Ternary polynomial relations and periods of cusp forms

Source abstract

Bachmann and van Ittersum conjectured that every polynomial $f\in\mathbb{Q}[x,y,z]$ satisfying $$f(x,y,z)=f(y,z,x),\qquad f(x,y,z)+f(-y,x-y,z)+f(y-x,-x,z)=0$$ is a sum of polynomials in two variables. We prove this conjecture over every field of characteristic zero and determine all homogeneous solutions. Moreover, restriction to $z=0$ identifies the degree $n$ solution space with the space of binary period polynomials of degree $n$ normalized to vanish on both coordinate axes. The solution corresponding to such a polynomial $q$ is $$f(x,y,z)=q(x,y)+q(y,z)+q(z,x).$$ There are no nonzero homogeneous solutions of odd degree. For even degrees $n$, the symmetric and alternating solution spaces are each isomorphic over $\mathbb{C}$ to the space of weight $n+2$ cusp forms on $SL_2(\mathbb{Z})$. The proof connects the polynomial relations with the cohomology of arithmetic groups through the rational Tits building for $GL_3$. Cusp forms enter both the description of the solutions and the proof that there are no others. The main theorems and the essential argument are formalized unconditionally in Lean~4 by AxiomProver.