Zoom ID: 919 6497 4060.

For password please contact Pieter Moree (moree@mpim...).

We will first consider the explicit formulation of the moduli

stacks of fibered algebraic surfaces as the moduli stacks of rational

curves on $\overline{\mathcal{M}}_{g,n}$.

This will lead us to exact arithmetic invariants on those moduli stacks

via motives in the Grothendieck ring of stacks introduced by the late

Torsten Ekedahl that will, in turn, render their point counts over

finite fields. We then enumerate elliptic & hyperelliptic curves over

$\mathbb{P}_{\mathbb{F}_q}^{1}$ with precise lower order terms ordered

by bounded discriminant height.

Along the way, we will glance at 2 important analogies in number theory

& geometry that are

1. Global fields analogy,

2. Rational points & Rational curves.

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