Number Theory Seminar: Counting primitive integral solutions to generalized Fermat equations

Let F ⁣:Axa+Byb+Czc=0 F \colon A x^a + B y^b + C z^c = 0 F:Axa+Byb+Czc=0 be a generalized Fermat equation with nonzero integer coefficients. A solution (x,y,z)∈Z3 (x, y, z) \in \mathbb{Z}^3(x,y,z)∈Z3 is called $\text{primitive}$ if $\gcd (x,y,z)=1$. We prove that when χ=1a+1b+1c−1>0 \chi = \tfrac{1}{a} + \tfrac{1}{b} + \tfrac{1}{c} – 1 > 0 χ=a1​+b1​+c1​−1>0, the counting function $N(F;h)$ of primitive integral solutions of height at most $h$ satisfies N(F;h)∼κ(F)⋅hχ, N(F; h) \sim \kappa(F) \cdot h^{\chi}, N(F;h)∼κ(F)⋅hχ, for some constant $\kappa (F)\ge 0$, as $h\to \infty$. This result refines a theorem of Beukers, and the proof relies on the stack-theoretic perspective introduced by Poonen–Schaefer–Stoll in their study of x2+y3+z7=0 x^2 + y^3 + z^7 = 0 x2+y3+z7=0.

During the pre-talk, I will introduce torsors and quotient stacks.