CS688 - Computational Arithmetic-Geometry & Applications (Sem I, 2013-14)

 

SYLLABUS


Fundamentals.

- Algebraic curves & function fields
- The Riemann-Roch theorem
- Zeta function of curves over finite fields
- The relevant Riemann hypothesis and its proof

Applications.

- Exponential sums
- Counting points on curves
- Integer factoring via curves
- Algebraic geometric codes
- Hyperelliptic curve cryptography
- Computing roots of unity in finite fields

 

Schedule.


[12-Nov-2013] [pdf] Computing the zeta function of hyperelliptic curves by Chinese remaindering. Action of the Frobenius on the l-torsion points of the Jacobian. 

[11-Nov-2013] [pdf] Computing the zeta function of curves. Hint of the cohomological interpretation of the zeta function.

[07-Nov-2013] [pdf] Bound the degree of the numerator of G. Optimally fix the parameters to finish the Riemann Hypothesis!

[05-Nov-2013] [pdf] Construct the rational function G, using the L(.) spaces and Riemann-Roch, whose zeros include the fixed points of σ-1F.

[31-Oct-2013] [pdf] Began the Bombieri-Stepanov proof of Weil's RH: For a Galois covering C of the projective line, N1(C,σ) <= q+1+(2g+1)q1/2. Motivated the rational function G that covers these points.

[29-Oct-2013] [pdf] The Galois covering C' of a curve C and its K-automorphisms. Their interaction with the Frobenius morphism. The corresponding relation between the number of points N1(C',σ) and N1(C).

[28-Oct-2013] [pdf] Introducing the Riemann Hypothesis: All the zeros of ζ(s,C) indeed lie on the Re(s) = 1/2 axis! It is equivalent to showing |Nn - (qn+1)| <= 2gqn/2 . Motivated the Galois covering of the curve. 

[24-Oct-2013] [pdf] Proving the base change. Writing Z(t) as L(t) / (1-t)(1-qt), where L is a degree 2g integral polynomial. Consequences to counting GF(qn)-points on the curve: Nn - (qn+1) = error-term (precisely given by the complex roots of L).

[22-Oct-2013] [pdf] The functional equation of the zeta function (using the Riemann-Roch theorem). The symmetry of ζ(s,C) around the Re(s)=1/2 axis! The poles of Z(t) and the base change theorem.

[21-Oct-2013] [pdf] The zeta function Z(t) of a smooth projective curve over Fq. It converges absolutely for |t| < q-1

[17-Oct-2013] [pdf] The canonical divisor W of K. The Riemann-Roch theorem. The Jacobian variety and its abelian group structure.

[15-Oct-2013] [pdf] The differentials of K, ΩK/k form a K-vector space of dimension one. Every differential w has a unique associated divisor (w).

[14-Oct-2013] [pdf] The k-dimension of A/(A(D)+K) is δ(D). Study its dual space; the differentials of K wrt D, ΩK/k(D).

[03-Oct-2013] [pdf] The adele ring A (``complete'' version of the fn.field K), the k-vector space A(D) (analog of L(D)). Study A(D')/A(D).

[01-Oct-2013] [pdf] The Riemann theorem. Genus intuition. The degree of speciality δ(D).

[30-Sep-2013] [pdf] The degree of principal divisors and the related (x)0 , (x) divisors. The group of divisor classes (resp. divisor classes of deg-0) Cl(C) (resp. Cl0(C)). The corresponding exact sequence.

[26-Sep-2013] [pdf] How big is the k-vector space L(D)? Defined its dimension as l(D). Study l(D)-d(D).

[24-Sep-2013] [pdf] Degree of a divisor, the group of deg-0 divisors Div0(C), the group of principal divisors Diva(C). The k-vector space L(D) of rational functions divisible by the divisor -D.

[23-Sep-2013] [pdf] Redefining varieties with maximal ideals as points. Motivating degree of a point. The group of divisors Div(C) of a smooth projective curve C.

[12-Sep-2013] [pdf] Approximation theorem for valuations at preassigned finitely many points.

[10-Sep-2013] [pdf] A smooth projective curve covers the projective line. Every point defines a valuation, residue field, and degree. Functions with preassigned zeros/poles.

[09-Sep-2013] [pdf] An abstract curve is isomorphic to a non-singular projective curve. Thus, it suffices to study the latter.

[05-Sep-2013] [pdf] The abstract curve and its morphisms.

[03-Sep-2013] [pdf] All the valuations on an affine curve.

[02-Sep-2013] [pdf] Uniformizer, integral closure. Equivalence of dvr, valuation and integral closure. All the valuations on the affine line.

[29-Aug-2013] [pdf] Resolving singularity, discrete valuation, DVR. Examples.

[27-Aug-2013] [pdf] Simple points, non-singular variety, tangent space, its dual, relationship with the germs at a point. Examples.

[26-Aug-2013] [pdf] Every variety is birational to a hypersurface. Primitive element theorem.

[22-Aug-2013]
[pdf] Defined birational isomorphism. Its criterion via function fields of the varieties.

[20-Aug-2013]
[pdf] The projective version of the previous lecture. Defined a more ``local morphism'', rational maps, dominant map, dense sets.

[19-Aug-2013] [pdf] Defined the distinguished open subsets Xf. Redefined OX(X) resp. OX(Xf) resp. OX,P . Showed that in the affine case these are simply A(X) resp. A(X)f resp. A(X)-localized at Mp.

[13-Aug-2013] [pdf] Defined the localization of a ring at a subset, or at a prime ideal. Use it to study O(Y) and Op(Y).

[12-Aug-2013] [pdf] Morphisms definition. Defined the ring of regular functions O(Y), function field K(Y), germs Op(Y) at a point p, and its unique maximal ideal Mp.

[8-Aug-2013] [pdf] 1-1 correspondences (between various homogeneous ideals and the corresponding closed subsets of the projective space). Open covering. Morphisms intuition. Defined regular functions.

[5-Aug-2013] [pdf] Proof of dimension vs. trdeg. The graded polynomial ring and homogeneous ideals. Definitions leading to projective varieties.

[1-Aug-2013] [pdf] Strong Hilbert Nullstellensatz. 1-1 correspondences (between various ideals and the corresponding closed subsets of the affine space). Dimension of a closed subset. Relation with the trdeg of the coordinate ring.

[29-Jul-2013] [pdf] Introduction. Definitions leading to affine varieties.