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 purposes thought of are: the matter of counting the variety of ideas of equations over finite fields; Bombieri's facts of the Reimann speculation for functionality fields, with results for the estimation of exponential sums in a single variable; Goppa's concept 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 booklet are few, and it may be used for graduate classes for arithmetic scholars. electric engineers who have to comprehend the fashionable advancements within the conception of error-correcting codes also will make the most of learning this paintings.

Show description

Read Online or Download Algebraic Curves over Finite Fields PDF

Similar algebraic geometry books

The legacy of Niels Henrik Abel: the Abel bicentennial, - download pdf or read online

This e-book incorporates a sequence of analysis papers on topics regarding the paintings of Niels Henrik Abel, written by way of a few of the premier experts of their fields. many of the authors were particularly invited to provide papers, discussing the impression of Abel in a mathematical-historical context.

Read e-book online Tata Lectures on Theta III PDF

The second one in a sequence of 3 volumes surveying the speculation of theta capabilities, this quantity supplies emphasis to the detailed houses of the theta services linked to compact Riemann surfaces and the way they result in recommendations of the Korteweg-de-Vries equations in addition to different non-linear differential equations of mathematical physics.

Get Logarithmic forms and diophantine geometry PDF

There's now a lot interaction among reviews on logarithmic kinds and deep points of mathematics algebraic geometry. New gentle has been shed, for example, at the recognized conjectures of Tate and Shafarevich with regards to abelian forms and the linked celebrated discoveries of Faltings setting up the Mordell conjecture.

Download PDF by Richard Evan Schwartz: Mostly surfaces

This booklet provides a few issues regarding surfaces, comparable to Euclidean, round and hyperbolic geometry, the basic staff, common masking surfaces, Riemannian manifolds, the Gauss-Bonnet Theorem, and the Riemann mapping theorem. the most proposal is to get to a few fascinating arithmetic with no an excessive amount of formality.

Additional resources for Algebraic Curves over Finite Fields

Sample text

We now want to show that Q^ /t (also known as the dualizing module) may also be viewed as a vector space over K. In fact think of K as the subset of principal pre-adeles in A and for x e K and u> e £%/jt(D) put xft)(r) = co(xr) for any r e A/(A (D) + K). K/k(D - (x)). The following properties are immediate consequences of the definitions: for x, y e K and co, co' e i^ /jk we have (i) (xy)a> = x(yco), (ii) (x + y)a> = xa> + yu> (iii) x(w + a)') = xo) + xw'. This shows that Q^ /t is a vector space over K.

If we let C1(C) = Div(C)/Diva(C) and C1O(C) = Divo(C)/Diva(C) then we obtain the exact sequence 0 -> C1O(C) -> C1(C) -> dT -»0, which is of fundamental importance in the study of the zeta function of the curve C. It will be shown later that indeed 8=1. 2 For any integer d, the number ofdivisor classes in C1(C) of degree d is independent of d and is equal to the cardinality of C1O(C). Proof. We first show that for a fixed positive integer d0 there can be at most a finite number of closed points P on C with bounded degree d{P) < d0.

If x, y e Kx, then (x><) = (x) + (y) and the set Divo(C) = {(x): x e K*} 34 The Riemann-Roch theorem of all principal divisors is a subgroup of Div(C). 1 shows that d((x)0) < IK : k(x)l and d((x)J < [K : fc(x)]. 2 If x is a non-constant element in Kx, then Proof. e. y satisfies a monic polynomial equation ym + fm-i W / " " 1 + • • • + fo{x) = o, with fj€ k [x], then P is a pole of y only if it is a pole of x; equivalently, P appears in (y)x only if it appears in {x)x. Observe that if P does not appear in (x)x, then ordP(x) > 0 and therefore /m-l \ mordp(y) = ordP(ym) = o r d J X &x)yJ \j=0 ^ ) min {jordp(y),0} OsjSm-l = ; 0 ord P (y), with 0

Download PDF sample

Algebraic Curves over Finite Fields by Carlos Moreno


by Steven
4.2

Rated 4.58 of 5 – based on 41 votes