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

research.

(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].