Title: Totality of Rational points on Moduli stacks - Counting Families of Varieties

Abstract: The study of fibrations of curves and abelian varieties over a smooth algebraic curve lies at the heart of the classification theory of algebraic surfaces and rational points on varieties. For the case of elliptic curves, it is natural to want to count elliptic curves over global fields such as the field Q of rational numbers or the field Fq(t) of rational functions over the finite field Fq.  To this end, we consider the fact that each E/K corresponds to a K-rational point on the fine moduli stack Mbar_{1,1} of stable elliptic curves, which in turn corresponds to a rational curve on Mbar_{1,1}.  In these lectures, I will explain the exact counting formula for all elliptic curves over Fq(t) along with an explanation for the geometric origin of lower order main terms, as well as basic context, relevant ideas and methods.

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