The Birch and Swinnerton-Dyer Conjecture

A polynomial relation f(x, y) = 0 in two variables defines a curve C0. If the coefficients of the polynomial are rational numbers then one can ask for solutions of the equation f(x, y) = 0 with x, y ∈ Q, in other words for rational points on the curve. The set of all such points is denoted C0(Q). If we consider a non-singular projective model C of the curve then topologically C is classified by its genus, and we call this the genus of C0 also. Note that C0(Q) and C(Q) are either both finite or both infinite. Mordell conjectured, and in 1983 Faltings proved, the following deep result

[1]  Ju. Manin,et al.  CYCLOTOMIC FIELDS AND MODULAR CURVES , 1971 .

[2]  James S. Milne,et al.  Arithmetic Duality Theorems , 1987 .

[3]  Correction: "Modularity of the Rankin-Selberg $L$-series, and multiplicity one for SL(2)" , 2000, math/0007203.

[4]  J. Tate,et al.  The arithmetic of elliptic curves , 1974 .

[5]  Don Zagier,et al.  Heegner points and derivatives ofL-series , 1986 .

[6]  J. Tate,et al.  Algebraic cycles and poles of zeta functions , 1965 .

[7]  André Weil,et al.  Basic number theory , 1967 .

[8]  J. Coates,et al.  The Conjecture of Birch and Swinnerton-Dyer , 1977, Open Problems in Mathematics.

[9]  L. Dickson.  History of the Theory of Numbers , 1924, Nature.

[10]  N. D. Elkies.  On A4+B4+C4=D4 , 1988 .

[11]  G. Faltings.  Endlichkeitssätze für abelsche Varietäten über Zahlkörpern , 1983 .

[12]  S. Bloch.  Height pairings for algebraic cycles , 1984 .

[13]  F. Jarvis.  On Galois representations associated to Hilbert modular forms. , 1997 .

[14]  Noam D. Elkies.  On A 4 + B 4 + C 4 = D 4 , 1988 .

[15]  A. Beilinson.  Notes on absolute Hodge cohomology , 1986 .

[16]  W. B..  History of the Theory of Numbers , Nature.

[17]  Henri Poincaré,et al.  Sur les propriétés arithmétiques des courbes algébriques , 1901 .

[18]  Wiles’ Theorem and the Arithmetic of Elliptic Curves , 1997 .

[19]  Jerrold B. Tunnell,et al.  A classical Diophantine problem and modular forms of weight 3/2 , 1983 .

[20]  S. Lang.  Number Theory III , 1991 .

[21]  Kenneth A. Ribet,et al.  Modular elliptic curves and fermat's last theorem , 1993 .

[22]  Irving Reiner,et al.  Methods of Representation Theory , 1981 .

[23]  R. Taylor,et al.  On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises , 2001, Journal of the American Mathematical Society.

[24]  J. Milne.  On the arithmetic of abelian varieties , 1972 .

[25]  D. Hilbert,et al.  Über die diophantischen Gleichungen vom Geschlecht Null , 1890 .

[26]  A. Wiles,et al.  Ring-Theoretic Properties of Certain Hecke Algebras , 1995 .

[27]  G. Faltings.  The General Case of S. Lang's Conjecture , 1994 .

[28]  G. L..  Collected Papers , 1912, Nature.

[29]  J. Tate,et al.  On the conjectures of Birch and Swinnerton-Dyer and a geometric analog , 1966 .

[30]  H. Swinnerton-Dyer,et al.  Notes on elliptic curves. II. , 1963 .