Totality of Rational Points on Moduli Stacks - Counting Families of Varieties
1. Organizational remarks, Motivations; Projective line P^1_{k}, Riemann sphere CP^1, 2-sphere S^2 / Rational function field Fq(t), Rational number field Q, Gauss-Bonnet formula, Grothendieck-Lefschetz trace formula
2. Point counts over finite fields, Étale cohomology with Weil conjecture, Practice of {Algebraization} via Grothendieck motivic classes
3. Rational points on P^1 over Fq(t) as coprime polynomial functions with coefficients in the finite field Fq
4. Bridge & Tension between Integers and Polynomials, Global Fields Analogy
5. Buffer class 1 - Discussion of Batyrev-Manin conjecture on Fano varieties
6. Basics on Invariant theory, Morse theory in AG via Stratifications and Algebraic stacks via Quotient stacks
7. Elliptic curves, Modular curves and Universal family, Automorphisms over non-algebraically closed field k
8. Elliptic fibrations via Weierstrass models as weighted linear series, Hodge line bundle, Discriminants, Reductions of elliptic curves
9. Failure of Valuative Criterion of Properness for Proper stacks, Tate's algorithm for elliptic curves, Stacky and Faltings heights
10. Hom stacks for Morphisms ( Integral points ) and Rat stacks for Rational maps ( Rational points )
11. Geometry of rational points on modular curves via twisted maps ( Morphisms with Characters ), Resolution of singularities, Minimal models
12. Buffer class 2 - Discussion on Shafarevich conjecture for families of abelian varieties or algebraic curves over a global field K
13. Arithmetic of Algebraic stacks over Fq - Weighted and Unweighted point counts, and Motives of Inertia stacks as Topology over residue field k
14. Height moduli spaces W^{min}_{n} of minimal weighted linear series, Vanishing conditions, Motives of strata with specified additive reduction
15. Ambient weighted projective stacks of all weighted linear series, Minimality condition, Rationality of motivic height zeta function
16. Motivic stabilization, PGL_2 stack quotients, Poincaré duality of open moduli loci
17. Enumerating all minimal elliptic curves over a rational function field, the origin and nature of lower order main terms
18. Buffer class 3 - Discussion on Mordell conjecture with Summary of Faltings’s proof
19. Rank of E(K) - Discussion on Boundedness conjecture with Summary of Ulmer's proof, Goldfeld and Katz-Sarnak Distribution conjecture