### Algebraic Curves over Finite Fields

This webpage is devoted to the graduade course Algebraic Curves over Finite Fields for which I am delivering lectures in Fall 2018 at the University of South Florida.

Time and location of class:

MW: 2:30–3:45 pm in CMC 109

Office hours (CMC 110):

MW: 5-6 pm

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Principal references:

• FUL: Algebraic Curves: An Introduction to Algbebraic Geometry, William Fulton.
• SHA: Basic Algebraic Geometry 1: Varieties in Projective Space, Igor Shafarevich.
• SIL: The Arithmetic of Elliptic Curves, Joseph H. Silverman.
• STI: Algebraic Function Fields and Codes (2nd edition), Henning Stichtenoth.
• TVN: Algebraic Geometric Codes: Basic Notions, Michael Tsfasman, Serge Vlǎduţ and Dmitry Nogin.

Announcements
• The second homework is due on Wedenesday, November 28 at the beginning of the class.
• The first homework is due on Monday, October 8 October 15 at the beginning of the class.
Here below you can find the material covered in each day of class. For each class the corresponding references are indicated.

#### Logbook:

• 08/20/18: From Diophantus to algebraic curves over finite fields

Introduction to the course.

• 08/22/18: Why do we normally do algebraic geometry over an algebraically closed field?

Quick review about commutative rings: commutative rings, integral domains, GCD domains, UFD, PID, ED, fields. Notation: the affine plane $\mathbb A^2(K)$ and the ring of polynomials in two variables $K[x,y]$. Definitions of an algebraic plane curve in $\mathbb A^2(K)$, degree of a curve, irreducible curves and decomposition into irreducible components. Examples in the real plane. Some definitions are not well defined if the ground field is not algebraically closed... Review of the definition of an algebraically closed field and examples. Proposition: An algebraically closed field is infinite - with proof. Lemma (Ch. 1 - SHA) - with proof.

Reference: SHA - Section 1.1 (pages 3-6).

• 08/27/18: Rational plane curves

A classical example $y^2=x^2+x^3$: geometrical construction of a parametrization. Definition of a rational plane curve. Examples of rational curves: curves of degree 1 - lines; curves of degree 2 - conics (geometrical construction of a parametrization and rational points on conics).

Reference: SHA - Section 1.2 (pages 6-9).

• 08/29/18: From algebraic curves to field theory

Rational functions defined on a (plane) curve (examples on the unit circle). The function field of a curve and examples (lines, rational curves). Theorem: $X$ is a rational curve if and only if $K(X)\cong K(t)$ (with proof). Properties of the parametrization arising from the isomorphism $K(X)\cong K(t)$. Definition of rational and birational maps between (plane) curves. Theorem: $X$ and $Y$ are birationally equivalent if and only if $K(X)\cong K(Y)$

Reference: SHA - Section 1.3, 1.4 (pages 9-13).

• 09/05/18: Affine algebraic sets

Review (commutative algebra): ideal, ideal generated by a set, sum and product of ideals, radical ideal, Noetherian ring). Definitions: affine $n$-space over $k$, hypersurface, hyperplane, affine algebraic set. Properties of algebraic sets and the Zariski topology on $\mathbb A^n(k)$. The ideal of a set of points and its properties. The Hilbert Basis Theorem. Proposition: every algebraic set is an intersection of a finite number of hypersurfaces.

Reference: FUL - Section 1.2, 1.3, 1.4 (pages 4-7).

• 09/10/18: Irreducible affine algebraic sets

Definition of a reducible affine algebraic set (example). Review: prime ideals of a ring. Proposition: $V$ irreducible $\Leftrightarrow$ $I(V)$ prime (with proof). Proposition: every algebraic set is the finite union of irreducible algebraic sets (with proof). Weak Nullstellensatz (with proof).

Reference: FUL - Section 1.5, 1.7 (pages 7,8,10).

• 09/12/18: Hilbert's Nullstellensatz

Hilbert's Nullstellensatz (with proof) and some corollaries. Correspondence between algebraic sets in $\mathbb A^n(k)$ and radical ideals is $k[X_1,\ldots, X_n]$. Defintion of an affine variety. Polynomial function on a variety $V$. The coordinate ring $k[V]$ (examples). Polynomial maps (morphisms) between two affine varieties $V$ and $W$ and corresponding ring homomorphisms between $k[W]$ and $k[V]$.

• Reference: FUL - Section 1.7, 2.1, 2.2 (pages 11, 17, 18).

• 09/17/18: Morphisms between varieties

Examples of morphisms. Definition: isomorphism between varieties and examples (affine change of coordinates). Function field $k(v)$ of a variety $V$. Definition: rational function regular at a point $P$. Definition of the pole set of a rational function. Proposition: the pole set of a rational function is an algebraic set of $V$ (with proof).

Reference: FUL - Section 2.2, 2.3, 2.4 (pages 19-20).

• 09/19/18: Function field of a variety

Value of a polynomial function at a point and evaluation map on $k[V]$. Example. Definition: the local ring $\mathcal O_P(V)$ of $V$ at a point $P$. Proposition: $k[V]$ is the set of rational functions on $V$ which are regular at all points of $V$ (with proof). The evaluation map on $\mathcal O_P(V)$. Proposition: $\mathcal O_P(V)$ is a local ring (proof).

Reference: FUL - Section 2.4 (pages 20-21).

• 09/24/18: How to show that a polynomial in two variables is irreducible?

Methods for showing the irreducibility of a polynomial in two variables: 1) using the definition of an irreducible element; 2) Eiseinstein's criterion (generalized version in an integral domain $R$). Application of the previous two methods to the polynomial $x^2+y^2-1$.

• 09/26/18: Rational maps bewteen affine varieties

Definition of a rational map $\phi:V\rightarrow W$ between affine varieties and regularity at point. Proposition: a rational map of affine varieties is a morphism if and only if it is regular. Problem for the composition of rational maps. The closure of an algebraic set in the Zariski topology. Definition of a dominant rational map. Theorem: the category of affine varieties with dominant rational maps and the category of function fields are contravariantly equivalent. Definition of birational equivalence. Theorem: two affine varieties $V$ and $W$ are birational equivalent if and only if $k(V)\cong k(W)$.

Reference: Lecture notes of A. Sutherland (Lecture #15 - Introduction to Arithmetic Geometry).

• 10/01/18: The dimension of a variety

Definition of a subvariety of a variety $V$. Correspondence between subvarieties of $V$ and prime ideals of $k[V]$. Equivalent definitions for the dimension of a variety $V$: topological definition in terms of chain of subvarieties, $\dim(V)$ is the Krull dimension of $k[V]$, $\dim(V)$ is the the transcendence degree of $k(V)$. Terminology: curves ($\dim=1$), surface ($\dim=2$), $n$-fold ($\dim=n$). The case of a hypersurface. Theorem: the dimension is invariant under birational equivalence. Definition of nonsingularity at a point in terms of the Jacobian matrix. The tangent space at a point. Examples.

Reference: Mix of TVN - Sec. 2.1.1 (page 71) and SIL - Sec. I.1 (page 8).

• 10/03/18: Towards an intrinsic definition of nonsingularity

Proposition: the set of singular points Sing($V$) of a variety $V$ is a proper algebraic subset of $V$ (quick proof). Corollary: a curve has a finite number of singular points (proof). Construction of the isomorphism between the $k$-vector spaces $\frac{M_p(V)}{M_p^2(V)}$ and $T_p(V)^*$ when $V=\mathbb A_n(k)$. Proposition: $P\in V$ is nonsingular if and only if $\dim_k \frac{M_p(V)}{M_p^2(V)}=\dim V$. Examples. Definition of the Zariski cotangent/tangent space. Definition of a regular ring. Theorem: $P\in V$ is nonsingular if and only if $\mathcal O_P(V)$ is regular.

Reference: Mix of SHA - Sec. II.1.3 (pages 86-88) and SIL - Sec. I.1 (page 9).

• 10/08/18: Nonsingularity for curves

A morphism $\phi: V\rightarrow W$, such that $\phi(P)=Q$, induces a linear map $J_{\phi}: T_p(V)\rightarrow T_q(W)$. Theorem: under isomorphism of varieties the tangent spaces at correponding points are isomorphic. Recall: definition of a regular ring. Case of curves, i.e. of rings of dimension 1. Equivalent definitions for a discrete valuation ring (discrete valuation). Theorem: a point $P$ on a curve $V$ is nonsingular if and only if $\mathcal O_P(V)$ is a discrete valuation ring. Construction of a discrete valuation on $k(V)$ corresponding to a nonsingular point $P\in V$: the order of a function $f\in k(V)$ at a point $P \in V$ (zeros and poles of a function). Examples.

Reference: Mix of SHA - Sec. II.1.3 (pages 88-89) and SIL - Sec. I.2 (pages 10-13).

• 10/10/18: Projective spaces

Quick introduction. Definitions: projective space $\mathbb P^{n}(k)$, homogeneous coordinates, projective algebraic set, what does it mean for a polynomial to vanish on a point of $\mathbb P^{n}(k)$ (examples), homogeneous polynomial (examples), homogeneous ideal, the ideal $I(X)$ of $X \subset\mathbb P^{n}(k)$. Examples: line in $\mathbb P^{2}(k)$, hyperplane in $\mathbb P^{n}(k)$. Embeddings of $\mathbb A^n(k)$ in $\mathbb P^{n}(k)$ and bijection with $U_i=\{[x_0:\ldots:x_n]\in \mathbb P^{n}(k) : x_i \neq 0 \}$ (description of the maps $\phi_i$, $\phi_i^{-1}$). The affine variety $V\cap \mathbb A^{n}:= \phi_i^{-1}(V\cap U_i)$ (process of dehomogenization with respect to $x_i$).

Reference: Mix of SIL - Sec. II.1 (pages 21-22), and SHA - Section 4.1 (pages 41,45,46).

• 10/15/18: Affine varieties and projective closures

Review of the previous class: the map $\phi_i$ in dimension 1 and 2 ($n=1,2$). Points at infinity. Process of homogenization/dehomogenization showed on an example. The projective closure $\overline{V}$ of an affine variety $V$. Covering of projective variety through affine varieties. Functions on a projective variety: polynomial function (i.e. constant function), rational functions.

Reference: SIL - Sec. I.2 (pages 12,13).

• 10/17/18: Rational functions on projective varieties

Definition of a rational function on a projective variety $V$. The function field of $V$. Definition of a regular function at a point $P\in V$. The $k$-algebra $k[V]$ of regular functions on $V$. Proposition: if $V$ is a projective variety, then $k[V]=k$ (proof for $V=\mathbb P^n$). Defintions: the dimension of $V$, the local ring of $V$ at a point $P$. Rational maps between projective varieties (two equivalent definitions) and regularity at a point.

Reference: SIL - Sec. I.3 (pages 14,15).

• 10/22/18: Maps between projective varieties

Example of a rational map between $\mathbb P^1$ and $V(X_0X_2^2-X_0X_1^2-X_1^3)$. Definition: isomorphic projective varieties. Dominant rational maps. Equivalence of categories: {projective varieties, dominant rational maps} and {function fields, k-homomorphisms}. Definitions: birational equivalence of projective varieties; rational varieties. Proposition: $C$ a projective curve, $P\in C$ a nonsingular point, $V$ a variety, and $\phi: C\rightarrow V$ a rational map; then $\phi$ is regular at $P$. Corollaries and counterexamples if we remove either smoothness or completion.

Reference: SIL - Sec. I.3 (pages 16,17).

• 10/24/18: Maps between projective curves

Proposition: $C$ a projective curve, $P\in C$ a nonsingular point, $V$ a variety, and $\phi: C\rightarrow V$ a rational map; then $\phi$ is regular at $P$ (with proof). Theorem: $\phi:C_1\rightarrow C_2$ a morphism, then either $\phi$ constant or surjective. A non-constant rational map $\phi:C_1\rightarrow C_2$ between curves induces a finite field extension $\phi^*(k(C_2))\rightarrow k(C_1)$. The degree of a rational map between curves. Equivalence of categories: {smooth curves, non-constant rational maps} and {extensions of $k$ of transcendent degree 1, field injections fixing $k$}. $k$ now an arbitrary field. The characteristic of a field. Fields of characteristic $0$ ($\mathbb Q\subseteq k$) and fields of characteristic $p$ (e.g.: finite fields).

Reference: SIL - Sec. II.1 (pages 23-26).

• 10/29/18: Review of some concepts of field theory

$k$ an arbitrary field. Definition: separable polynomial of $k[x]$. Remarks in characteristic 0 and $p$. Separable elements and separable extensions. Given field extensions $k\subseteq L_1$ and $k\subseteq L_2$, definition of a $k$-embedding of $L_1$ into $L_2$. Remarks for separable extensions. Definition of a perfect field. Example of perfect fields and non perfect fields. Galois extensions and the corresponding Galois group. The Galois correspondence. The Galois group $\operatorname{Gal}_{\mathbb F_q}(\mathbb F_{q^n})$.

Reference: STI - Appendix A (pages 327-331,334).

• 10/31/18: Algebraic sets defined over perfect fields

Let $k$ be a perfect field and $\overline{k}$ be its algebraic closure. The set $\mathbb A^n(k)$ of $k$-rational points of $\mathbb A^n(\overline{k})$. The action of $\operatorname{Gal}_k(\overline{k})$ on $\mathbb A^n(\overline{k})$. Definition of an algebraic set $V$ defined over $k$ and the set $V(k)$ of $k$-rational points on $V$. Examples. The ideal $I(V/k)$. Definitions: closed point of an algebraic set defined over $\mathbb F_q$, the degree of a closed point, the notation $B_d(V)$. Example: for $V=V(y^2+2x^3+x+2)$, describe $V(\mathbb F_3)$.

Reference: SHA - Sec. I.1 (pages 5-7).

• 11/05/18: Algebraic closure of finite fields and the Galois group $\operatorname{Gal}_{\mathbb F_q}\overline{\mathbb F}_q$

Theorem: $\overline{\mathbb F}_q=\bigcup_{n=1}^{\infty}\mathbb F_{q^n}$ (with proof). Definitions: directed set, inverse system of finite groups and homomorphism over a directed set, inverse limit. Examples:

• $(\mathbb N,\leq)$, with $n\leq m \Leftrightarrow n|m$, $\frac{\mathbb Z}{i\mathbb Z}$, $\hat{\mathbb Z}:=\varprojlim\frac{\mathbb Z}{i\mathbb Z}$;
• $(\mathbb N,\leq)$, with $n\leq m \Leftrightarrow n|m$, $\operatorname{Gal}_{\mathbb F_q}(\mathbb F_{q^i})$.
Theorem: $\operatorname{Gal}_{\mathbb F_q}\overline{\mathbb F}_q\cong \varprojlim\operatorname{Gal}_{\mathbb F_q}(\mathbb F_{q^i})$ (with proof).

Reference: http://assets.press.princeton.edu/chapters/s9103.pdf.

• 11/07/18: Affine varieties defined over perfect fields

Definition of irreducibility and absolutely irreducibility for an algebraic set defined over $k$. Examples of algebraic sets which are irreducible over $k$ but not absolutely irreducible ($k=\mathbb Q$ and $k=\mathbb F_2$). Definition of a variety defined over $k$. For $V$ a variety defined over $k$, definition of the coordinate ring $k[V]$ and the function field $k(V)$. Galois action over $\overline{k}[V]$ and $\overline{k}(V)$. If $P\in V(\overline{k})$, definition of $k(P)$, the smallest extension of $k$ containing the coordinates of $P$, and the degree of $P$. Evaluation map $\theta_P:k[V]\rightarrow k(P)$ and remarks on $\ker(\theta_P)$. Example with $k=\mathbb F_3$ and $V=\mathbb A^1(\overline{\mathbb F}_3)$.

• 11/14/18: Morphisms and rational maps defined over $k$

The local ring of a variety defined over $k$ at a closed point $P$ and condition for the non-singularity of $P$. Example in $\mathbb F_3(x)$ and comparison with the situation in $\overline{\mathbb F_3}(x)$. Morphisms and rational maps defined over $k$. Two varieties $V$ and $W$ defined over $k$ are isomorphic over $k$ if and only if $k[V]\cong k[W]$. If $V$ and $W$ are isomorphic over $k$ then there is a bijection between the set of their $k$-rational points. Quick remark on projective algebraic sets defined over $k$. Definition of an algebraic function field of one variable over $k$. Equivalence of categories: absolutely irreducible smooth projective algebraic curves defined over $k$ with non-constant rational maps defined over $k$ and algebraic function fields of one variable over $k$ with $k$-homomorphisms.

• 11/19/18: Algebraic function fields of one variable over $K$

Review of the definition of an algebraic function field $F$ of one variable over $K$. The field of constants of $F$ and function fields with full constant field. For a function field over $k$: discrete valuation rings, places, degree of a place, valuation associated to a place and properties, zeros and poles of a function at a place. The case of the rational function field over $K$. Definitions: the divisor group $\operatorname{Div}(F)$ of $F$, the support of a divisor $D$, prime divisors, partial ordering on $\operatorname{Div}(F)$, effective divisors, the degree of a divisor, principal divisor $(x)$ associate to an element $x$ in $F$, the subgroup $\operatorname{Princ}(F)$, the divisor class group $\operatorname{Cl}(F)$, the Riemann-Roch space associate to a divisor.

Reference: STI - Sec. 1.1, 1.2, 1.4.

• 11/21/18: The Riemann-Roch space associate to a divisor

Quick review of the definitions for the divisor group associate to a function field $F$. The Riemann-Roch space $\mathcal L(A)$ associate to a divisor $A$. Prescribed zeros and allowed poles for a function in $\mathcal L(A)$. $\mathcal L(A)$ is a vector space (proof) of finite dimension (proof), equivalent divisors have same Riemann-Roch space (proof). $\mathcal L(A)$ when $A=0$ and $A<0$. Notation for the dimension of $\mathcal L(A)$.

Reference: STI - Sec. 1.4.

• Notes:
• Homework: