STUDY OUTLINE FOR ACADEMIC YEAR 2015-16

Source.  “Elliptic Tales (Curves, Counting & Number Theory)” by A. Ash and R. Gross

Authors’ Objective.  Providing details leading up to the limited scope of understanding the BSD conjecture within algebraic geometry.

(Ch00) The Prologue provides an historical progression of the subject, and the best map of the topic contents through the entire book.

Key terms:    elliptic curve                          polynomial equations

                        cubic equation                     group structure

                        diophantine equation         rational solution

(Ch01) Two disparate definitions of the degree (algebraic and geometric) of an algebraic curve, and its graph.

Key terms:    algebraic variety                   parameter

                        degree of an equation        intersection

                        root of a polynomial

(Ch02) Enlarging the set of numbers to include complex numbers, and a first attempt at resolving the definitions of degree.

Key terms:    complex numbers                ring                             algebraic closure

                        conjugate                              field                            congruence mod n

                        prime numbers

(Ch03) Progressing from the affine plane to the projective plane.

Key terms:    affine                                      Riemann sphere

                        projective                               projective coordinates [x:y:z]

                        point at infinity                      homothety

                        homogeneous polynomialcoordinate patch

(Ch04) Introduction of multiplicity to count the points of intersection more than once.

Key terms:    multiplicity                             smooth curve

                        perturbation                          singularity / singular points

                        derivative of f(x)                    tangent line

                        gradient

(Ch05) Bezout’s theorem for counting the intersections of two curves.

(Ch06) Using the previous development with a projective homogenized plane curve in order to count the intersection points multiplicities used to determine the number of solutions of E(Q).

Key terms:    singular                     number of solutions

                        non-singular

(Ch07) Consider the field over which E(F) is a finite generated Abelian group for non-singular solutions defined by the rank of the Abelian group.

Key terms:    Abelian group           group generators

                        subgroups                 rank

                        torsion                        finitely generated

(Ch08) Using the geometry to generate the Abelian group operation on the points of E(C): finding the identity and the inverse of the group.

Key terms:    elliptic curve              discriminant of the curve

                        inflection point

(Ch09) Counting the number of points on a singular cubic equation using the same group laws as for a non-singular cubic.

Key terms:    additive reduction    split multiplicative reduction

                        Isomorphism             non-split multiplicative reduction

(Ch10) Describing the torsion and rank of a finitely generated Abelian group to count the rational points of E(Q).

Key terms:    Mordell’s theorem

(Ch11) Taking a sequence of numbers to construct generating functions, using the concepts of algebra of elliptic curves and Abelian groups.

Key terms:    generating functions           Riemann zeta function

                        Dirichtlet series                    gamma function

                        Pole of order one                 Euler product

(Ch12) Extending the domain of the L-function to count points on an elliptic curve.

Key terms:    analytic function                  Taylor series of f(x)

                        analytic continuation          entire function

                        monodromy                           Laurent series

                        pole of order k

(Ch13) Using elliptic curves to create a generating function called a (computable) L-function.

Key terms:    Hasse-Weil zeta function

(Ch14) Analytic properties of the L-function continues analytically to the whole complex s-plane.

Key terms:    modular form

(Ch15) Relating the properties of the L-function to the properties of an elliptic curve used to construct the L-function.

Key terms:    algebraic rank of E(Q)         analytic rant of E(Q)

                        BSD conjecture