Ternary polynomial relations and periods of cusp forms
Abstract
Bachmann and van Ittersum conjectured that every polynomial f∈ℚ[x,y,z] satisfying f(x,y,z)=f(y,z,x), 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 ℂ to the space of weight n+2 cusp forms on SL2(ℤ). The proof connects the polynomial relations with the cohomology of arithmetic groups through the rational Tits building for GL3. 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.
Related
- An explicit refined Gan--Gross--Prasad identity for Fourier--Jacobi periods of degree 2 Siegel cusp forms
- Spectral fourth moments of Hecke--Maaß cusp forms
- Noncongruence cusp forms, Veech groups and enumeration
- When GL(2) cusp forms meet CM points in GSp(4)
- On Decomposition of Drinfeld cusp forms of level t
BibTeX
@misc{ilin2026ternary,
title = {{Ternary polynomial relations and periods of cusp forms}},
author = {Vasily Ilin and Ken Ono},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19309},
eprint = {2610.19309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19309},
note = {CC0},
}
RIS
TY - UNPB
TI - Ternary polynomial relations and periods of cusp forms
AU - Vasily Ilin
AU - Ken Ono
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19309
UR - https://arxiv.org/abs/2610.19309
AB - Bachmann and van Ittersum conjectured that every polynomial f∈ℚ[x,y,z] satisfying f(x,y,z)=f(y,z,x), 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 ℂ to the space of weight n+2 cusp forms on SL_2(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.
ER -