Modular forms from three perspectives

Three worlds

Automorphic representations

\(\pi\) is a tempered balanced cuspidal automorphic representation of $GL(d, A_\Q)$.

Modular forms

\(f\) is a function on a symmetric space, with particular analytic and invariance properties
(adjectives omitted!)

L-functions

\(L\) is a Dirichlet series with a functional equation and an Euler product
(adjectives omitted!)

$\Gamma_\R(s) = \pi^{-s/2}\Gamma(s/2)$

$\Gamma_\C(s) = (2\pi)^{-s} \Gamma(s)$

Functional equations only involve \[ \Gamma_\R(s + \mu) \ \ \ \text{for}\ \ \ \mu\in \{0,1\} \] \[ \Gamma_\C(s + \nu) \ \ \ \text{for}\ \ \ \nu\in \left\{\frac12,\ 1,\ \frac32,\ 2,\ \frac52,\ 3, \dots\right\} \] and also "shifted": $\Gamma_\R(s + \mu + it)$ and $\Gamma_\C(s + \nu + it)$ for $t\in\R$.

Degree 1

Dirichlet characters:
     $N^{s/2}\Gamma_\R(s)$ for even characters
     $N^{s/2}\Gamma_\R(s+1)$ for odd

Degree 2

Two cases: $2=2$ and $2 = 1+1$.

$N^{s/2}\Gamma_\C\left(s+\frac{\ell}{2}\right)$ for $\ell \in \{1,\ 2,\ 3,\ldots\}$

equivalently

$\displaystyle f(z) = \sum_{n\ge 1} a_n n^{\ell/2} e^{2 \pi i n z} $

or

$N^{s/2} \Gamma_\R(s + \mu_1 + i\lambda)\Gamma_\R(s + \mu_2 - i\lambda)$
So 3 sub-cases.

equivalently

$\displaystyle f(x,y) = \sum_{n \not = 0} a_n y K_{i\lambda} (2\pi |n|y)e^{2 \pi i n x} $
Classify as even/odd, and $+1/-1$ under Fricke

How to construct examples?

What do they look like?

Degree 3

Two cases: $3=2+1$ and $3 = 1+1+1$.

$N^{s/2}\Gamma_\R(s+\mu+ i\lambda)\Gamma_\C\left(s+\frac{\ell}{2} - i\lambda/2\right)$ for $\mu\in \{0,1\}$ and $\ell \in \{1,\ 2,\ 3,\ldots\}$

or

$N^{s/2} \Gamma_\R(s + \mu_1 + i\lambda_1)\Gamma_\R(s + \mu_2 + i\lambda_2)\Gamma_\R(s + \mu_2 + i\lambda_3)$
4 sub-cases from the $\mu_j$.
2 parameters because $\lambda_1 + \lambda_2 + \lambda_3 = 0$.
Can assume $0\le \lambda_1 \le \lambda_2$

equivalently $\displaystyle f(x_1, x_2, x_3,y_1, y_2) = \text{a very complicated sum} $

Examples in the LMFDB

Degree 4

Three cases: $4=2+2$, $4=2+1+1$, and $4 = 1+1+1+1$.

$N^{s/2}\Gamma_\C\left(s+\frac{\ell_1}{2}+ i\lambda\right)\Gamma_\C\left(s+\frac{\ell_2}{2} - i\lambda\right)$ for $\ell_j \in \{1,\ 2,\ 3,\ldots\}$

$\lambda=0$, $\ell_1=\ell_2 = 1$ come from:
Genus 2 curves, elliptic curves over quadratic fields, abelian surfaces,
Hilbert modular forms, Bianchi modular forms, and Siegel paramodular cusp forms.
Every example with integer coefficients and conductor up to 600.

or

$N^{s/2} \Gamma_\C(s + \mu_1 + i\lambda_1)\Gamma_\R(s + \mu_1 + i\lambda_2)\Gamma_\R(s + \mu_2 + i\lambda_3)$
4 sub-cases from the $\mu_j$.
2 parameters because $2\lambda_1 + \lambda_2 + \lambda_3 = 0$.

or

$N^{s/2} \Gamma_\R(s + \mu_1 + i\lambda_1)\Gamma_\R(s + \mu_2 + i\lambda_2)\Gamma_\R(s + \mu_3 + i\lambda_3)\Gamma_\R(s + \mu_4 + i\lambda_4)$
5 sub-cases from the $\mu_j$.
2 parameters because $\lambda_1 + \lambda_2 + \lambda_3 = 0$.
Can assume $0\le \lambda_1 \le \lambda_2$
Examples in the LMFDB

Theorem. (Stephen D. Miller) Every L-function of real archimedean type has a zero with imaginary part in $[0,\ 14.135]$.

In other words, the Riemann $\zeta$-function has the "highest lowest zero".

A "Maass form on $Sp(4, \Z)$": LMFDB

Some topics of interest