By Carlos Moreno

ISBN-10: 052134252X

ISBN-13: 9780521342520

During this tract, Professor Moreno develops the speculation of algebraic curves over finite fields, their zeta and L-functions, and, for the 1st time, the idea of algebraic geometric Goppa codes on algebraic curves. one of the functions thought of are: the matter of counting the variety of strategies of equations over finite fields; Bombieri's evidence of the Reimann speculation for functionality fields, with effects for the estimation of exponential sums in a single variable; Goppa's conception of error-correcting codes made out of linear platforms on algebraic curves; there's additionally a brand new evidence of the TsfasmanSHVladutSHZink theorem. the must haves had to stick to this ebook are few, and it may be used for graduate classes for arithmetic scholars. electric engineers who have to comprehend the trendy advancements within the idea of error-correcting codes also will reap the benefits of learning this paintings.

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**Extra resources for Algebraic Curves over Finite Fields**

**Example text**

Verify that the curve y 2 — y = x3 is non-singular over a field of characteristic 2. 2. (Automorphism groups of algebraic curves) It is well known that over the field of complex numbers the full automorphism group of the Klein curve xy 3 + yz3 + zx3 is PSL 2 (F 7 ). Verify that the same is true over the algebraic closure of any finite field Fp, p # 7. What can you say about the automorphism group of the singular curve over the finite field F7? 3. Show that the automorphism group of the curve y 2 — y = x 3 is isomorphic to SL2(F3).

S; hence dim t L(D + Q)S/L(D)S < d. , ad in k, the element y = YJ=I a,*;" does not belong to L(D)S. i=\ aj*'j ^ ®> an<* hence ordgfyu"1) = 0 or equivalently ord 0 >' + ord 0 (D + Q) = ordQ(y) + ord 0 (D) + 1 = 0. On the other hand if y e L(D)S then ord Q y + ord e (D) > 0, which is impossible. 1. 2 The vector space L(D) The statement of the Riemann-Roch theorem refers to a vector space which generalizes L(D)S and whose definition we now present. 2 Let S be the set of all closed points of C. For D a divisor in Div(C) we put = {xe Kx: ordp(x) + ordP(D) > 0 for all P e S} and L{D) is a vector space over k and its dimension is denoted by = dim t L(D).

With A there is associated a smooth point on a curve C with function field K. If K is of degree n over k(x), then we can think of it as the function field of a curve C which is a covering of the projective line P 1 of degree n. Heuristically this gives us an idea of where the points of C come from. Since P 1 has an infinite number of points, so does the curve C. Example Let C be the curve y2 — y = x3 — x2 and let K = ¥tl(x,y) be its function field. 1, above each point on P 1 there are two points on C except over x = — 3 where ramification occurs.

### Algebraic Curves over Finite Fields by Carlos Moreno

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