# Intro


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Some additional resources:

- <https://math.stackexchange.com/questions/48981/the-langlands-program-for-beginners>

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Some vague definitions:

- The congruence subgroups $\Gamma(N), \Gamma_0(N), \Gamma_1(N)$.
\[
\Gamma(N) &\da \ts{M\in \SL_2(\ZZ) \st M\equiv I \mod N} \\
\Gamma_0(N) &\da \ts{M\in \SL_2(\ZZ) \st M\equiv \matt * * 0 * \mod N}
.\]
  - A **congruence subgroup of level $N$** is any $H \contains \Gamma(N)$
  - The **level** is the smallest $N$ such that $H \contains \Gamma(N)$.
  - Recovers $\SL_2(\ZZ) = \Gamma_0(1)$.
  - Principal congruence subgroups of **level** $N$:
  \[
  1\to \Gamma(N) \to \SL_2(\ZZ) \mapsvia{\mod N} \SL_2(\ZZ/N\ZZ)\to 1
  .\]

- Fuchsian groups: discrete subgroups of $\SL_2(\RR)$.
- $Y(\Gamma)$: modular curves of the form $\dcosetl{\Gamma}{\HH}$ for $\Gamma$ a Fuchsian group of the first kind.
  For congruence subgroups, abbreviated $Y(N)$.
- $X(\Gamma)$: the compactification of $Y(\Gamma)$ obtained by adding cusps.
  For congruence subgroups, abbreviated $X(N)$.
- Shimura-Taniyama-Weil theorem: for $E$ an elliptic curve, there is a cover $X_0(N) \to E$ where $N$ is the conductor of $E$.
- Bad reduction for an elliptic curve: primes $p$ for which the equation reduces $\mod p$ to a singular curve.
- Factors of automorphy: for $\gamma = \matt a b c d\in \Gamma$, $j(\gamma, \tau) \da (c\tau + d)$.
- Slash operators: $f\mid [\gamma]_k \da f(\gamma(\wait))\cdot j(\gamma,\wait)^{-k}$
- Classical automorphic forms of weight $k$ and level $N$:
  - $f\in \Mero(\HH, \CC)$,
  - $f\mid [\gamma]_k = f$ for all $\gamma \in \Gamma$ a congruence subgroup of level $N$,
  - Meromorphic at cusps, so the corresponding Fourier expansion at the cusps has a finite tail.
  - Note that these conditions guarantee the corresponding $L$ function will be meromorphic with known poles, or holomorphic for cusp forms.
 - Classical modular forms as automorphic forms:
  - $f\in \Hol(\HH, \CC)$
  - $f\mid[\gamma]_k = f$ for all $\gamma$,
  - Holomorphic at cusps, so Fourier expansion as no negative terms.
- **Cusp form**: modular forms with vanishing constant Fourier coefficient at every cusp.

 - Automorphic forms $f$ of weight $2k$ correspond to meromorphic differential forms $\omega = f(z) (dz)\tensorpower{\CC}{k}$.

- Weakly modular forms of **weight** $k$ for $\SL_2(\ZZ)$: $f(\gamma(\tau)) = (c\tau + d)^k f(\tau)$ for all $\gamma\in \Gamma$ and $\tau \in \HH$.
  - Automorphic forms: meromorphic at $\infty$, form spaces $\mca_k(\Gamma)$
  - Modular forms: holomorphic on $\HH \union\ts{\infty}$, form spaces $\mcm_k(\Gamma)$
  - Cusp forms: vanishing at cusp points, i.e. $f(\infty) = 0$, form spaces $\mcs_k(\Gamma)$.

  - The containment:
  \[
  \mca_k(\Gamma) \supseteq \mcm_k(\Gamma) \supseteq \mcs_k(\Gamma)
  .\]
  - The Eisenstein space:
  \[
  0 \to \mcs_K(\Gamma) \to \mfm_k(\Gamma) \to \mce_k(\Gamma) \to 0
  .\]



- Eisenstein series:
\[
G_{k}(\tau):=\sum_{(c, d) \in \mathbb{Z}^{2}}^{\prime} \frac{1}{(c \tau+d)^{k}}
,\]
where $G_k(\infty) = 2\zeta(k)$.
  - Normalization:
  \[
  E_{k}\da \frac{G_{k}}{2 \zeta(k)} \in \mcm_{k}(\Gamma)
  .\]
- The discriminant form:
\[
\Delta: \HH\to \CC \\
\Delta &= g_2^3-27 g_3^2, \quad g_2 \da 60 G_4,\quad g_3 \da 14- G_6
.\]
  - Facts: $\Delta/(2\pi)^{12}\in \mcm_{12}(\SL_2(\ZZ))$ and has a Fourier expansion $\sum_k \tau(k) q^k$ for $q=\exp(2\pi i \tau)$ where $\tau$ is the Ramanujan $\tau$ function.
    Moreover $\Delta \in \mcs_{12}(\SL_2(\ZZ))$

- Modular curves $Y(N)$: ?
  - $X(N)$: ?

- Geometric interpretations:
  - Modular forms of weight $k$ and level $N$: meromorphic differential forms on $X_0(N)$ which are multiples of a certain divisor, i.e. holomorphic sections of lines bundles on modular curves, so $\mcm_k(\Gamma) \cong H^0(X; \mcl_{\Gamma, k})$ for some line bundle.
  - Automorphic form: for $G$ an algebraic group, a function $f: G\to \CC$ which is invariant with respect to some subgroup $\Gamma \leq G$

- Petersson inner product: for $f,g$ modular forms of weight $k$ for $\Gamma$, at least one cuspidal, and $F$ a fundamental domain for $\Gamma$,
\[
\ip{f}{g} \da \iint_F f\bar{g} y^k \, {\dx \dy \over y^2}
.\]

- Mellin transform:
\[
M(f)(s) \da \int_{\RR\geq 0} f(y) y^s {\dy\over y}
.\]
  - Note that $M(e^{-x})(s) = \Gamma(s)$

- Riemann zeta is the L function associated to the Jacobi theta function, which is modular of weight $1/2$ with respect to $\gens{z\mapsto z+2,\matt 0 {-1} 1 0 }$.
- First instance of Langlands: Taniyama-Shimura-Weil theorem.
  For $E\slice \QQ$ an elliptic curve of conductor $N$, there is a weight 2 cusp form which is a Hecke eigenform on $\Gamma_0(N)$ with $L(E, s) = L(f, s)$.
  More generally, all $L$ functions attached to algebraic varieties should arise as $L$ functions coming from automorphic forms.


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