In collaboration with Han, Padurariu, and Park, we show that the number of $5$-isogenies of elliptic curves defined over $\mathbb{Q}$ with naive height bounded by $H > 0$ is asymptotic to $C_5\cdot H^{1/6} (\log H)^2$ for some explicitly computable constant $C_5 > 0$. This settles the asymptotic count of rational points on the genus zero modular curves $X_0(m)$. We leverage an explicit $\mathbb{Q}$-isomorphism between the stack $\mathscr{X}_0(5)$ and the generalized Fermat equation $x^2 + y^2 = z^4$ with $\mathbb{G}_m$ action of weights $(4, 4, 2)$.

Pretalk: I will explain how to count isomorphism classes of elliptic curves over the rationals. On the way, I will introduce some basic stacky notions: torsors, quotient stacks, weighted projective stacks, and canonical rings.

[pre-talk at 3:00PM]