Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Arithmetic geometry of the moduli stack of Weierstrass fibrations over
  $\mathbb{P}^1$   Coarse moduli spaces of Weierstrass fibrations over the (unparameterized)
projective line were constructed by the classical work of [Miranda] using
Geometric Invariant Theory. In our paper, we extend this treatment by using
results of [Romagny] regarding group actions on stacks to give an explicit
construction of the moduli stack $\mathcal{W}_n$ of Weierstrass fibrations over
an unparameterized $\mathbb{P}^{1}$ with discriminant degree $12n$ and a
section. We show that it is a smooth algebraic stack and prove that for $n \geq
2$, the open substack $\mathcal{W}_{\mathrm{min},n}$ of minimal Weierstrass
fibrations is a separated Deligne-Mumford stack over any base field $K$ with
$\mathrm{char}(K) \neq 2,3$ and not dividing $n$. Arithmetically, for the
moduli stack $\mathcal{W}_{\mathrm{sf},n}$ of stable Weierstrass fibrations, we
determine its motive in the Grothendieck ring of stacks to be
$\{\mathcal{W}_{\mathrm{sf},n}\} = \mathbb{L}^{10n - 2}$ in the case that $n$
is odd, which results in its weighted point count to be
$\#_q(\mathcal{W}_{\mathrm{sf},n}) = q^{10n - 2}$ over $\mathbb{F}_q$. In the
appendix, we show how our methods can be applied similarly to the classical
work of [Silverman] on coarse moduli spaces of self-maps of the projective
line, allowing us to construct the natural moduli stack and to compute its
motive.

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The categories of this paper are: q-alg
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: funct-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ex