6_lfunctions.qmd

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# L-functions {#sec-lfunctions}

In this chapter we study the connection of modular forms with
$L$-functions.

## Basic definitions {#sec:levelone}

Let $f\in M_k(\Gamma_1(N))$ be a modular form, given by a $q$-expansion
$f=\sum_{n=0}^\infty a_nq^n$.

::: {#def-}
The of $f$ is the function of $s\in\CC$ given formally as
$$L(f,s)=\sum_{n=1}^\infty a_n n^{-s}.$$
:::

::: {#prp-}
If $f\in S_k(\Gamma_1(N))$ is a cusp form then $L(f,s)$ converges
absolutely for all $s$ such that $\Re(s)>k/2+1$. If
$f\in M_k(\Gamma_1(N))$ is not a cusp form then $L(f,s)$ converges
absolutely for all $s$ with $\Re(s)>k$.
:::

::: {.proof}
We have seen in
@thm-growthcusps and
@cor-growthmodforms that $|a_n|\leq Mn^{r(k)}$ for
some constant $M$, where $r(k)=k/2$ when $f$ is a cusp form and
$r(k)=k-1$ when $f$ is not a cusp form. Although those results were
stated and proven only for level $1$, they hold true (with essentially
the same proofs) for higher levels. Therefore if $\Re(s)>r(k)+1$ then
$$|\sum_{n\geq 0} a_nn^{-s}|\leq M \sum_{n\geq 0} n^{r(k)-\Re(s)} < \infty.$$
:::

The $L$-functions attached to normalized eigenforms have a very
remarkable decomposition, known as . In fact, having this property
characterizes normalized eigenforms, as the following result states.

::: {#thm-}
Let $f\in M_k(\Gamma_0(N),\chi)$ be a modular form with $q$-expansion
$f=\sum_{n\geq 0} a_n q^n$. Then $f$ is a normalized eigenform if and
only if $L(f,s)$ has an Euler product expansion
$$L(f,s)=\prod_{p\text{ prime}} (1-a_pp^{-s}+\chi(p)p^{k-1-2s})^{-1}.$$
:::

::: {.proof}
By
@prp-coeffsnormalizedeigen we need to show that
conditions $(1)$, $(2)$ and $(3)$ of loc.cit. are equivalent to $L(f,s)$
having an Euler product. For a fixed prime $p$, condition $(2)$ says
$$a_{p^r}(f) = a_p(f)a_{p^{r-1}}(f) - p^{k-1}\chi(p) a_{p^{r-2}}(f).$$
Multiplying by $t^r$ and summing over all $r\geq 2$ we see that $(2)$ is
equivalent to
$$\sum_{r=2}^\infty a_{p^r}(f)t^r=a_p(f)t\sum_{r=1}^\infty a_{p^{r}}(f)-p^{k-1}\chi(p)t^2\sum_{r=0}^\infty a_{p^{r}}(f),$$
or
$$\left(\sum_{r\geq 0} a_{p^r}(f)t^r\right)(1-a_p(f)t+\chi(p)p^{k-1}t^2) = a_1(f)+a_p(f)t(1-a_1(f)).$$
Since we are assuming that $a_1(f)=1$ we get, by substituting
$t=p^{-s}$, the equality
$$
\sum_{r=0}^{\infty} a_{p^r}(f)p^{-rs} = (1-a_p(f)p^{-s}+\chi(p)p^{k-1-2s})^{-1}.
$$ {#eq-eulerfactor}
Conversely, if this equality holds then letting $s$
approach $\infty$ we get $a_1(f)=1$, and the other implications can also
be reversed to show that
@eq-eulerfactor is equivalent to conditions $(1)$ and $(2)$
for the $a_n(f)$'s.

The Fundamental Theorem of Arithmetic implies that if $g$ is any
function of prime powers, then
$$\prod_p \sum_{r=0}^\infty g(p^r) = \sum_{n=1}^\infty \prod_{p^r\| n} g(p^r).$$
Using this fact, it is easy to see that
@eq-eulerfactor and condition $(3)$ are equivalent to the
existence of the Euler product, thus finishing the proof.
:::

## L-functions of Eisenstein series

Let $\chi\colon\ZZ\to\CC$ be a primitive Dirichlet character modulo $N$.
One can attach an L-function to $\chi$ via the formula
$$L(\chi,s)=\sum_{n=1}^\infty \chi(n)n^{-s},\quad \Re(s)>1.$$

::: {#prp-}
The L-function of $\chi$ extends to an entire function of on $\CC$
unless $\chi=1$, in which case $L(1,s)=\zeta(s)$ has a simple pole at
$s=1$.
:::

::: {.proof}
Omitted.
:::

We also have an Euler product:

::: {#prp-}
There is an Euler product decomposition
$$L(\chi,s)=\prod_{p\text{ prime}} \frac{1}{1-\chi(p)p^{-s}}.$$
:::

::: {.proof}
Exercise.
:::

We have defined the L-function of any modular form. In particular, if
$\chi_1$ and $\chi_2$ are primitive Dirichlet characters modulo $N_1$
and $N_2$ respectively, then we can consider the L-function
$L(E_k^{\chi_1,\chi_2},s)$.

::: {#exm-}
Consider the Eisenstein series for the full modular group
$E_k(z)\in M_k(\SL_2(\ZZ))$. In
@prp-Ek-is-eigen we have seen that $E_k$ is an eigenform
for all the Hecke algebra, satisfying $T_p E_k = \sigma_{k-1}(p) E_k$.
If we normalize $E_k$ using its first coefficient (instead of the
zero-th) and call the resulting Eisenstein series $\bar E_K$, then we
have $a_p(\bar E_k) = \sigma_{k-1}(p) = 1+p^{k-1}$. Therefore
\begin{align*}
L(\bar E_k,s) &= \prod_{p\text{ prime}}\frac{1}{1-(1+p^{k-1})p^{-s}+p^{k-1-2s}}\\
&=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}}\frac{1}{1-p^{k-1-s}}=\zeta(s)\zeta(s-k+1).
\end{align*}
:::

The factorization of the example holds in much more generality. Denote
by $\bar E_k^{\chi_1,\chi_2}=\frac{1}{2} E_k^{\chi_1,\chi_2}$, where
$E_k^{\chi_1,\chi_2}$ were defined in
@thm-eisenstein-gamma1.

::: {#prp-}
The L-function attached to the Eisenstein series
$\bar E_k^{\chi_1,\chi_2}$ has a factorization
$$L(\bar E_k^{\chi_1,\chi_2},s)=L(\chi_1,s)L(\chi_2,s-k+1).$$
:::

::: {.proof}
Exercise.
:::

The idea that one can extract from this is that the Eisenstein series
are quite simple, and their L-functions are not too interesting since
they can be understood from the (simpler) L-functions attached to
characters. In stark contrast, the L-functions attached to cusp forms
have much deeper connections.

## L-functions of cusp forms

We focus from now on on cusp forms. The next striking property of
L-functions of cusp forms is known as , a symmetry property of deep
consequences. In order to state it precisely, we first define the ,
which appears often in number theory, as
$$\Gamma(s)=\int_0^\infty t^s e^{-t}\frac{dt}{t}.$$ Note that
$\Gamma(n+1)= n!$ for all integers $n\geq 1$, so we can think of
$\Gamma$ as an analytic function interpolating the factorials. The
Gamma-function enters also in the definition of another complex
function, for which the symmetry property is more apparent.

::: {#def-}
The of $f\in S_k(\Gamma_1(N))$ is
$$\Lambda(f,s)=(2\pi)^{-s}\Gamma(s)L(f,s),\quad \Re(s)>k/2+1.$$
:::

The next result gives an integral formula for the completed L-function.

::: {#prp-}
We have
$$\Lambda(f,s)=\int_0^\infty f(it)t^s\frac{dt}{t},\quad \Re(s)>k/2+1.$$
This is called the of $f$.
:::

::: {.proof}
We first remark that the integral makes sense, since
$$\left|\int_0^\infty f(it)t^s\frac{dt}{t}\right| <\!\!\!\!< \int_0^\infty t^{-k/2+s} \frac{dt}{t},$$
which converges for $\Re(s)>k/2+1$. Now we compute
$$\Lambda(f,s) = (2\pi)^{-s} \left(\int_0^\infty t^s e^{-t}\frac{dt}{t}\right)\sum a_n n^{-s} = \sum_{n=1}^\infty a_n\int_0^\infty \left(\frac{t}{2\pi n}\right)^s e^{-t}\frac{dt}{t}.$$
By doing a change of variables $t\mapsto t/(2\pi n)$ in each term, the
above expression becomes
$$\sum_{n=1}^\infty a_n\int_0^\infty t^s e^{-2\pi n t}\frac{dt}{t} = \int_0^\infty \left(\sum_{n=1}^\infty a_n e^{-2\pi nt}\right) t^s\frac{dt}{t} = \int_0^\infty f(it)t^s\frac{dt}{t},$$
which gives the desired equality.
:::

In order to extend $\Lambda(f,s)$ (and thus $L(f,s)$) to $s\in\CC$ we
need to avoid integrating near the real axis. We will also need to
consider the operator $W_N$ given by
$$W_N(f)= i^k N^{1-k/2} f|_k \mtx{0}{-1}{N}{0}.$$ It is an idempotent
operator: $W_N^2=W_N$, and one easily sees that it is self-adjoint:
$\langle W_Nf,g\rangle = \langle f,W_Ng\rangle$ for
$f,g\in S_k(\Gamma_1(N))$. Consider the $+$ and $-$-eigenspaces
$$S_k(\Gamma_1(N))^{\pm} = \{f\in S_k(\Gamma_1(N)) ~|~ W_Nf = \pm f\},$$
which gives an orthogonal decomposition of $S_k = S_k^+\oplus S_k^-$.

::: {#thm-}
Suppose that $f\in S_k(\Gamma_1(N))^\pm$. Then the function
$\Lambda(f,s)$ extends to an entire function on $\CC$, which satisfies
the functional equation $$\Lambda(f,s)=\pm N^{s-k/2}\Lambda(f,k-s).$$ In
particular, the $L$-function $L(f,s)$ has an analytic continuation to
all of $\CC$.
:::

::: {.proof}
Define $\Lambda_N(s)=N^{s/2}\Lambda(f,s)$, and note that we
must show that $\Lambda_N(s) = \pm \Lambda_N(k-s)$. By changing
$t\mapsto t/\sqrt{N}$ we get
$$\Lambda_N(s)=N^{s/2}\int_0^\infty f(it)t^s\frac{dt}{t} = \int_0^\infty f(it/\sqrt{N}) t^s\frac{dt}{t}.$$
We break the integral at $t=1$. Note that the piece
$$\int_1^\infty f(it/\sqrt{N}) t^s\frac{dt}{t}$$ converges to an entire
function of $s$, because $f(it/\sqrt{N})=O(e^{-2\pi t/\sqrt{N}})$ when
$t\to \infty$. As for the other part, use that
$(W_Nf)(i/(\sqrt N t)) = t^k f(it/\sqrt{N})$ to get
$$\int_0^1 f(it/\sqrt{N}) t^s\frac{dt}{t} = \int_0^1 (W_Nf)(i/(\sqrt{N}t)) t^{s-k}\frac{dt}{t} = \int_1^\infty (W_Nf)(it/\sqrt{N})t^{k-s}\frac{dt}{t}.$$
Again, since $W_nf = \pm f$ this converges to an entire function. As for
the functional equation, note that we have obtained
$$\Lambda_N(s) = \int_1^\infty\left(f(it/\sqrt{N})t^s \pm f(it/\sqrt{N})t^{k-s}\right)\frac{dt}{t} = \pm\Lambda_N(k-s).$$
:::

## Relation to elliptic curves

Let $E/\QQ$ be an elliptic curve. It can be thought of as the set cut
out by an equation of the form
$$E\colon Y^2 = X^3+AX+B,\quad A,B\in\ZZ,$$ such that the discriminant
$\Delta_E$ of $X^3+AX+B$ is nonzero. The coefficients of this equation
can be reduced modulo any prime $p$ and the conductor $N_E$ of $E$ is an
integer whose prime divisors are precisely the prime divisors of $N_E$
(although in general $N_E\neq \Delta_E$. One can define an L-function
attached to $E$ via the following Euler product:
$$L(E,s)=\prod_{p\mid N_E}(1-a_p(E)p^{-s})^{-1} \prod_{p\nmid N_E} (1-a_p(E)p^{-s} + p^{1-2s})^{-1},\quad \Re(s)>3/2.$$
where $a_p(E) = 1+p-\# E(\FF_p)$. Here, by $E(\FF_p)$ we mean the set of
points of (the reduction of) $E$ over the finite field $\FF_p$, where we
always include the "point at infinity".

It turns out that elliptic curves arise from modular forms, thanks to
results of Eichler and Shimura.

::: {#thm-}
Let $f\in S_2(\Gamma_0(N))$ be a normalized eigenform whose Fourier
coefficients $a_n(f)$ are all integers. Then there exists an elliptic
curve $E_f$ defined over $\QQ$ such that $L(E_f,s)=L(f,s)$.
:::

::: {.proof}
*Construction of $E_f$.* Consider the differential form
$\omega_f=2\pi i f(z)dz$, and write $\HH^*=\HH\cup\PP^1(\QQ)$. To a
point $\tau\in\HH^*$ we attach the following complex number
$$\varphi(\tau) = \int_{\infty}^\tau \omega_f \in\CC.$$ Let
$\gamma\in\Gamma_0(N)$. Then note that
$$\beta_{\gamma} =\varphi(\gamma\tau)-\varphi(\tau) = \int_{\tau}^{\gamma_\tau} \omega_f$$
does not depend on $\tau$: \begin{align*}
\int_\tau^{\gamma\tau}\omega_f &= \int_\tau^\infty\omega_f+\int_\infty^{\gamma\infty}\omega_f+ \int_{\gamma\infty}^{\gamma\tau}\omega_f\\
&= \int_\tau^\infty\omega_f+\int_\infty^{\gamma\infty}\omega_f+ \int_{\infty}^{\tau}\omega_f\\
&=\int_\infty^{\gamma\infty}\omega_f.
\end{align*} Therefore if denote by $\Lambda_f$ the following subset
of complex numbers
$$\Lambda_f=\left\{\beta_\gamma=\int_{\infty}^{\gamma\infty}\omega_f ~|~ \gamma\in\Gamma_0(N)\right\}\subset\CC,$$
we get a well-defined map
$$\Gamma_0(N)\backslash \HH^*\to \CC/\Lambda_f.$$ One can show that
$\Lambda_f$ is a lattice, and define $E_f$ to be the elliptic curve
corresponding to the complex torus $\CC/\Lambda_f$. It is considerably
harder to show that $E_f$ is defined over $\QQ$, and that
$L(E_f,s)=L(f,s)$.
:::

We may wonder about a converse to the previous result. That is, given an
elliptic curve $E$ of conductor $N_E$, can we find a cusp form of level
$N_E$ having the same L-function as that of $E$? Let us give a name to
the elliptic curves $E$ satisfying this property.

::: {#def-}
We say that $E$ is if there is a newform $f\in S_2(\Gamma_0(N_E))$ with
$a_p(E)=a_p(f)$. Equivalently, if $L(E,s)=L(f,s)$.
:::

The following theorem, which gives a positive answer to the question we
asked, is one of the hallmarks of XX-century number theory. Its proof,
spanning hundreds of pages of difficult mathematics, relies on
breakthrough work of Andrew Wiles in the nineties, although the full
proof needed extra work of Taylor--Wiles and
Breuil--Conrad--Diamond--Taylor.

::: {#thm-}
Let $E/\QQ$ be an elliptic curve. Then $E$ is modular.
:::

Thanks to the above theorem, the L-function of $E$ extends to an entire
function, which satisfies a functional equation relating $L(E,s)$ with
$L(E,2-s)$. In fact, there is no known proof of these two facts that
does not need modularity of $E$. Finally, the Birch--Swinnerton-Dyer
conjecture is a prediction about the behavior of $L(E,s)$ near $s=1$.
Recall that the set of points $E(\QQ)$ of $E$ which have coordinates in
the rational numbers has a structure of a finitely-generated group (this
is the Mordell--Weil theorem).

::: {#cnj-}
Let $E$ be an elliptic curve defined over $\QQ$. Let $L(E,s)$ be its
L-function. Then
$$\ord_{s=1} L(E,s) = \operatorname{rank}_{\ZZ} E(\QQ).$$
:::

This conjectures is one of the ten "Millennium" problems proposed in
2000 by the Clay Mathematics Institute, and it is worth $1\text{M}\$$.
Very little is known of it. For instance, one does not yet know how to
show the particular case
$$L(E,1)=0\stackrel{?}{\implies} E(\QQ)\text{ infinite}.$$ However,
thanks to work of B.Gross, D.Zagier and V.Kolyvagin, one has the
following result.

::: {#thm-}
Let $E/\QQ$ be a modular elliptic curve.

1.  Suppose that $L(E,1)\neq 0$. Then $E(\QQ)$ is finite.

2.  Suppose that $L(E,1)=0$ and $L'(E,1)\neq 0$. Then $E(\QQ)$ has rank
    one.

That is, BSD holds if we assume *a priori* that $\ord_{s=1} L(E,s)$ is
at most one.
:::

The proof of this is also very difficult and uses crucially the modular
form $f_E$ attached to $E$ by modularity. This is nowadays no
restriction, since by the modularity theorem we know that all elliptic
curves over $\QQ$ are modular. However, the result of Gross--Zagier and
Kolyvagin was proven in the eighties, *before* modularity was proven (or
even thought to be attainable). A crucial ingredient that goes in the
proof is to be able to produce, in the case of $L(E,1)=0$, a point
$P \in E(\QQ)$ which has infinite order, as predicted by BSD. It is an
open problem to find a point of infinite order in $E(\QQ)$ knowing that
$\ord_{s=1} L(E,s)\geq 2$. This is an example of the recurring
phenomenon in mathematics: it is easy to construct objects that are
uniquely defined, in what could be thought of as a perverse
manifestation of the "axiom of choice".