Endomorphism Algebras of

Modular Motives

Eknath Ghate AMS-INDIA meeting

Bangalore, 2003.

This is joint work with Alex Brown. Ann. Inst. Fourier (2003).

Overview

Let f be a cusp form, M_{f} be the associated motive, X_{f} the endomorphism algebra of M_{f}. We wish to study X_{f}. There are two aspects to our study:

1. Geometric: • Construction of endomorphisms of M_{f} • Crossed product structure of X_{f}

• The Tate conjecture.

2. Arithmetic: • The Brauer class of X_{f}.

• Ramification of X_{f} ↔ slopes of f.

• Numerical examples.

Cusp forms

Let f = ∑_{n=1}^{∞} a_{n} q^{n} be a cusp form of • weight k ≥ 2,

• level N ≥ 1,

• and character ϵ : (ℤ/N)^{×} → ℂ^{×}. f(\frac{az+b}{cz+d} ) = ϵ(d)(cz+d)^{k}f(z) for all ({a \atop c} {b \atop d}) ∈ Γ_{0}(N). Assume f is a newform, normalized eigenform. Recall: f has CM if there is a character θ such that a_{p} = a_{p} · θ(p) for all but finitely many primes p. Assume f is NOT of CM type.

Modular motives

There is a geometric object (motive) attached to f:

M_{f} = \begin{cases}Abelian variety, if k = 2 (Shimura)  , \\ Grothendieck motive, if k > 2 (Scholl)  . \end{cases}

M_{f} is a pure motive over ℚ of weight k − 1. Let

X_{f} = End (M_{f}) ⊗ ℚ

be the ℚ-algebra of all endomorphisms of M_{f} which are defined over \overline{ℚ}. When k > 2 we work modulo cohomological equivalence. Set E = ℚ(a_{n}) = number field! Then

E ↪ X_{f} via a_{n} ↦ T_{n}.

We say M_{f} has coefficients in E.

Scholl’s construction of M_{f}

Write k + 2 for the weight, so assume k > 0. Also assume N ≥ 3.

Let M_{N} be the modular curve representing isomorphism classes of pairs (E, σ); E an elliptic curve, σ : E[N] ≅ (ℤ/N)^{2} a level N-structure. Let M̄_{N} be the compactification of M_{N}.

Let π : X_{N} → M_{N} denote the universal elliptic curve and let π̄ : X̄_{N} → M̄_{N} denote the universal generalized elliptic curve.

Let X̄_{N}^{k} → M̄_{N} denote the k-fold fiber product of X̄_{N} with itself over M̄_{N}.

Let ¯ X̄_{N}^{k} denote the canonical desingularization of X̄_{N}^{k} (Deligne).

The group

Γ_{k} = ((ℤ/N)^{2} ⋊ μ_{2})^{k} ⋊ Σ_{k}

acts on X̄_{N}^{k} and on ¯ X̄_{N}^{k}.

Let ε : Γ_{k} → {±1} be the homomorphism which is trivial on (ℤ/N)^{2k}, is the product map on μ_{2}^{k} and is the sign character on Σ_{k}. Let

Π_{ε} = \frac{1}{|Γ_{k}|} _{g∈Γ_{k}}^{∑} ε(g) · g ∈ ℤ [\frac{1}{2N · k!} ] [Γ_{k}]

denote the projector attached to ε and let

W_{N} = ( ¯X̄_{N}^{k}, Π_{ε})

be the associated motive.

Let Γ_{N}^{1} be the subgroup of GL_{2}(ℤ/N) of matrices of the form ( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} ). Let I_{f} be the annhilator of f in the Γ_{1}(N) Hecke algebra.

Scholl defines M_{f} to be the sub-motive of W_{N}^{Γ_{N}^{1}}

which is the kernel of I_{f} acting on W_{N}^{Γ_{N}^{1}} .

Extra Endomorphisms of M_{f} Are there any more endomorphisms of M_{f} other than the Hecke operators (action of E)? Let γ ∈ Aut(E); χ_{γ} be an E^{×}-valued character. The pair (γ, χ_{γ}) is called an extra twist for f if

a_{p}^{γ} = a_{p} · χ_{γ}(p)

for all but finitely many primes p. Theorem. There is a natural construction: extra twist of f ⇝ extra endomorphism of M_{f}. It is possible to describe the sub-algebra of X_{f} generated over E, by these endomorphisms. For each pair of extra twists (γ, χ_{γ}) and (δ, χ_{δ}) consider the 2 cocycle:

c(γ, δ) = \frac{G(χ_{γ}^{−1})G(χ_{δ}^{−γ})}{G(χ_{γδ}^{−1})} ∈ E^{×}.

Let X be the crossed product algebra:

X = ⨁ E · x_{γ}

associated to the cocycle c, with relations:

x_{γ} · x_{δ} = c(γ, δ) x_{γδ}, x_{γ} · e = γ(e) x_{γ}, for e ∈ E. Restatement of the theorem: Theorem′. If k ≥ 2 then X_{f} contains a subalgebra isomorphic to X. Previous results: If k = 2 then X_{f} ≅ X: Ribet ‘80, Momose ‘81. If k > 2 then X ⊂ End (H_{B}(M_{f})): Momose ‘81. If k = 2, f has CM, then X_{f} ≅ matrix algebra over an imaginary quadratic field: Shimura ‘71. Question: If k > 2, f has CM, what is X_{f} ??

Sketch of proof of Theorem: Assume a_{p} = 0 if p|N. Step 1: For each extra twist (γ, χ_{γ}) set

H_{B}(η_{γ}) = _{u mod r}^{∑} χ_{γ}^{−1}(u) ( \begin{matrix}1 & u/r \\ 0 & 1 \end{matrix} )

where r is the conductor of χ_{γ}. Check:

H_{B}(η_{γ}) · f^{σ} = G(χ_{γ}^{−σ}) · f^{γσ}

for each σ : E ↪ ℂ. So H_{B}(η_{γ}) is an endomorphism of the Betti realization H_{B}(M_{f}) ⊗ ℂ. Step 2: Construct an endomorphism

η_{γ} : M_{f} → M_{f}

whose Betti realization is H_{B}(η_{γ}). This uses the cohomological equivalence relation. Step 3: Finally show that the algebra generated by the η_{γ} over E is isomorphic to X.

Tate Conjecture: K = a number field, G_{K} = Gal(\overline{ℚ}/K) ℓ = prime, M_{ℓ} = ℓ-adic realization of M_{f}. There is a natural map

α_{K} : End_{K}(M_{f}) ⊗ ℚ_{l} → End_{G_{K}}(M_{ℓ}). The Tate conjecture for M_{f} predicts that

(??) α_{K} is an isomorphism.

If k = 2, proved by Ribet ‘80, Faltings ’83. Corollary. If k > 2 and α_{K} is injective for K sufficiently large then

1. α_{K} is an isomorphism, and,

2. X_{f} is isomorphic to X.

Brauer Class of X

Let F be the center of X. Then F is real and

F = ℚ(a_{p}^{2}ϵ(p)^{−1} : p | N̸) ⊂ E. Claim: Class of X in Br(F) is 2-torsion.

Proof: X = M_{n}(D) by Wedderburn; dim_{F} (D) = (\frac{[E:F]}{n})^{2}. Now X acts on M_{B} = H_{B}(M_{f}). Hence 2[E : F] = dim_{F} (M_{B}) = dim_{D}(M_{B}) · dim_{F} (D), so,

2 = \frac{dim_{D}(M_{B})}{n} · \frac{[E : F]}{n} ⟹ \frac{[E : F]}{n} = 1 or 2. □

So X is either

1. a matrix algebra over F, or,

2. a matrix algebra over a quaternion division algebra over F.

Question. (Ribet) Which of the above cases occur?

Recall

Br(F)[2] ↪ _{v}^{⨁} Br(F_{v})[2] = _{v}^{⨁} ℤ/2
X ↦ X_{v} = X ⊗_{F} F_{v}.
Write \begin{cases}X_{v} ∼ 0 & if X_{v} splits, \\ X_{v} ∼ 1 & if X_{v} ramifies. \end{cases}

Infinite places: (Momose) v|∞ ⟹ X_{v} ∼ k. Good finite places: Say v|p but p | N̸. Theorem. The ramification of X_{v} is essentially determined by the normalized slope of f at p. More precisely, set

m_{v} := [F_{v} : ℚ_{p}] · v(a_{p}^{2}ϵ(p)^{−1}) ∈ ℤ ∪ {∞}. Assume a_{p} ≠ 0 and p > 2 if F ≠ ℚ. Then

X_{v} ∼ m_{v}

except in an ‘exceptional case’ where X_{v} ∼ 0.

This was proved with A. Brown if all extra twists are quadratic; the general case is treated in forthcoming work with E. Gonzalez, J. Quer. Corollary. (Ordinary case; Ribet ‘81 if k = 2)

v(a_{p}^{2}ϵ(p)^{−1}) = 0 ⟹ X_{v} ∼ 0. Bad finite places: Say v|p and p|N. Let C denote the conductor of ϵ. Then C|N. Let N_{p} = v_{p}(N) and C_{p} = v_{p}(C). So N_{p} ≥ C_{p}. The local factor at p of the automorphic representation attached to f is one of three types:

I  II  III
N_{p} = C_{p}  N_{p} = 1  Other N_{p} ≠ C_{p}
C_{p} = 0  cases
Ramified Unramified twists  Supercuspidal,
Principal  of Steinberg  extraordinary,
Series  representation  twists of I, II

Type I: We have a_{p}ā_{p} = p^{k−1}.

Theorem. Let α ∈ ℚ be such that 0 ≤ α < \frac{k−1}{2} and α has odd denominator. If for each place w|v of E, either w(a_{p}) = α or w(ā_{p}) = α, then

X_{v} ∼ 0.

Corollary. (Potentially ordinary case). For each place w|v of E say

w(a_{p}) = 0 or w(ā_{p}) = 0.

Then X_{v} ∼ 0.

Idea of proof:

Each of the local representations

ρ_{f,w}|_{G_{p}} → GL_{2}(E_{w})

has potentially good reduction over a cyclotomic extension of ℚ_{p}.

Let M_{v} denote the direct sum over w|v of the filtered (ϕ, N)-modules attached to the ρ_{f,w}.

Here are some possibilities for the slopes of M_{v}:

[figure: ordinary; α = 1/2; α = 1; super-; singular; α = 0; α = \frac{k−1}{2}]

Under the hypothesis 0 ≤ α < \frac{k−1}{2}, the slopes splits M_{v} into two pieces. Some linear algebra now shows that X_{v} must split as well. □

(The odd denominator condition ensures that the proof works after a base change, necessary if there are endomorphisms not defined over ℚ_{p}).

Type II: If k = 2 then X_{v} ∼ 0. In fact X is everywhere unramified (Ribet ‘81).

Warning: If k > 2 then X_{v} ∼ 1 is possible!

Eg: if k = 3, there is a form f of level N = 35 with ϵ of conductor 7, such that F = ℚ. Then p = 5 is a ‘Steinberg’ place but X_{p} ∼ 1.

However, one can prove:

Theorem. If k ≥ 2 is even, then X is everywhere unramified.

In general one can say:

Theorem. Say k ≥ 2. Then X is unramified everywhere except possibly at the places of F lying above p and ∞.

Corollary. If f is ‘Steinberg’ at two places then X can ramify only at ∞.

Type III: In the case that the local representation is a twist of I or II, one is reduced to the previous cases, since twisting does not change ramification.

Question: If the local factor is supercuspidal (or extraordinary if p = 2), then

Is X_{v} ramified or split?

Typical numerical example: k = 5, level N = 75, ϵ = ^{(\underline{−3})}. Turns out that there is a form f with

E = ℚ(√10 , √−26 ) and F = ℚ.

The algebra X is ramified exactly at

Prime  Reason
2  a_{2}^{2} = −10 = −2 · 5, m_{v} = 1 (odd!)
5  III: a_{5} = 0
13  a_{13}^{2} = 59904 = 2^{9} · 3^{2} · 13, m_{v} = 1 (odd!)
∞ k = 5 (odd!)

and X is unramified, for example, at

Prime  Reason
3  I: Slopes are 1 and 3 (≠ \frac{k−1}{2})
7  a_{7}^{2} = 234 = 2 · 3^{2} · 13, ordinary
11  a_{11}^{2} = −9360 = −2^{4} · 3^{2} · 5 · 13, ordinary
⋮  ⋮

Tables

E. Gonzalez has computed the Brauer class of X for all forms of small weight k ≤ 5 and level N ≤ 100 in the case F = ℚ. For example:

Table 1: k = 2 and N ≤ 100

Label  ord(ϵ)  E  Extra twists  Ram  Slope
28A_{[1,1]}  6  ℚ(√−1, √3)  [1, 0], [1, 5]  {2 \atop 3}  \begin{matrix}RP S \\ 1 \end{matrix}
35A_{[1,3]}  4  ℚ(√10 , √−1)  [0, 3], [3, 3]  {2 \atop 5}  \begin{matrix}1 \\ RP S \end{matrix}
44A_{[1,5]}  2  ℚ(√2, √−3)  [1, 5], [1, 0]  {2 \atop 3}  \begin{matrix}RP S \\ 1 \end{matrix}
56B_{[1,1,3]}  2  ℚ(√−1, √6) [1, 1, 0], [1, 1, 3]  {2 \atop 3}  \begin{matrix}RP S \\ 1 \end{matrix}
57A_{[1,9]}  2  ℚ(√2, √−5)  [1, 9], [0, 9]  {2 \atop 5}  {1 \atop 1}
60A_{[0,1,1]}  4  ℚ(√5, √−1) [0, 0, 3], [0, 1, 0]  {2 \atop 5}  \begin{matrix}a_{p} = 0 \\ RP S \end{matrix}
63A_{[3,1]}  6  ℚ(√−2, √6)  [3, 5], [0, 5]  {2 \atop 3}  \begin{matrix}1 \\ a_{p} = 0 \end{matrix}
77B_{[3,5]}  2  ℚ(√10 , √−2)  [0, 5], [3, 5]  {2 \atop 5}  {1 \atop 1}
80B_{[1,0,1]}  4  ℚ(√−1, √3) [0, 0, 3], [1, 0, 3]  {2 \atop 3}  \begin{matrix}a_{p} = 0 \\ 1 \end{matrix}
92A_{[1,11]}  2  ℚ(√−1, √14 )  [0, 11], [1, 11]  {2 \atop 7}  \begin{matrix}RP S \\ 1 \end{matrix}
93D_{[1,5]}  6  ℚ(√2, √−3)  [1, 0], [1, 25]  {2 \atop 3}  \begin{matrix}1 \\ RP S \end{matrix}
95A_{[1,3]}  12  ℚ(√−1, √3)  [3, 15], [3, 0]  {2 \atop 3}  {1 \atop 1}
95B_{[1,3]}  12  ℚ(√−1, √3)  [3, 15], [3, 0]  {2 \atop 3}  \begin{matrix}∞ \\ 1 \end{matrix}

Table 2: k = 3 and 75 ≤ N ≤ 100

Label  ord(ϵ)  E  Extra twists  Ram Slope
75A_{[0,5]}^{3}  4  ℚ(√−1, √6)  [0, 15], [0, 5]  3  St
75B_{[0,5]}^{3}  4  ℚ(√−1, √6)  [0, 15], [0, 5]  3  St
75C_{[1,0]}^{3}  2  ℚ(√−11 )  [1, 0]  11  1
76A_{[0,9]}^{3}  2  ℚ(√−29 )  [0, 9]  29  1
77A_{[0,5]}^{3}  2  ℚ(√−7)  [0, 5]  7  St
77A_{[2,5]}^{3}  6  ℚ(√−3, √21 )  [4, 5], [2, 0]  7  RP S
78A_{[0,1,6]}^{3}  2  ℚ(√2, √−5)  [0, 6], [1, 0]  2  St
78B_{[0,1,6]}^{3}  2  ℚ(√2, √−5)  [0, 6], [1, 0]  2  St
80B_{[1,0,2]}^{3}  2  ℚ(√2, √−3)  [0, 0, 2], [1, 0, 0]  2  a_{p} = 0
84A_{[0,0,1]}^{3}  6  ℚ(√−3)  [0, 0, 5]  3  St
84A_{[0,0,3]}^{3}  2  ℚ(√−3)  [0, 0, 3]  3  St
84B_{[0,1,2]}^{3}  6  ℚ(√−3, √15 )  [0, 0, 2], [0, 1, 4]  5  1
84C_{[0,1,2]}^{3}  6  ℚ(√−3, √105 )  [0, 0, 4], [0, 1, 4]  5  1
84A_{[1,0,2]}^{3}  6  ℚ(√−3)  [1, 0, 4]  3  St
86A_{[0,21]}^{3}  2  ℚ(√−2)  [21]  2  St
90A_{[0,0,1]}^{3}  4  ℚ(√−1)  [0, 3]  2  St
90B_{[0,0,1]}^{3}  4  ℚ(√−1)  [0, 3]  2  St
90A_{[0,3,2]}^{3}  2  ℚ(√2, √−1)  [0, 2], [3, 2]  2  St
91A_{[3,4]}^{3}  6  ℚ(√−3, √39 )  [0, 4], [3, 0]  13  RP S
91C_{[3,6]}^{3}  2  ℚ(√26 , √−1)  [3, 0], [3, 6]  13  RP S
93A_{[0,15]}^{3}  2  ℚ(√−3)  [0, 15]  3  St
96A_{[0,0,1]}^{3}  2  ℚ(√−2, √3)  [0, 0, 1], [1, 0, 0]  3  RP S
96B_{[0,0,1]}^{3}  2  ℚ(√7, √−2)  [0, 0, 1], [1, 0, 1]  7  1
99C_{[0,5]}^{3}  2  ℚ(√−138 , √−3)  [3, 0], [0, 5]  23  1
100B_{[0,5]}^{3}  4  ℚ(√−1, √6)  [0, 15], [0, 5]  3  3(!)