Algebraic Function Fields

In large parts of this book, the basic theory of algebraic function fields isassumed. In this appendix we collect the main definitions, notations and resultsof this theory. For a detailed exposition the reader is referred to the books “Al-gebraic Function Fields and Codes” by H. Stichtenoth (Springer Universitext,1993) and “Rational Points on Curves over Finite Fields” by H. Niederreiterand C. P. Xing (London Math. Soc. Lecture Notes Ser. 285, 2001).

(1) An algebraic function field F/K is a finite field extension of the rationalfunction field K(x) where K is a perfect field. We always assume implicitelythat the field K is algebraically closed in F (i.e.; every element z ∈ F whichis algebraic over K is already in K). The field K is called the constant fieldof F . We consider in this book mostly function fields F/Fq where Fq is thefinite field with q elements. Such function fields are also called global functionfields.

(2) A place of F is, by definition, the maximal ideal of some valuation ringO of F/K. To every place P there corresponds a unique normalized discretevaluation, denoted vP or νP , which is a surjective map from F to Z ∪ {∞}satisfying the following properties:

(i) vP (x) = ∞ if and only if x = 0.(ii) vP (xy) = vP (x) + vP (y) for all x, y ∈ F .(iii) vP (x+ y) ≥ min(vP (x), vP (y)) for all x, y ∈ F .(iv) vP (a) = 0 for all a ∈ K×.

In terms of the valuation vP , the corresponding valuation ring O = OP of theplace P is then given as OP = {x ∈ F | vP (x) ≥ 0}, and the place P isgiven as P = {x ∈ F | vP (x) > 0}. The residue class field OP /P is a finiteextension of the constant field K, and the degree of the place P is defined as

degP = [OP /P : K].

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Appendix:

196

The place P is said to be a rational place if degP = 1. In this case we havethe residue class map at P as follows:

OP → K , f �→ f(P ) ,

where f(P ) ∈ K is the residue class of f in K = OP /P .

(3) A divisor D of F/K is a formal sum D =∑

P aPP of places P withinteger coefficients aP , and aP �= 0 for only finitely many P . One aften writesvP (D) for the coefficient aP , hence

D =∑

P

vP (D)P.

The support of the divisor D is the finite set of places

Supp(D) = {P | vP (D) �= 0},

and the degree of D is defined as

deg(D) =∑

P

vP (D) degP.

Let P be a place of F and x a nonzero element of F . The place P is called azero of x if vP (x) > 0 and a pole of x if vP (x) < 0. The zero divisor of theelement x is defined as

(x)0 =∑

vP (x)>0

vP (x)P,

and the pole divisor of x is defined as

(x)∞ = (x−1)0 = −∑

vP (x)<0

vP (x)P.

The principal divisor of x is given by

div(x) = (x)0 − (x)∞.

All principal divisors have degree zero.

(4) With a divisor D of F one associates its Riemann-Roch space

L(D) = {x ∈ F× | div(x) ≥ −D} ∪ {0}.

This is a finite-dimensional vector space over K, its dimension is denoted by�(D). One of the main results in the theory of function fields is the followingtheorem which gives a formula for the dimension of Riemann-Roch spaces:

Appendix: Algebraic Function Fields

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(5) (Riemann-Roch theorem) Let F/K be a function field. Then there is anon-negative integer g = g(F ) such that:

(i) �(D) ≥ deg(D) + 1− g for all divisors D of F/K.(ii) For all divisorsDwith degD > 2g−2 we have �(D) = deg(D)+1−g.

The integer g is uniquely determined by the conditions in (i) and (ii), and it iscalled the genus of the function field F . The rational function field K(x) hasgenus g(K(x)) = 0.

(6) Let F/K and E/K be function fields with F ⊆ E. Then the extensionE/F is a finite field extension. Let P be a place of F and let Q be a place ofE. We say that Q lies above P (and write then Q|P ), if the valuation ring ofthe place Q contains the valuation ring of P . We have the following facts:

(i) For all places P of F , the set of places Q of E which lie above P is finiteand non-empty.

(ii) Let Q be a place of E. Then there exists exactly one place P of F suchthat Q|P , namely P = Q ∩ F .Now letQ|P be places as above. Then there is a unique integer e = e(Q|P ) ≥ 1such that vQ(z) = e · vP (z) for all elements z ∈ F . The number e is calledthe ramification index of Q|P . The place Q is said to be ramified over P ife(Q|P ) > 1, otherwise Q|P is unramified.Also, there is an integer f = f(Q|P ) ≥ 1 such that deg(Q) = f(Q|P ) ·deg(P ), and we call f(Q|P ) the relative degree ofQ|P . The following formula(“fundamental equality”) holds for any place P of the function field F :

∑

Q|Pe(Q|P ) · f(Q|P ) = [E : F ].

(7) Let F/K and E/K be function fields such that E ⊇ F is a finite andseparable extension. Then almost all (i.e., all but finitely many) places of Eare unramified in E/F . Let P be a place of F and let Q be a place of E lyingaboveP . Then one defines the different exponent d(Q|P ); this is a non-negativeinteger which has the following property :

d(Q|P ) ≥ e(Q|P )− 1 ,with equality if and only if e(Q|P ) is not divisible by the characteristic of K.The divisor

Diff(E/F ) =∑

Q

d(Q|P )Q

is called the different ofE/F . Note that the different ofE/F is a divisor of thefunction field E/K, and we have Diff(E/F ) ≥ 0. The support of Diff(E/F )contains exactly the places of E which are ramified in E/F .

Appendix: Algebraic Function Fields

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(8) Consider again a separable extension E/F of function fields as in (7), andlet P and Q be places of F and E with Q|P . We say that Q|P is wildlyramified if the ramification index e(Q|P ) is divisible by the characteristic ofK. Otherwise, Q|P is said to be tame. By (7) we have that

d(Q|P ) = e(Q|P )− 1, if Q|P is tame,

andd(Q|P ) ≥ e(Q|P ), if Q|P is wild.

(9) LetE/F be a separable extension of function fields having the same constantfield K. Then one has the following “Hurwitz genus formula” which relatesthe genera of F and E:

2g(E)− 2 = [E : F ](2g(F )− 2) + deg Diff(E/F ).

This formula is crucial in order to determine the genus of a function field, sincea function field F/K is often represented as a finite separable extension of arational subfield K(x). Then the Hurwitz genus formula becomes

2g(F )− 2 = −2[F : K(x)] + deg Diff(F/K(x)).

Appendix: Algebraic Function Fields

About the Authors

Arnaldo Garcia; Instituto de Matematica Pura e Aplicada (IMPA), Rio deJaneiro RJ, Brazil; E-mail: garcia@impa.br

Cem Guneri; Faculty of Engineering and Natural Sciences, Sabancı University,Istanbul, Turkey; E-mail: guneri@sabanciuniv.edu

Ram Murty1 2; Department of Mathematics, Queen’s University, Ontario,Canada ; E-mail: murty@mast.queensu.ca

Harald Niederreiter3; Department of Mathematics, National University ofSingapore, Singapore; E-mail: nied@math.nus.edu.sg

Ferruh Ozbudak4; Department of Mathematics, Middle East Technical Uni-versity, Ankara, Turkey; E-mail: ozbudak@metu.edu.tr

Igor Shparlinski1 5; Department of Computing, Macquarie University, Sydney,Australia; E-mail: igor@ics.mq.edu.au

Henning Stichtenoth; Department of Mathematics, University of Duisburg-Essen, Essen, Germany and Faculty of Engineering and Natural Sciences,Sabancı University, Istanbul, Turkey; E-mail: stichtenoth@uni-essen.de andhenning@sabanciuniv.edu

Alev Topuzoglu; Faculty of Engineering and Natural Sciences, Sabancı Uni-versity, Istanbul, Turkey; E-mail: alev@sabanciuniv.edu

1Thanks to Florian Luca and Francesco Pappalardi for a careful reading of the manuscript and many valuablecomments.2The author was supported in part by an NSERC grant.3The author was supported by Australian Research Council Discovery Grants and by the MOE-ARF researchgrant R-146-000-066-112.4The author was supported in part by the Turkish Academy of Sciences in the framework of Young ScientistsAward Programme (F.O./TUBA-GEBIP/2003-13).5The author was supported in part by an ARC grant.

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A. Garcia and H. Stichtenoth (eds.), Topics in Geometry, Coding Theory and Cryptography, 199–198.C© 2007 Springer.

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Huaxiong Wang3; Department of Computing, Macquarie University, Sydney,Australia ; E-mail: hwang@ics.mq.edu.au

Arne Winterhof6; Johann Radon Institute for Computational and Ap-plied Mathematics, Austrian Academy of Sciences, Linz, Austria; E-mail:arne.winterhof@oeaw.ac.at

Chaoping Xing3; Department of Mathematics, National University of Singa-pore, Singapore; E-mail: matxcp@nus.edu.sg

6The author was supported in part by Austrian Science Fund (FWF), grant S8313

About the Authors

Algebras and Applications

1. C.P. Millies and S.K. Sehgal: An Introduction to Group Rings. 2002

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Groups. 2003 ISBN 1-4020-1438-4 3. J.F. Carlson, L. Townsley, L. Valero-Elizondo and M. Zhang: Cohomology Rings

of Finite Groups. Calculations of Cohomology Rings of Groups of Order Dividing 64. 2003 ISBN 1-4020-1525-9

4. K. Kiyek and J.L. Vicente: Resolution of Curve and Surface Singularities. In Characteristic Zero. 2004 ISBN 1-4020-2028-7

5. U. Ray: Automorphic Forms and Lie Superalgebras. 2006 ISBN 1-4020-5009-7

6. A. Garcia and H. Stichtenoth: Topics in Geometry, Coding Theory and Cryptography. 2006 ISBN 1-4020-5333-9

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