_{2}(z_{nγ})} · z′_{n_{γ}}. We have

(z_{n_{γ}} , t_{γ})_{2} = (2, t_{γ})_{2}^{v_{2}(z_{nγ})} · (z′_{n_{γ}} , t_{γ})_{2}.

Since N is prime to p = 2 by hypothesis t_{γ} must be odd. One can easily check that (2, t_{γ})_{2} is equal to (−1)^{ω(t_{γ})} by (4.0.17) which may again be easily checked to be the same as (\frac{t_{γ}}{2} ) using the fact 2 splits in ℚ(\sqrt{t_{γ}}) if and only if t_{γ} ≡ 1 mod 8. On the other hand (z′_{n_{γ}} , t_{γ})_{2} = 1 by (4.0.16) since ε(t_{γ}) ≡ 0 mod 2. Thus

(z_{n_{γ}} , t_{γ})_{2} = (\frac{t_{γ}}{2} )^{v_{2}(z_{nγ})} .

Now the argument proceeds as in the case p ≠ 2 proving the theorem in this case as well. □

Remark 4.0.24. The assumption that F = ℚ when p = 2 could probably be removed if one had formulas for wild symbols other than in the case F = ℚ.

As in [BG03] it is possible to treat the case a_{p} = 0 (and p still prime to N) with minimal effort. The structure of X_{v} is not determined by the parity of m_{v} since m_{v} = ∞. Thus the notion of slope is not useful in measuring the ramification in this case. However as we now show the structure of X_{v} is still determined by the v-adic valuation of a Fourier coefficient at a prime p^{†}, closely related to p.

In fact we take p^{†} to be any prime such that pp^{†} ≡ 1 mod N and such that a_{p^{†}} ≠ 0. Serre’s result, quoted above, guarantees that one can always find such a p^{†}. Set

m_{v}^{†} := [F_{v} : ℚ_{p}] · v(a_{p^{†}}^{2}ϵ(p^{†})).

Theorem 4.0.18 now has the following avatar when a_{p} = 0.

Proposition 4.0.25. Let v be a place of F of residue characteristic p prime to N and assume a_{p} = 0. Let m_{v}^{†} be as above. If p ≠ 2 we have

X_{v} ∼ \begin{cases}0 & if ψ_{γ}(p) = 1 for all γ ∈ Γ_{0}, \\ m_{v}^{†} & otherwise. \end{cases}

If p = 2 then the same conclusion holds if F = ℚ.

Proof. Since pp^{†} ≡ 1 mod N we have

(\frac{t_{γ}}{p} ) = (\frac{t_{γ}}{p^{†}} )

so that the proof of Theorem 4.0.18 goes through replacing p with p^{†}. □

Let us record the following easy consequences of the above results.

Corollary 4.0.26. Let v be a place of F of residue characteristic p with p | 2̸N. If v is unramified in E then X_{v} is a matrix algebra over F_{v}.

Proof. This is immediate from Theorem 4.0.18 and Proposition 4.0.25 since in this case the integer m_{v} or m_{v}^{†} is necessarily even. It may also be proved directly by studying the formula in Theorem 4.0.10. Indeed if v | 2̸N is a finite prime of F for which X_{v} is ramified then the normalized v-adic valuation of at least one of the entries in the symbols appearing in that theorem must be odd. Since t_{γ}|N for all γ ∈ Γ the only possibility is that the normalized valuation v(z_{q}) must be odd for some prime q. One checks easily then that v ramifies in F (a_{q}) so that v ramifies in E. □

Corollary 4.0.27. If X is ramified at v then v must divide either the discriminant of the field E, or 2N, or ∞.

Proof. This is immediate from the previous corollary. □

5. Bad Places

In the previous section we showed how the ramification of X at the good places (v|p | N̸) is essentially determined by the normalized slopes of f at p. In this section we continue our investigation of the ramification of X_{v} at the bad places (v|p|N) that was begun in [BG03]. We recall some notation introduced in that paper.

Assume now that p|N. Let N_{p} be the exponent of the exact power of p that divides N. Let C denote the conductor of ϵ and let C_{p} denote the exponent of the exact power of p that divides C_{p}. Note N_{p} ≥ C_{p} and N_{p} ≥ 1. We consider three cases:

(1) N_{p} = C_{p}: in which case |a_{p}| = p^{(k−1)/2} (Ramified principal series),

(2) N_{p} = 1 and C_{p} = 0: in which case a_{p}^{2} = ϵ(p)p^{k−2} (Steinberg),

(3) N_{p} ≠ C_{p} and N_{p} ≥ 2 in which case a_{p} = 0 (Other).

In the second case the local factor at p in the automorphic representation corresponding to f is the Steinberg representation or a twist of it by an unramified character. This case is treated in [Rib81, Theorem 3] in the case k = 2, and in [BG03] for higher weight forms (see in particular Theorems 3.4.6 and 3.4.8 in [BG03]). Almost nothing is known in the third case, where a_{p} = 0. This case includes twists of previous cases and also cases where the local automorphic representation is supercuspidal. Here we will be concerned with the first case, in which the local automorphic representation is in the ramified principal series. The following theorem contains Theorems 3.4.1, 3.4.2 in [BG03] as special cases, and was stated without proof as [BG03, Theorem 3.4.4].

Theorem 5.0.28. Suppose p|N with N_{p} = C_{p} and let v be a place of F lying over p. Let α ∈ ℚ be such that

0 ≤ α < (k − 1)/2

and α has odd denominator. If for each place w of E lying over v either w(a_{p}) = α or w̄(a_{p}) = α then X_{v} is a matrix algebra over F_{v}.

Proof. Let M_{crys,v} denote the crystal attached to f and v. Let us recall the definition. Fix a place w|v of E. The local Galois representation ρ_{f}|_{G_{p}} : G_{p} → GL_{2}(E_{w}) is potentially crystalline. In fact if K = ℚ(μ_{p^{r}}) where r = N_{p} = C_{p} then ρ_{f}|_{G_{K}} is unramified. Let D_{w} = D_{st}(ρ_{f}|_{G_{K}}) be the associated filtered module. It is a free module of rank 2 over E_{w}. Set M_{crys,v} = ⊕_{w|v}D_{w}. This has dimension 2[E : F][F_{v} : ℚ_{p}] over ℚ_{p}. A study of the crystals D_{w} now show that the crystalline Frobenius ϕ : M_{crys,v} → M_{crys,v} has characteristic polynomial

H(x) = ∏_{w|v} Norm_{E_{w}/ℚ_{p}} ((x − a_{p})(x − ϵ′(p)ā_{p}))

where ϵ′ is the prime-to-p part of ϵ. By hypothesis the Newton polygon of H(x) has two distinct slopes, namely α and k − 1 − α each occurring with equal multiplicity, say n. Let M̄_{crys,v} = M_{crys,v} ⊗_{ℚ_{p}} ℚ_{p}^{un}. It follows that M̄_{crys,v} ≅ C_{α}^{n} ×C_{k−1−α}^{n} where C_{α} and C_{k−1−α} are the simple crystals over ℚ_{p}^{un} of slopes α and k − 1 − α. Now dim_{ℚ_{p}^{un}} C_{α} = dim_{ℚ_{p}^{un}} C_{k−1−α} = s where α = r/s as a fraction in lowest terms. It follows that

2[E : F][F_{v} : ℚ_{p}] = dim_{ℚ_{p}^{un}} M̄_{crys,v} = 2sn.

Now let V = Hom(C_{α}, M̄_{crys,v}) = Hom(C_{α}, C_{α})^{n}. This is a left X_{v}-module of dimension

dim_{F_{v}} V = \frac{s^{2}n}{[F_{v} : ℚ_{p}]} = s[E : F]

Since s is odd, it follows from the representation theory of the algebra X_{v} that X_{v} must split. □

6. Tables of QM Modular Motives

In this section we give complete tables of the endomorphism algebras of all modular motives of small weight (2 ≤ k ≤ 4) and small level (1 ≤ N ≤ 100) with F = ℚ. (For k = 5 see the version of this paper on the first author’s web page). An entry appears in the tables below only if the corresponding motive has quaternionic multiplication (QM for short), that is, only if the class of X is non-zero in the Brauer group of ℚ.

Recall that twisting a form by a Dirichlet character does not change the Brauer class of X (see [Rib81, Proposition 3] for the weight 2 case; the proof there works in all weights). In the interest of conserving space we do not list those entries that are obtained from forms of smaller level by twisting.

That X_{f} can have non-trivial Brauer class was discovered by Shimura: the example on page 166 of [Shi72] appears as the fifth entry in Table 1 below.

Table 1: Modular QM-abelian varieties of level ≤ 100

Label  ord(ϵ)  E  Extra twists Ramification Slope
28A_{[1,1]}  6  ℚ(\sqrt{−1}, \sqrt{3})  [1, 0], [1, 5]  {2 \atop 3}  \begin{matrix}RP S \\ 1 \end{matrix}
35A_{[1,3]}  4  ℚ(\sqrt{10}, \sqrt{−1})  [0, 3], [3, 3]  {2 \atop 5}  \begin{matrix}1 \\ RP S \end{matrix}
44A_{[1,5]}  2  ℚ(\sqrt{2}, \sqrt{−3})  [1, 5], [1, 0]  {2 \atop 3}  \begin{matrix}RP S \\ 1 \end{matrix}
56B_{[1,1,3]}  2  ℚ(\sqrt{−1}, \sqrt{6}) [1, 1, 0], [1, 1, 3]  {2 \atop 3}  \begin{matrix}RP S \\ 1 \end{matrix}
57A_{[1,9]}  2  ℚ(\sqrt{2}, \sqrt{−5})  [1, 9], [0, 9]  {2 \atop 5}  {1 \atop 1}
60A_{[0,1,1]}  4  ℚ(\sqrt{5}, \sqrt{−1}) [0, 0, 3], [0, 1, 0]  {2 \atop 5}  \begin{matrix}a_{p} = 0 \\ RP S \end{matrix}

Label  ord(ϵ)  E  Extra twists Ramification Slope
63A_{[3,1]}  6  ℚ(\sqrt{−2}, \sqrt{6})  [3, 5], [0, 5]  {2 \atop 3}  \begin{matrix}1 \\ a_{p} = 0 \end{matrix}
77B_{[3,5]}  2  ℚ(\sqrt{10}, \sqrt{−2})  [0, 5], [3, 5]  {2 \atop 5}  {1 \atop 1}
80B_{[1,0,1]}  4  ℚ(\sqrt{−1}, \sqrt{3}) [0, 0, 3], [1, 0, 3]  {2 \atop 3}  \begin{matrix}a_{p} = 0 \\ 1 \end{matrix}
92A_{[1,11]}  2  ℚ(\sqrt{−1}, \sqrt{14}) [0, 11], [1, 11]  {2 \atop 7}  \begin{matrix}RP S \\ 1 \end{matrix}
93D_{[1,5]}  6  ℚ(\sqrt{2}, \sqrt{−3})  [1, 0], [1, 25]  {2 \atop 3}  \begin{matrix}1 \\ RP S \end{matrix}
95A_{[1,3]}  12  ℚ(\sqrt{−1}, \sqrt{3})  [3, 15], [3, 0]  {2 \atop 3}  {1 \atop 1}
95B_{[1,3]}  12  ℚ(\sqrt{−1}, \sqrt{3})  [3, 15], [3, 0]  {2 \atop 3}  \begin{matrix}∞ \\ 1 \end{matrix}

Table 2: Modular QM-motives of weight 3 and level ≤ 100

Label  ord(ϵ)  E  Extra twists Ramification Slope
9A_{[1]}^{3}  6  ℚ(\sqrt{−3})  [5]  3  RP S
10A_{[0,1]}^{3}  4  ℚ(\sqrt{−1})  [3]  2  St
12A_{[1,0]}^{3}  2  ℚ(\sqrt{−3})  [1, 0]  3  St
15A_{[1,0]}^{3}  2  ℚ(\sqrt{−5})  [1, 0]  5  St
18A_{[0,3]}^{3}  2  ℚ(\sqrt{−2})  [3]  2  St
19B_{[9]}^{3}  2  ℚ(\sqrt{−13})  [9]  13  1
20A_{[0,1]}^{3}  4  ℚ(\sqrt{−1})  [0, 3]  2  a_{p} = 0
21A_{[0,1]}^{3}  6  ℚ(\sqrt{−3})  [0, 5]  3  St
21B_{[0,1]}^{3}  6  ℚ(\sqrt{−3})  [0, 5]  3  St
21C_{[0,1]}^{3}  6  ℚ(\sqrt{−3})  [0, 5]  3  St
21A_{[0,3]}^{3}  2  ℚ(\sqrt{−3})  [0, 3]  3  St
21B_{[1,2]}^{3}  6  ℚ(\sqrt{−3}, \sqrt{15})  [0, 4], [1, 4]  5  1
22A_{[0,5]}^{3}  2  ℚ(\sqrt{−2})  [5]  2  St
24A_{[0,0,1]}^{3}  2  ℚ(\sqrt{−2})  [0, 0, 1]  2  a_{p} = 0
24C_{[0,1,1]}^{3}  2  ℚ(\sqrt{2}, \sqrt{−7})  [0, 0, 1], [0, 1, 1]  2  RP S
25A_{[5]}^{3}  4  ℚ(\sqrt{−1}, \sqrt{6})  [15], [5]  3  1
26A_{[0,3]}^{3}  4  ℚ(\sqrt{−1})  [9]  2  St
28A_{[0,1]}^{3}  6  ℚ(\sqrt{−3})  [0, 5]  3  1
28A_{[0,3]}^{3}  2  ℚ(\sqrt{−6})  [0, 3]  3  1
30A_{[0,1,2]}^{3}  2  ℚ(\sqrt{2}, \sqrt{−17})  [0, 2], [1, 0]  2  St
31A_{[5]}^{3}  6  ℚ(\sqrt{−3})  [25]  3  1
31A_{[15]}^{3}  2  ℚ(\sqrt{−26})  [15]  13  1
33A_{[1,0]}^{3}  2  ℚ(\sqrt{−11})  [1, 0]  11  St
35A_{[0,3]}^{3}  2  ℚ(\sqrt{−5})  [0, 3]  5  St
35B_{[0,3]}^{3}  2  ℚ(\sqrt{−5})  [0, 3]  5  St
35C_{[2,3]}^{3}  2  ℚ(\sqrt{10}, \sqrt{−1})  [0, 3], [2, 3]  5  RP S
36A_{[1,2]}^{3}  6  ℚ(\sqrt{−3})  [1, 4]  3  RP S
38A_{[0,9]}^{3}  2  ℚ(\sqrt{−2})  [9]  2  St
39C_{[1,6]}^{3}  2  ℚ(\sqrt{−35}, \sqrt{3})  [1, 6], [1, 0]  7  1

Label  ord(ϵ)  E  Extra twists Ramification Slope
40A_{[0,0,1]}^{3}  4  ℚ(\sqrt{−1})  [0, 0, 3]  2  a_{p} = 0
42A_{[0,1,2]}^{3}  6  ℚ(\sqrt{−2}, \sqrt{6})  [0, 4], [1, 4]  2  St
45A_{[0,1]}^{3}  4  ℚ(\sqrt{−1}, \sqrt{10})  [3, 0], [3, 3]  5  RP S
45A_{[3,2]}^{3}  2  ℚ(\sqrt{7}, \sqrt{−2})  [0, 2], [3, 2]  7  1
47A_{[23]}^{3}  2  ℚ(\sqrt{−78})  [23]  13  1
48A_{[1,0,0]}^{3}  2  ℚ(\sqrt{−3})  [1, 0, 0]  3  St
50A_{[0,5]}^{3}  4  ℚ(\sqrt{−1})  [15]  2  St
54A_{[0,9]}^{3}  2  ℚ(\sqrt{−2})  [9]  2  St
55D_{[2,5]}^{3}  2  ℚ(\sqrt{−21}, \sqrt{5})  [2, 0], [2, 5]  3  1
56A_{[1,1,2]}^{3}  6  ℚ(\sqrt{−3})  [1, 1, 4]  2  RP S
57A_{[0,9]}^{3}  2  ℚ(\sqrt{−3})  [0, 9]  3  St
60A_{[0,1,0]}^{3}  2  ℚ(\sqrt{−5})  [0, 1, 0]  5  St
60A_{[0,1,2]}^{3}  2  ℚ(\sqrt{−1}, \sqrt{5})  [0, 0, 2], [0, 1, 0]  5  RP S
60A_{[1,0,2]}^{3}  2  ℚ(\sqrt{−1}, \sqrt{3})  [1, 0, 2], [1, 0, 0]  3  St
63E_{[0,1]}^{3}  6  ℚ(\sqrt{−3}, \sqrt{13})  [3, 5], [0, 5]  3  a_{p} = 0
64A_{[1,8]}^{3}  2  ℚ(\sqrt{−1}, \sqrt{3})  [1, 8], [1, 0]  3  1
72A_{[0,0,3]}^{3}  2  ℚ(\sqrt{−2})  [0, 0, 3]  2  a_{p} = 0
72C_{[1,1,0]}^{3}  2  ℚ(\sqrt{10}, \sqrt{−6}) [0, 0, 3], [1, 1, 0]  2  RP S
74A_{[0,9]}^{3}  4  ℚ(\sqrt{−1})  [27]  2  St
74B_{[0,9]}^{3}  4  ℚ(\sqrt{−1})  [27]  2  St
74C_{[0,9]}^{3}  4  ℚ(\sqrt{−1})  [27]  2  St
75A_{[0,5]}^{3}  4  ℚ(\sqrt{−1}, \sqrt{6})  [0, 15], [0, 5]  3  St
75B_{[0,5]}^{3}  4  ℚ(\sqrt{−1}, \sqrt{6})  [0, 15], [0, 5]  3  St
75C_{[1,0]}^{3}  2  ℚ(\sqrt{−11})  [1, 0]  11  1
76A_{[0,9]}^{3}  2  ℚ(\sqrt{−29})  [0, 9]  29  1
77A_{[0,5]}^{3}  2  ℚ(\sqrt{−7})  [0, 5]  7  St
77A_{[2,5]}^{3}  6  ℚ(\sqrt{−3}, \sqrt{21})  [4, 5], [2, 0]  7  RP S
78A_{[0,1,6]}^{3}  2  ℚ(\sqrt{2}, \sqrt{−5})  [0, 6], [1, 0]  2  St
78B_{[0,1,6]}^{3}  2  ℚ(\sqrt{2}, \sqrt{−5})  [0, 6], [1, 0]  2  St
80B_{[1,0,2]}^{3}  2  ℚ(\sqrt{2}, \sqrt{−3})  [0, 0, 2], [1, 0, 0]  2  a_{p} = 0
84A_{[0,0,1]}^{3}  6  ℚ(\sqrt{−3})  [0, 0, 5]  3  St
84A_{[0,0,3]}^{3}  2  ℚ(\sqrt{−3})  [0, 0, 3]  3  St
84B_{[0,1,2]}^{3}  6  ℚ(\sqrt{−3}, \sqrt{15}) [0, 0, 2], [0, 1, 4]  5  1
84C_{[0,1,2]}^{3}  6  ℚ(\sqrt{−3}, \sqrt{105}) [0, 0, 4], [0, 1, 4]  5  1
84A_{[1,0,2]}^{3}  6  ℚ(\sqrt{−3})  [1, 0, 4]  3  St
86A_{[0,21]}^{3}  2  ℚ(\sqrt{−2})  [21]  2  St
90A_{[0,0,1]}^{3}  4  ℚ(\sqrt{−1})  [0, 3]  2  St
90B_{[0,0,1]}^{3}  4  ℚ(\sqrt{−1})  [0, 3]  2  St
90A_{[0,3,2]}^{3}  2  ℚ(\sqrt{2}, \sqrt{−1})  [0, 2], [3, 2]  2  St
91A_{[3,4]}^{3}  6  ℚ(\sqrt{−3}, \sqrt{39})  [0, 4], [3, 0]  13  RP S
91C_{[3,6]}^{3}  2  ℚ(\sqrt{26}, \sqrt{−1})  [3, 0], [3, 6]  13  RP S
93A_{[0,15]}^{3}  2  ℚ(\sqrt{−3})  [0, 15]  3  St
96A_{[0,0,1]}^{3}  2  ℚ(\sqrt{−2}, \sqrt{3})  [0, 0, 1], [1, 0, 0]  3  RP S
96B_{[0,0,1]}^{3}  2  ℚ(\sqrt{7}, \sqrt{−2})  [0, 0, 1], [1, 0, 1]  7  1
99C_{[0,5]}^{3}  2  ℚ(\sqrt{−138}, \sqrt{−3})  [3, 0], [0, 5]  23  1
100B_{[0,5]}^{3}  4  ℚ(\sqrt{−1}, \sqrt{6})  [0, 15], [0, 5]  3  3

MODULAR BRAUER CLASSES 15

Table 3: Modular QM-motives of weight 4 and level ≤ 100

Label  ord(ϵ)  E  Extra twists  Ramification Slope
12A_{[1,1]}^{4}  2  ℚ(\sqrt{−5}, \sqrt{3})  [0, 1], [1, 0]  {3 \atop 5}  \begin{matrix}RP S \\ 1 \end{matrix}
21B_{[1,3]}^{4}  2  ℚ(\sqrt{−6}, \sqrt{102})  [1, 3], [0, 3]  {3 \atop 17}  \begin{matrix}RP S \\ 1 \end{matrix}
27C_{[0]}^{4}  1  ℚ(\sqrt{2})  [9]  {2 \atop 3}  \begin{matrix}1 \\ a_{p} = 0 \end{matrix}
35A_{[1,3]}^{4}  4  ℚ(\sqrt{5}, \sqrt{−1})  [0, 3], [3, 0]  {2 \atop 5}  \begin{matrix}3 \\ RP S \end{matrix}
36B_{[1,3]}^{4}  2  ℚ(\sqrt{30}, \sqrt{−2})  [0, 3], [1, 0]  {2 \atop 3}  \begin{matrix}RP S \\ a_{p} = 0 \end{matrix}
48B_{[1,0,1]}^{4}  2  ℚ(\sqrt{−2}, \sqrt{6})  [1, 0, 0], [1, 0, 1]  {2 \atop 3}  \begin{matrix}a_{p} = 0 \\ RP S \end{matrix}
56B_{[1,1,3]}^{4}  2  ℚ(\sqrt{−3}, \sqrt{21})  [0, 0, 3], [1, 1, 3]  {3 \atop 7}  \begin{matrix}1 \\ RP S \end{matrix}
57B_{[1,9]}^{4}  2  ℚ(\sqrt{17}, \sqrt{−10})  [1, 9], [0, 9]  {5 \atop 17}  {1 \atop 1}
63B_{[3,3]}^{4}  2  ℚ(\sqrt{−222}, \sqrt{−2})  [3, 0], [0, 3]  {2 \atop 3}  \begin{matrix}3 \\ a_{p} = 0 \end{matrix}
72C_{[0,1,0]}^{4}  2  ℚ(\sqrt{22}, \sqrt{−10})  [0, 0, 3], [0, 1, 0]  {2 \atop 5}  \begin{matrix}RP S \\ 1 \end{matrix}
80B_{[1,0,1]}^{4}  4  ℚ(\sqrt{−1}, \sqrt{35})  [1, 0, 0], [1, 0, 3]  {2 \atop 7}  \begin{matrix}a_{p} = 0 \\ 1 \end{matrix}
100D_{[1,5]}^{4}  4  ℚ(\sqrt{11}, \sqrt{−1}, \sqrt{5}) [1, 10], [0, 5], [1, 5]  {11 \atop 2}  \begin{matrix}1 \\ RP S \end{matrix}

Table 4: Modular QM-motives of weight 5 and level ≤ 100

Label  ord(ϵ)  E  Extra twists Ramification Slope
5A_{[1]}^{5}  4  ℚ(\sqrt{−1})  [3]  2  1
6A_{[0,1]}^{5}  2  ℚ(\sqrt{−2})  [1]  2  St
8B_{[1,1]}^{5}  2  ℚ(\sqrt{−15})  [1, 1]  5  1
9A_{[3]}^{5}  2  ℚ(\sqrt{−2})  [3]  2  1
10A_{[0,1]}^{5}  4  ℚ(\sqrt{−1})  [3]  2  St
10B_{[0,1]}^{5}  4  ℚ(\sqrt{−1})  [3]  2  St
11B_{[5]}^{5}  2  ℚ(\sqrt{−30})  [5]  2  1
15C_{[1,2]}^{5}  2  ℚ(\sqrt{10}, \sqrt{−26}) [0, 2], [1, 2]  2 13 5  \begin{matrix}1 \\ 1 \\ RP S \end{matrix}
16A_{[1,0]}^{5}  2  ℚ(\sqrt{−3})  [1, 0]  3  1
21A_{[0,1]}^{5}  6  ℚ(\sqrt{−3})  [0, 5]  3  St
21B_{[0,1]}^{5}  6  ℚ(\sqrt{−3})  [0, 5]  3  St
25B_{[5]}^{5}  4  ℚ(\sqrt{−1}, \sqrt{6})  [15], [5]  3  1
25C_{[5]}^{5}  4  ℚ(\sqrt{−1}, \sqrt{21})  [15], [10]  2 3 7  1 1 1
27B_{[9]}^{5}  2  ℚ(\sqrt{−6})  [9]  2  1

Label  ord(ϵ)  E  Extra twists Ramification Slope
28A_{[0,3]}^{5}  2  ℚ(\sqrt{−3})  [0, 3]  3  1
35C_{[2,3]}^{5}  2  ℚ(\sqrt{−6})  [2, 3]  2  1
36A_{[0,3]}^{5}  2  ℚ(\sqrt{−2})  [0, 3]  2  a_{p} = 0
36C_{[1,0]}^{5}  2  ℚ(\sqrt{7}, \sqrt{−1})  [1, 3], [1, 0]  7  1
40A_{[0,0,1]}^{5}  4  ℚ(\sqrt{−1})  [0, 0, 3]  2  a_{p} = 0
44B_{[0,5]}^{5}  2  ℚ(\sqrt{−206})  [0, 5]  2  a_{p} = 0
45A_{[0,1]}^{5}  4  ℚ(\sqrt{−1})  [0, 3]  2  1
45B_{[0,1]}^{5}  4  ℚ(\sqrt{−1})  [0, 3]  2  1
45D_{[0,1]}^{5}  4  ℚ(\sqrt{−1}, \sqrt{10})  [3, 0], [3, 3]  5  RP S
48A_{[1,0,0]}^{5}  2  ℚ(\sqrt{−3})  [1, 0, 0]  3  St
48B_{[1,0,0]}^{5}  2  ℚ(\sqrt{−3})  [1, 0, 0]  3  St
54A_{[0,9]}^{5}  2  ℚ(\sqrt{−2})  [9]  2  St
56A_{[1,1,0]}^{5}  2  ℚ(\sqrt{−7})  [1, 1, 0]  7  St
60A_{[0,1,0]}^{5}  2  ℚ(\sqrt{−5})  [0, 1, 0]  5  St
63C_{[0,3]}^{5}  2  ℚ(\sqrt{10}, \sqrt{−106}) [3, 0], [3, 3]  2  1
64A_{[1,8]}^{5}  2  ℚ(\sqrt{−1}, \sqrt{3})  [1, 8], [1, 0]  3  1
64B_{[1,8]}^{5}  2  ℚ(\sqrt{51}, \sqrt{−1})  [1, 8], [0, 8]  3  1
72A_{[0,0,3]}^{5}  2  ℚ(\sqrt{−2})  [0, 0, 3]  2  a_{p} = 0
72B_{[0,0,3]}^{5}  2  ℚ(\sqrt{−2})  [0, 0, 3]  2  a_{p} = 0
72B_{[1,1,0]}^{5}  2  ℚ(\sqrt{−15})  [1, 1, 0]  5  1
75B_{[0,5]}^{5}  4  ℚ(\sqrt{−1}, \sqrt{6})  [0, 15], [0, 5]  3  St
75C_{[0,5]}^{5}  4  ℚ(\sqrt{−1}, \sqrt{6})  [0, 15], [0, 5]  3  St
75C_{[1,0]}^{5}  2  ℚ(\sqrt{−14})  [1, 0]  2  1
75D_{[1,0]}^{5}  2  ℚ(\sqrt{−35})  [1, 0]  5  a_{p} = 0
80D_{[0,0,1]}^{5}  4  ℚ(\sqrt{−1})  [0, 0, 3]  2  a_{p} = 0
81D_{[9]}^{5}  6  ℚ(\sqrt{2}, \sqrt{−3})  [27], [45]  2  1
90A_{[0,0,1]}^{5}  4  ℚ(\sqrt{−1})  [0, 3]  2  St
90B_{[0,0,1]}^{5}  4  ℚ(\sqrt{−1})  [0, 3]  2  St
99B_{[0,5]}^{5}  2  ℚ(\sqrt{−30})  [0, 5]  2  1
100A_{[0,5]}^{5}  4  ℚ(\sqrt{−1}, \sqrt{6})  [0, 15], [0, 5]  3  1
100B_{[0,5]}^{5}  4  ℚ(\sqrt{−1}, \sqrt{−69}) [0, 5], [0, 10]  2 3 23  \begin{matrix}a_{p} = 0 \\ 1 \\ 1 \end{matrix}

Let us describe how the above tables are labeled. The format is similar to that used in the tables in the Appendix of [BGGP]. The first column contains Galois conjugacy classes of primitive forms of given level, weight and nebentypus. The ordering we use to list these forms is described by a function which maps a primitive form f to a label of the form NX_{ϵ}^{k} (for example 19B_{[9]}^{3}), where N is the level of f, X is a letter or string of letters in {A, B, …, Z, AA, BB, … }, ϵ is an encoding of the nebentypus of f (described in more detail below) and k is the weight of f. When k = 2 we omit the superscript 2. To construct X assume that k, N and ϵ are fixed. To f = ∑ a_{n}q^{n} associate the infinite sequence of integers t_{f} = (Tr_{E/ℚ}a_{1}, Tr_{E/ℚ}a_{2}, …). Choose X ∈ {A, B, …, Z, AA, BB, … } according to the position of t_{f} in the set { t_{g} : g primitive of weight k, level N and nebentypus ϵ } sorted in increasing dictionary order. Notice that t_{f} determines the Galois conjugacy class of f. The above ordering was introduced by J. Cremona in the case of trivial nebentypus and weight 2 and by W. Stein in the general situation.

The encoding of the nebentypus ϵ : (ℤ/N)^{∗} → ℂ^{∗} is done as follows. Let N = ∏ p_{n}^{α_{n}} be the prime-ordered factorization of N. Then for each p_{n} there exists a unique Dirichlet character ϵ_{p_{n}} : (ℤ/p_{n}^{α_{n}})^{×} → ℂ^{∗} such that ϵ = ∏ ϵ_{p_{n}}. Fix p = p_{n} momentarily and write ϵ_{p} for ϵ_{p_{n}}. If p is odd let g_{p} be the smallest positive integer that generates (ℤ/p^{α})^{×}, and if p = 2 and α ≤ 2, let g_{p} = −1. In the above cases ϵ_{p} is determined by the integer e_{p} ∈ [0, φ(p^{α})) such that ϵ_{p}(g_{p}) = e^{2πie_{p}/φ(p^{α})}. If p = 2 and α > 2, then (ℤ/2^{α})^{×} ≅ ℤ/2 × ℤ/2^{α−2} where the first factor is generated by −1 and the second factor by 5. Thus in this case ϵ_{2} is determined by a pair of integers e′_{2} ∈ [0, 2), e′′_{2} ∈ [0, 2^{α−2}) such that ϵ_{2}(−1) = e^{2πie′_{2}/2} and ϵ_{2}(5) = e^{2πie′′_{2}/2^{α−2}}. We denote the pair e′_{2}, e′′_{2} by e_{2}. Finally we denote ϵ by [e_{p_{n}} : p_{n}|N].

The middle columns are as follows: column 2 contains the order of ϵ, column 3 contains the Hecke field E (recall that F = ℚ), column 4 lists a generating set for the extra twists (encoded in a similar manner as described above for ϵ), and column 5 lists the primes where the endomorphism algebra X ramifies.

The last column lists the numbers m_{v} (normalized slope) if p is prime to N and a_{p} ≠ 0. If a (finite) integer occurs in this column it is always ODD as predicted by the main result of this paper (Theorem 2.0.3). If a_{p} = 0 and p is still prime to N then m_{v} = ∞ and the ramification is controlled by Proposition 4.0.25. On the other hand if some p|N is a prime of ramification then we give some further information as follows. Recall that C denotes the conductor of ϵ, and N_{p} and C_{p} denote the exponent of p of the exact power of p diving N and C respectively. If N_{p} = C_{p} we write RP S for ramified principal series. If N_{p} = 1 and C_{p} = 0 then we write St for Steinberg. Finally a_{p} vanishes in the remaining cases N_{p} ≠ C_{p}. These include the cases which are twists of previous cases, in which case the ramification can be sometimes explained, but also includes the cases where the local representation is supercuspidal. In either case we simply write a_{p} = 0. It is worth mentioning that while a_{p} can vanish both when p | N̸ and p|N, in the former case it only occurs once (see 95B_{[1,3]} below) within the scope of the tables.

Corrections to [BG03]: We take this opportunity to correct some errors in [BG03]. The last two errors do not occur in the electronic version of the article.

• page 1655, line 8: N_{p} ≥ C_{p} ≥ 1 should be N_{p} ≥ C_{p} ≥ 0

• page 1669, line 12: \frac{1}{2}(u^{2} − 1) should be (u^{2} − 1)/8

• page 1670, last few lines: ‘supersingular primes’ should read ‘primes p for which a_{p} = 0’, and the last phrase should read ‘for non-CM forms the density of primes p for which a_{p} = 0 is 0’

• page 1672, line 9 from the bottom: ‘ordinary primes’ should read ‘primes p for which a_{p} = 0’.

Acknowledgements: The first author would like to thank N. Fakhruddin, M. Nori, and V. Srinivas for useful discussions concerning the material in Section 3. He would also like to thank the third author for arranging visits to Barcelona in 2002 and 2004. The second author held the position of Contratado Doctor C.A.M. (02/0294/2002) during the preparation of this work. The third author is supported by research projects BFM-2003-0678-C02-01 and 2002SGR 00148. The tables in this paper were prepared using W. Stein’s Magma modular forms package.

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School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India.

   E-mail address: eghate@math.tifr.res.in

Departamento de Matemáticas, Facultad de Ciencias, Universidad de Alcalá, E-28871 Madrid, Spain.

   E-mail address: enrique.gonzalezj@uah.es

Departament Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Jordi Girona, 1-3, Campus Nord, Edifici Omega, Despatx 438, — 08034 Barcelona, Spain

   E-mail address: Jordi.Quer@upc.es