SERVICIOS ESTUDIANTILES.

The representationsρ_f,λ are uncountable objects. This implies that we will not be able to

compute them precisely, except in some special cases. So if we want to compute them then we have to approximate them, like one approximates real numbers by floating point numbers. The approximations that we will study are representationsρ=ρ_f,λ : Gal(_Q/_Q)→GL2(Fλ)

that have charpol(ρ(Frobp))congruent toX2−ap(f)X+ε(p)pk−1modλ.

LetGbe a compact group, letKbe an`-adic field with residue fieldkand letρ:G→GLn(K)

be a semi-simple representation. From the compactness ofGit follows thatKnhas aG-stable OK-sublattice: if Λ⊂Kn is any OK-lattice, then the OK-module generated by GΛ is aG-

stableOK-lattice. Reducing this lattice modulo the primeλ ofK we obtain a 2-dimensional

representation ofGover k. This representation depends in general on the choice of the lat- tice. However, the Brauer-Nesbitt theorem shows that its semi-simplificationρ is unique up

to isomorphism (note that sincekis finite, the representation factors through a finite quotient ofG). This semi-simple representationρ is called thereductionofρ moduloλ.

This shows that the above mentioned representations ρ_f,λ at least do exist. We can also

find them concretely. Assume for this thatρ_f,λ is absolutely irreducible, which is the most

interesting case anyway.

The casek=2

The above mentioned construction ofρ_f,λ suggests that we should look inside Jacobians of

modular curves.

Theorem 1.15(Boston-Lenstra-Ribet [9, Theorem 2]). Let f ∈S₂(Γ1(N))be a newform and letλ be a prime of Kf such thatρf,λ is absolutely irreducible. LetTbe the Hecke algebra

hdi 7→εf(d)modλ. Letm=mf ⊂T be the kernel ofθf,λ. Then the(T/m)[Gal(Q/Q)]-

module J1(N)(Q)[m]is a direct sum of copies ofρ_f,λ.

If we take m as in the theorem then from the construction of ρ_f,λ it follows a priori that ρ_f,λ is an irreducible constituent of J1(N)(Q)[mr]for some r>0. An argument of Mazur

[52, Section 14] shows that we can in fact taker=1 here, showing that the number of copies in Theorem 1.15 is positive. The mapθ_f,λ mentioned in Theorem 1.15 need not be surjective.

So it may happen that the representationρ_f,λ is actually defined over a field that is smaller

than_Fλ. The casek6=2

If we writeN =N0`n with`_-N0 then it can be shown that there is a newform f0 of weight kand level dividing N0, a primeλ0 of Kf0 and embeddings ofFλ and F

0

λ intoF` such that

for allncoprime toN we have in_F`an equality ofan(f)modλ withan(f0)modλ0. For a

proof of this see [61, Theorem 2.1] and [12, Proposition 1.1]. In other words, without loss of generality we can and do assume`_-N.

If we let the weight vary, we can find more congruences. In fact, [61, Theorem 2.2] states that for k≤`+1 there is a newform f0 of level dividingN` and weight 2 such that in the notation as above, an(f)modλ is equal to an(f0)modλ0 for n coprime to N`. So also in

this case, we can find the representation inside the Jacobian of a modular curve. If we have k> `+1 then the representationρ =ρ<sub>f,λ</sub> might not always be present inside the`-torsion

of someJ1(M)but there is a twist

ρ⊗χn_{`</sub>:σ 7→ρ(σ)χn<sub>`}(σ)

which does belong to a form of weight at most`+1, hence can be reduced to weight 2 again; see [27, Section 7].

In conclusion, ifρ_f,λ is absolutely irreducible, we can always reduce to weight 2 and work

inside the`-torsion of the Jacobian of some modular curveX₁(M). Multiplicity one

The number of copies ofρ_f,λ in Theorem 1.15 is called themultiplicityofρ_f,λ. In general,

let f ∈S_k(Γ₁(N)) be a newform and λ is a prime of Kf such that ρ =ρf,λ is absolutely

irreducible. Then we define the multiplicity ofρ_f,λ as the multiplicity of its twist that is

associated to a weight 2 form of minimal level. This multiplicity is equal to 1 in most cases, exceptions are only possible if a list of very strong conditions are satisfied.

Theorem 1.16(Multiplicity one theorem, cf. [13, Theorem 6.1]). Let N and k be positive integers and let f ∈S_k(Γ1(N))be a newform. Furthermore, let`-N be a prime and suppose

2≤k≤`+1. Take a primeλ of Kf above`such thatρ =ρf,` is an absolutely irreducible

representation of multiplicity not equal to one. Then k is equal to`, the representationρ is

With some possible exceptions for`=2, the converse of the theorem also holds; for a proof of this, see [87, Corollary 4.5]. For computational examples on representations of multiplic- ity not equal to one, see [41].