10 1. Modular Forms
(1) Fermat’s last theorem: Wiles’ proof [Wil95] of Fermat’s last
theorem uses modular forms extensively. The work of Wiles et al.
on modularity also massively extends computational methods for
elliptic curves over Q, because many elliptic curve algorithms, e.g.,
for computing L-functions, modular degrees, Heegner points, etc.,
require that the elliptic curve be modular.
(2) Diophantine equations: Wiles’ proof of Fermat’s last theorem
has made available a wide array of new techniques for solving cer-
tain diophantine equations. Such work relies crucially on having
access to tables or software for computing modular forms. See,
e.g., [Dar97, Mer99, Che05, SC03]. (Wiles did not need a com-
puter, because the relevant spaces of modular forms that arise in
his proof have dimension 0!) Also, according to Siksek (personal
communication) the paper [BMS06] would “have been entirely im-
possible to write without [the algorithms described in this book].”
(3) Congruent number problem: This ancient open problem is to
determine which integers are the area of a right triangle with ra-
tional side lengths. There is a potential solution that uses modular
forms (of weight 3/2) extensively (the solution is conditional on
truth of the Birch and Swinnerton-Dyer conjecture, which is not
yet known). See [Kob84].
(4) Topology: Topological modular forms are a major area of current
(5) Construction of Ramanujan graphs: Modular forms can be
used to construct almost optimal expander graphs, which play a
role in communications network theory.
(6) Cryptography and Coding Theory: Point counting on elliptic
curves over finite fields is crucial to the construction of elliptic curve
cryptosystems, and modular forms are relevant to eﬃcient algo-
rithms for point counting (see [Elk98]). Algebraic curves that are
associated to modular forms are useful in constructing and studying
certain error-correcting codes (see [Ebe02]).
(7) The Birch and Swinnerton-Dyer conjecture: This central
open problem in arithmetic geometry relates arithmetic proper-
ties of elliptic curves (and abelian varieties) to special values of
L-functions. Most deep results toward this conjecture use modu-
lar forms extensively (e.g., work of Kolyvagin, Gross-Zagier, and
Kato). Also, modular forms are used to compute and prove results
about special values of these L-functions. See [Wil00].