Proof of a conjecture by Ahlgren and Ono on the non-existence of certain partition congruences (Q2847126)

Let \(p(n)\) denote the number of partitions of \(n\) and let \(\ell \geq 5\) be prime. \textit{S. Ahlgren} and \textit{K. Ono} [Contemp. Math. 291, 1--10 (2001; Zbl 1009.11059)] conjectured that if the Ramanujan-type congruence NEWLINE\[NEWLINE p(An+B) \equiv 0 \pmod{\ell} \; \; \; \; \forall n\in \mathbb{N} NEWLINE\]NEWLINE holds for some natural numbers \(A\) and \(B\) with \(A >B\), then \(\ell \mid A\) and NEWLINE\[NEWLINE \left(\frac{24B-1}{\ell}\right) \neq \left( \frac{-1}{\ell} \right). NEWLINE\]NEWLINE In this paper, the author gives a proof of this conjecture. A key role is played by a result of \textit{P. Deligne} and \textit{M. Rapoport} [Lect. Notes Math. 349, 143--316 (1973; Zbl 0281.14010)] which says that certain divisibility properties of a modular form \(f\) of (positive integral) weight \(k\) are inherited by \(f |_k \gamma\).