By Álvaro Lozano-Robledo

ISBN-10: 0821852426

ISBN-13: 9780821852422

Many difficulties in quantity concept have easy statements, yet their recommendations require a deep figuring out of algebra, algebraic geometry, advanced research, staff representations, or a mix of all 4. the unique easily said challenge should be obscured within the intensity of the idea built to appreciate it. This e-book is an advent to a few of those difficulties, and an summary of the theories used these days to assault them, provided in order that the quantity conception is often on the vanguard of the dialogue. Lozano-Robledo supplies an introductory survey of elliptic curves, modular kinds, and $L$-functions. His major objective is to supply the reader with the large photograph of the excellent connections between those 3 households of mathematical gadgets and their which means for quantity idea. As a for instance, Lozano-Robledo explains the modularity theorem and its well-known outcome, Fermat's final Theorem. He additionally discusses the Birch and Swinnerton-Dyer Conjecture and different smooth conjectures. The booklet starts off with a few motivating difficulties and comprises a number of concrete examples through the textual content, frequently regarding genuine numbers, similar to three, four, five, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular varieties, and $L$-functions are too immense to be lined in one quantity, and their proofs are outdoor the scope of the undergraduate curriculum. even if, the first items of research, the statements of the most theorems, and their corollaries are in the grab of complicated undergraduates. This ebook concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and offering plenty of examples, instead of residing at the demanding proofs. The ebook succeeds if, after analyzing the textual content, scholars think forced to check elliptic curves and modular kinds in all their glory

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**Extra resources for Elliptic curves, modular forms, and their L-functions**

**Example text**

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. The point (x0 , y0 ) = 66469980 ( 164184 289 , 4913 ) is rational and on E. We can write x0 − e1 = 164184 −6=2· 289 285 17 2 293 2 2 and, similarly, x0 − e2 = ( 398 17 ) and x0 − e3 = 2 · ( 17 ) .

Let P1 , P2 , . . , Pr be r rational points on E(Q). The elliptic height matrix associated to {Pi }ri=1 is H = H({Pi }ri=1 ) := ( Pi , Pj )1≤i≤r, 1≤j≤r . The determinant of H is called the elliptic regulator of the set of points {Pi }ri=1 . If {Pi }ri=1 is a complete set of generators of the free 48 2. Elliptic curves part of E(Q), then the determinant of H({Pi }ri=1 ) is called the elliptic regulator of E/Q. 5. Let E/Q be an elliptic curve. , (1) For all P, Q ∈ E(Q), P, Q = Q, P . (2) For all P, Q, R ∈ E(Q) and all m, n ∈ Z, P, mQ + nR = m P, Q + n P, R .

13. Suppose that E/Q is an elliptic curve that has bad reduction at a prime p. How many points does the singular curve E have over Fp ? 1. 14. Let E/Q be the elliptic curve y 2 = x3 + 3. Its minimal discriminant is ΔE = −3888 = −24 · 35 . Thus, the only primes of bad reduction are 2 and 3 and E/Fp is smooth for all p ≥ 5. 6. Elliptic curves over ﬁnite ﬁelds 41 where all the coordinates should be regarded as congruences modulo 5. 5278 . . 4721 . . Similarly, one can verify that N7 = 13. The connections between the numbers Np and the group E(Q) are numerous and of great interest.

### Elliptic curves, modular forms, and their L-functions by Álvaro Lozano-Robledo

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