By Alvaro Lozano-Robledo

ISBN-10: 0821852426

ISBN-13: 9780821852422

Many difficulties in quantity concept have uncomplicated statements, yet their suggestions require a deep knowing of algebra, algebraic geometry, advanced research, workforce representations, or a mix of all 4. the unique easily acknowledged challenge may be obscured within the intensity of the idea constructed to appreciate it. This ebook is an creation to a few of those difficulties, and an outline of the theories used these days to assault them, awarded in order that the quantity concept is usually on the leading edge of the dialogue. Lozano-Robledo offers an introductory survey of elliptic curves, modular kinds, and $L$-functions. His major aim is to supply the reader with the massive photograph of the striking connections between those 3 households of mathematical items and their which means for quantity conception. As a working example, Lozano-Robledo explains the modularity theorem and its well-known end result, Fermat's final Theorem. He additionally discusses the Birch and Swinnerton-Dyer Conjecture and different smooth conjectures. The ebook starts off with a few motivating difficulties and contains a number of concrete examples during the textual content, frequently concerning genuine numbers, reminiscent of three, four, five, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular types, and $L$-functions are too tremendous to be coated in one quantity, and their proofs are outdoors the scope of the undergraduate curriculum. even though, the first gadgets of research, the statements of the most theorems, and their corollaries are in the seize of complicated undergraduates. This publication concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and delivering plenty of examples, instead of living at the demanding proofs. The ebook succeeds if, after interpreting the textual content, scholars believe forced to check elliptic curves and modular types in all their glory.

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Additional info for Elliptic Curves, Modular Forms, and Their L-functions (Student Mathematical Library, Volume 58)

Example text

Since x3 +Ax+B does not have an x2 term, we conclude that e1 + e2 + e3 = 0. Suppose, then, that E : y 2 = (x − e1 )(x − e2 )(x − e3 ), where the roots satisfy ei ∈ Z and e1 + e2 + e3 = 0. e. y02 = (x0 − e1 )(x0 − e2 )(x0 − e3 ). Thus, each term (x0 − ei ) must be almost a square, and we can make this precise by writing (x0 − e1 ) = au2 , (x0 − e2 ) = bv 2 , (x0 − e3 ) = cw2 , y02 = abc(uvw)2 where a, b, c, u, v, w ∈ Q, the numbers a, b, c ∈ Q are square-free, and abc is a square (in Q). 1. Let E : y 2 = x3 − 556x + 3120 = (x − 6)(x − 20)(x + 26), so that e1 = 6, e2 = 20 and e3 = −26.

For instance, 3 and 25 2 12 represent the same element of Q× /(Q× )2 because 25 = 3 · 25 . The following theorem constructs such a homomorphism. 14. 8. 3. Let E/Q be an elliptic curve y 2 = x3 + Ax + B = (x − e1 )(x − e2 )(x − e3 ) with distinct e1 , e2 , e3 ∈ Z and e1 + e2 + e3 = 0. There is a homomorphism of groups δ : E(Q) → Q× /(Q× )2 × Q× /(Q× )2 × Q× /(Q× )2 defined for P = (x0 , y0 ) by:   (1, 1, 1)       (x0 − e1 , x0 − e2 , x0 − e3 ) δ(P ) = ((e1 − e2 )(e1 − e3 ), e1 − e2 , e1 − e3 )     (e2 − e1 , (e2 − e1 )(e2 − e3 ), e2 − e3 )    (e − e , e − e , (e − e )(e − e )) 3 1 3 2 3 1 3 2 if P = O; if y0 = 0; if P = (e1 , 0); if P = (e2 , 0); if P = (e3 , 0).

6. Let E/Q : y 2 = x3 − 2, so that A = 0 and B = −2. The polynomial x3 − 2 does not have any rational roots, so E(Q) does not contain any points of order 2. Also, 4A3 + 27B 2 = 27 · 4. Thus, if (x(P ), y(P )) are the coordinates of a torsion point in E(Q) then y(P ) is an integer and y(P )2 divides 27 · 4. This implies that y(P ) = ±1, ±2, ±3, or ±6. In turn, this implies that x(P )3 = 3, 6, 11 or 38, respectively. However, x(P ) is an integer, and none of 3, 6, 11 or 38 is a perfect cube. e. the only torsion point is O).