Introduction to modular motives

William Stein, UCB

November 23, 1999

1 Introduction

WARNING – These are informal notes, and I am not an expert! Long-term goal: Learn how to compute enough invariants of specific motives associated to modular forms of weight greater than 2 to verify the Bloch and Kato conjecture in a few special cases.

\underline{As far as I know,} the above has not been done for even a single example.

Questions:

1. What are motives? (Grothendieck, etc.)

2. What are modular motives? (Scholl)

3. How are L-functions related to motives? (Galois representations)

4. What did Bloch and Kato conjecture? (Grothendieck Feitschrift)

5. How can we compute examples? (????????)

2 Birch and Swinnerton-Dyer conjecture

Conjecture: Let A/K be an abelian variety over a number field K. Then

L(1) = \frac{#� (A) · ∏ c_{v} · Ω_{A} · Reg}{A(K)_{tor} · A^{∨}(K)_{tor}}

where

• L(1) is the leading coefficient of the L-series at 1 (that this is defined is part of the conjecture).

• � = ker(H^{1}(K, A) → ∏_{v} H^{1}(K_{v}, A)) measures failure of the local-to-global principle.

• c_{v} = local volumes at finite places = orders of component groups.

• Ω_{A} = product of volumes at infinite places with respect to basis of integral differentials. (I don’t know how to define this when O has lots of units.)

• Reg = regulator = discriminant of canonical height pairing A(K) × A^{∨}(K) → R.

• A(K)_{tor} = torsion subgroup.

3 Modular abelian varieties

Eichler-Shimura: Let

f = q + a_{2}q^{2} + a_{3}q^{3} + a_{4}q^{4} + a_{5}q^{5} + ⋯

be a weight-2 modular eigenform with character ε. Let 𝔭_{f} be the annihilator of f in the Hecke algebra. Shimura associated to f an abelian variety A_{f} = J_{1}(N)/𝔭_{f}J_{1}(N). Eichler-Shimura proved the following nontrivial theorem: the characteristic polynomial of Frob_{p} on the ℓ-adic Tate module of A_{f} tensored with Q_{ℓ}(… a_{n} …) is X^{2}− a_{p}X + pε(p).

See Brian Conrad’s notes.

Writing down abelian varieties: In dimension greater than two, there appears to be no satisfactory way to “write down” an abelian variety. In contrast, the modular abelian varieties (up to isogeny) can be “written down” by giving the corresponding modular forms! For example,

f = q + aq^{2} + (−a^{2} − a + 3)q^{3} + ⋯,

where a^{3} + a^{2} − 3a − 1 = 0, defines a three-dimensional quotient of the four-dimensional abelian variety J_{0}(53). The other part of J_{0}(53) is an elliptic curve, which we can write down, in the classical sense.

For many purposes the modular form is just as good a starting point as explicit equations.

4 The Bloch-Kato conjecture

Question: Do abelian varieties fit into a “universal” category of objects possessing L-functions, arithmetic invariants and BSD-like relations?

Idea of Bloch and Kato: Try to approach the invariants appearing in the BSD conjecture from the point of view of cohomology.

• The L-function packages together the characteristic polynomials of Frobenius acting on the Tate module (=ℓ-adic cohomology).

• � is a kernel of a map between Galois cohomology groups. (Mazur defined � using flat cohomology.)

• c_{p} and Ω are volumes with respect to a measure. Somehow these can be recovered via de Rham and Betti cohomology and a choice of lattice, but I don’t understand this yet.

• Torsion (?)

• Regulator (?)

My understanding is that Bloch and Kato succeed, modulo “standard conjectures in K-theory.”

5 Grothendieck Motives

Grothendieck dreamed up a category of motives while searching for a category \underline{M} equipped with a “universal cohomology functor”

H : \{proj. nonsing. alg. var. over k\} → \underline{M}.

This H should be univeral in the sense that any other (reasonable) cohomology theory, for example, étale cohomology, should factor through it.

Homomorphisms: Consider the category whose objects are algebraic varieties, but with the following weird morphisms:

Hom(X, Y) := \frac{\{algebraic cycles on X × Y of dim. dim X\}}{(numerical equivalence)}

(Numerically equivalent cycles are indistinguishable from the point of view of the intersection pairing into .) These morphisms form a group so our first thought is: “abelian category!!!!”

Unfortunately, this category is far from abelian. There are lots of projectors, i.e., maps that equal their square, whose kernels are not algebraic varieties.

Motives: Formally “add in all kernels of projectors”, to obtain a pseduo-abelian category. Motives are represented by pairs (X, π) where X is a k-variety and π ∈ Hom(X, X), π^{2} = π.

6 Realizations of motives

Quote: Manin gives a remark of Herbert Read:

Cézanne’s method of painting was first to choose his visual apprehension of this motif; and in this process loose nothing of the vital intensity that the motif possessed in actual existence.

Remark: There is a special Cézanne exhibit at

http://metalab.unc.edu/wm/paint/auth/cezanne/.

Realization: Cohomology functors on the category of algebraic varieties, such as étale, Hodge, etc., (should?) factor through the category of motives. Thus, given a motive M, there should be finite dimensional vector spaces H_{ét}(M), H_{Hodge}(M), etc. attached to M. These are the realizations of the motive M.

7 Modular motives

Let f be a modular eigenform of weight greater than 2.

Deligne: Associated a compatible system of ℓ-adic Galois representations to f satisfying a characteristic polynomial condition as in Eichler-Shimura.

Shimura: Using period integrals, Shimura constructs an abelian variety over C attached to f and equipped with action of higher-weight Hecke operators. It has no apparent arithmetic structure and have been ignored (?), as far as I can tell.

Scholl: Using Kuga-Sato varieties, Scholl constructs a motive that is naturally linked up with f. The Kuga-Sato varieties are desingularized compactifications of products of universal elliptic curves. I will describe Scholl’s result more precisely next week.