Arithmetic geometry of the moduli stack of Weierstrass fibrations over $${\mathbb {P}}^1$$

Abstract Coarse moduli spaces of Weierstrass fibrations over a smooth conic curve 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 $${\mathcal {W}}_{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> of Weierstrass fibrations over a smooth conic curve with discriminant degree 12 n and a section. We show that it is a smooth algebraic stack and prove that for $$n\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , the open substack $${\mathcal {W}}_{\textrm{min},n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mtext>min</mml:mtext> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> of minimal Weierstrass fibrations is a separated Deligne–Mumford stack over any base field K with $$\textrm{char}(K) \ne 2,3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>char</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>K</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≠</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> and $$\not \mid n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> . Arithmetically, for the moduli stack $${\mathcal {W}}_{\textrm{sf},n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mtext>sf</mml:mtext> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> of stable Weierstrass fibrations, we determine its motive in the Grothendieck ring of stacks to be $$\{{\mathcal {W}}_{\textrm{sf},n}\} = {\mathbb {L}}^{10n - 2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mtext>sf</mml:mtext> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>10</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> in the case that n is odd, which results in its weighted point count to be $$\#_q({\mathcal {W}}_{\textrm{sf},n}) = q^{10n - 2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mo>#</mml:mo> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mrow> <mml:mtext>sf</mml:mtext> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow> <mml:mn>10</mml:mn> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> over $${\mathbb {F}}_q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:math> . 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.