Title: Inner arithmetic of rational points on moduli stacks

Abstract: The study of elliptic fibrations f : X → C of an elliptic surface X over a curve C (i.e. family of elliptic curves over a function field) lies at the heart of surface classification.

John Tate (1925-2019) gave us wealth of essential mathematical ideas and constructions in algebraic number theory and arithmetic geometry, such as ' Tate's algorithm ' to determine the structure of elliptic singular fibers.

In this talk, we will unpack the classical Tate's algorithm by explicitly

1. working with monodromy factorizations in the mapping class group SL(2,Z) of torus that corresponds to elliptic Lefschetz fibrations;

2. working with global section homogeneous polynomials that corresponds to k(t)-rational point on the fine moduli stack Mbar_{1,1} of stable elliptic curves; we will supplement conceptual understanding by Online MAGMA utilizing the following code:

KK<t> := FunctionField(GF(5));

E := EllipticCurve([0, t^5]);

&*BadPlaces(E);

LocalInformation(E);

MordellWeilGroup(E);

KK<t> := FunctionField(GF(4007));

E := EllipticCurve([-27, 54 - 2^6*3^6*t^1]);

&*BadPlaces(E);

LocalInformation(E);

and see the geometric meaning of representable classifying morphisms.