Math 726: L-functions and modular forms Fall 2011

Lecture 24 : Appplications of Rankin-Selberg

Instructor: Henri Darmon Notes written by: Jason K.C. Polak We recall briefly part of the morning and continue.

Theorem 1. Let f = ∑_{n=1}^{∞} a_{n}q^{n} be a normalized newform of weight k, level N and character χ. Then the sum ∑_{p∤N} |a_{n}|^{2}p^{−s} converges absolutely for ℜ(s) > k and

∑_{p∤N}|a_{n}|^{2}p^{−s} = log (\frac{1}{s − k} ) + O(1)

as s → k. Proof. Consider

log L_{N}(f ⊗ \overline{f}, s) = − ∑_{p∤N}[∑_{i,j=1}^{2} log (1 − α_{pi}\overline{α_{pj}}p^{−s})]

= ∑_{p∤N}(∑_{m=1}^{∞} \frac{(α_{p,1}^{m} + α_{p,2}^{m})(\overline{α_{p,1}}^{m} + \overline{α_{p,2}}^{m})}{mp^{ms}}) .

If we let

g_{m}(s) = ∑_{m=1}^{∞} \frac{(α_{p,1}^{m} + α_{p,2}^{m})(\overline{α_{p,1}}^{m} + \overline{α_{p,2}}^{m})}{mp^{ms}}

then

log L_{N}(f ⊗ \overline{f}) = ∑_{m=1}^{∞} g_{m}(s).

Note that

g_{1}(s) = ∑_{p∤N} \frac{|α_{p,1} + α_{p,2}|^{2}}{p^{s}}

= ∑_{p∤N} \frac{|a_{p}|^{2}}{p^{s}}

≤ ∑_{m=1}^{∞} g_{m}(s)

= log (\frac{1}{s − k} ) as s → k.

Here, we have used that L_{N}(f ⊗ \overline{f}, s) has a simple pole at s = k.

Definition 2. Let ℙ be the set of primes and let X ⊆ ℙ. Define the superior density of X to be

dens. sup(X) = lim sup_{s→1^{+}} (\frac{∑_{p∈X} p^{−s}}{log (\frac{1}{s−1})} ) .

Recall that

∑ p^{−s} = log (\frac{1}{s − 1}) + O(1)

as s → 1^{+} so that dens. sup(X) ∈ [0, 1].

Note that the image of an Artin representation G_{ℚ} → GL_{2}(ℂ) has finite image so that the set

{a_{p} : p ∈ ℙ} ⊆ ℂ

is a finite set.

Proposition 1. Let f be a normalized newform of weight one, level N, and character χ. Then for each η > 0 there exists sets X_{η}, Y_{η} ⊆ ℂ such that |Y_{η}| < ∞ and

dens. sup(X) ≤ η

and a_{p} ∈ Y_{η} if p ∉ X_{η}.

Before proving this, we will state some facts without proof, which we already did for level N = 1 when we introduced algebraic modular forms. The general weight case follows from similar arguments.

Proposition 2. Let f be a normalized newform of weight k, level N, and character χ. Write

f = ∑ a_{n}q^{n}.

Then

1. The field K = ℚ(a_{n} : n ∈ ℕ) is a finite extension of ℚ and each a_{n} is an algebraic integer (i.e. a_{n} ∈ O_{K} for each n).

2. If σ : K_{f} ↪ ℂ is an embedding then σ(f) := ∑_{n=1}^{∞} σ(a_{n})q^{n} is a normalized newform of weight k, level N, and character σ ∘ χ.

3. S_{k}(Γ_{1}(N)) has a ℤ-basis.

Now we prove Proposition 1 Proof. Fix c > 0 let

Y (c) := {α ∈ O_{K_{f}} : |σ(α)|^{2} ≤ c ∀σ ∈ Hom(K_{f}, ℂ)}.

Since the image O_{K_{f}} ↪ K_{f} ⊗_{ℚ} ℝ is discrete and cocompact, the set Y (c) is finite. Define

X(c) = {p ∈ ℙ : a_{p} ∉ Y (c)}.

By Theorem 2 of Lecture 23 and Part 2 of Proposition 2 in this lecture,

∑_{p} |a_{p}|^{2}p^{−s} = log (\frac{1}{s − 1}) + O(1)

= ∑_{p} |σ(a_{p})|^{2}p^{−s}

for all embeddings σ : K_{f} → ℂ, so

∑_{σ∈Hom(K_{f} ,ℂ)}(∑_{p} |σ(a_{p})|^{2}p^{−s}) = [K_{f} : ℚ] log (\frac{1}{s − 1}) + O(1).

Since ∑_{σ} |σ(a_{p})|^{2} ≥ c whenever p ∈ X(c),

c ∑_{p∈X(c)} |σ(a_{p})|^{2}p^{−s} ≤ ∑_{p∈X(c)} ( ∑_{σ∈Hom(K_{f} ,ℂ)}|σ(a_{p})|^{2}) p^{−s}

≤ [K_{f} : ℚ] log (\frac{1}{s − 1}) + O(1)

and so dens. sup(X(c)) = [K_{f} : ℚ]/c. Thus we can select X_{η} = X(c), Y_{η} = Y (c). Remark 1. We can also derive the result of today’s lecture assuming the Petersson conjecture. By this conjecture, we know that |a_{p}| ≤ 2p^{\frac{k−1}{2}} for all p ∤ N. Then

log L_{N}(f ⊗ \overline{f}, s) = log (\frac{1}{s − k} ) + O(1)

and

log L_{N}(f ⊗ \overline{f}, s) = ∑_{p∤N}|a_{p}|^{2}p^{−s} + ∑_{m=2}^{∞} g_{m}

so by the Petersson conjecture ∑_{m=2}^{∞} g_{m}(s) converges absolutely at s = k and so

∑_{p∤N}|a_{p}|^{2}p^{−s} = log (\frac{1}{s − k} ) + O(1).