The Taniyama-Shimura conjecture, since its proof now sometimes known as the modularity theorem, is very general and important conjecture (and now theorem) connecting topology and number theory which arose from several problems proposed by Taniyama in a 1955 international mathematics symposium.
Let E be an elliptic curve whose equation has integer coefficients, let N be the so-called j-conductor of E and, for each n, let a_n be the number appearing in the L-function of E. Then, in technical terms, the Taniyama-Shimura conjecture states that there exists a modular form of weight two and level N which is an eigenform under the Hecke operators and has a Fourier series suma_nq^n.
In effect, the conjecture says that every rational elliptic curve is a modular form in disguise. Or, more formally, the conjecture suggests that, for every elliptic curve y^2=Ax^3+Bx^2+Cx+D over the rationals, there exist nonconstant modular functions f(z) and g(z) of the same level N such that
[f(z)]^2=A[g(z)]^2+Cg(z)+D.
Equivalently, for every elliptic curve, there is a modular form with the same Dirichlet L-series.
The Taniyama-Shimura conjecture, since its proof now sometimes known as the modularity theorem, is very general and important conjecture (and now theorem) connecting topology and number theory which arose from several problems proposed by Taniyama in a 1955 international mathematics symposium.Let E be an elliptic curve whose equation has integer coefficients, let N be the so-called j-conductor of E and, for each n, let a_n be the number appearing in the L-function of E. Then, in technical terms, the Taniyama-Shimura conjecture states that there exists a modular form of weight two and level N which is an eigenform under the Hecke operators and has a Fourier series suma_nq^n.In effect, the conjecture says that every rational elliptic curve is a modular form in disguise. Or, more formally, the conjecture suggests that, for every elliptic curve y^2=Ax^3+Bx^2+Cx+D over the rationals, there exist nonconstant modular functions f(z) and g(z) of the same level N such that [f(z)]^2=A[g(z)]^2+Cg(z)+D. Equivalently, for every elliptic curve, there is a modular form with the same Dirichlet L-series.
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