Work

On the parity of coefficients of eta powers

Year: 2025

Type: article

Abstract: We study a new notion of mod-p twisted density for a modular form f supported on an arithmetic progression: the proportion of primes $$\ell $$ ℓ for which the mod-p order of infinity of $$U_\ell f$$ U ℓ f is minimal. We show that for classical modular forms, twisted density is always defined and rational, connecting it with an earlier notion of density studied by Bellaïche. Finally, we specialize to $$p = 2$$ p = 2 and f a positive power of the Dedekind eta function, studying densities via Galois-theoretic techniques developed by Bellaïche in level 1 and extending them to level 9. In particular, we explicitly calculate twisted densities for certain eta powers corresponding to CM/dihedral mod-2 modular forms in the sense of Nicolas and Serre. En passant we take the opportunity to communicate proofs of two of Bellaïche’s unpublished results on densities of mod-2 modular forms.

Source: Research in the Mathematical Sciences

DOI: 10.1007/s40687-025-00507-9

Authors: Steven Charlton, Lukas Mauth, Anna Medvedovsky

Institutions: Max Planck Institute for Mathematics, University of Cologne, University of Arizona

Language: English

Topic: Advanced Algebra and Geometry

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,090

OpenAlex ID: W4413236310