Notes on modular motives

William Stein

November 18, 1999

Introduction

Goals:

• Meta:

— Get my head around modular motives.

— Help friends to do the same.

— Generate excitement, enthusiasm, and ideas.

• Precise:

• Describe how modular motives fit into the rest of mathematics, and argue that they are worth paying attention to; motivate using the BSD conjecture.

• Gain a first understanding of many of the basic objects:

— Chow motive

— Grothendieck motive

— Betti and ℓ-adic cohomology

— crystalline representation

— ϕ-filtered module

— Kuga-Sato varieties

— Geometry of modular curves and their desingularizations.

• State Scholl’s main theorem.

The rest of these notes will record details of everything I figure out. Their purpose is to make it painless for me to later understand this material.

1 Scholl’s theorem

Here is a statement of Scholl’s theorem:

Theorem 1.1 (Scholl). Let f = ∑_{n=1}^{∞} a_{n}q^{n} be a weight-k normalized newform of level N and character ε, and let K_{f} = Q(… a_{n} …). Then there is a Grothendieck motive M(f) over Q with coefficients in K_{f} such that:

1. If p ∤ Nℓ and λ is a prime of K_{f} dividing ℓ, then the λ-adic realization H_{λ}(M(f)) of M(f) is unramified at p, and the characteristic polynomial of a geometric Frobenius at p is T_{p}(X) = X^{2} − a_{p}X + ε(p)p^{k−1}.

2. If p ∤ N and π is a prime of K_{f} dividing p, then the π-adic realization H_{π}(M(f)) is a crystalline representation of Gal(\overline{Q}_{p}/Q_{p}), and the characteristic polynomial of ϕ on the associated ϕ-filtered module is equal to T_{p}(X).