The L-series is a Dirichlet series, commonly written The generating function of the coefficients an is then If we make the substitution we see that we have written the Fourier expansion of a function f(E,τ) of the complex variable τ, so the coefficients of the q-series are also thought of as the Fourier coefficients of f. The function obtained in this way is, remarkably, a cusp form of weight two and level N and is also an eigenform (an eigenvector of all Hecke operators); this is the Hasse–Weil conjecture, which follows from the modularity theorem.
The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible Abelian varieties, corresponding to Hecke eigenforms of weight 2.
Yutaka Taniyama[1] stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikkō as the twelfth of his set of 36 unsolved problems.
[3][4] The conjecture attracted considerable interest when Gerhard Frey[5] suggested in 1986 that it implies Fermat's Last Theorem.
He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply the existence of at least one non-modular elliptic curve.
[8] For example, Wiles's Ph.D. supervisor John Coates states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".
In 2013, Freitas, Le Hung, and Siksek proved that elliptic curves defined over real quadratic fields are modular.
[13] For example,[14][15][16] the elliptic curve y2 − y = x3 − x, with discriminant (and conductor) 37, is associated to the form For prime numbers l not equal to 37, one can verify the property about the coefficients.
The conjecture, going back to the 1950s, was completely proven by 1999 using the ideas of Andrew Wiles, who proved it in 1994 for a large family of elliptic curves.
The functions x(z) and y(z) are modular of weight 0 and level 37; in other words they are meromorphic, defined on the upper half-plane Im(z) > 0 and satisfy and likewise for y(z), for all integers a, b, c, d with ad − bc = 1 and 37 | c. Another formulation depends on the comparison of Galois representations attached on the one hand to elliptic curves, and on the other hand to modular forms.