Non-ordinary primes: A story. (English) Zbl 0887.11020

This paper deals with normalized eigenforms of level 1 with Fourier expansions of the form \[ f=q+ \sum^\infty_{n=2} a_nq^n \] with integer coefficients \(a_n\), where \(q= e^{2\pi iz}\) and \(\text{Im} (z)>0\). If a prime \(p\) divides the \(p\)th coefficient \(a_p\), the form \(f\) is called ordinary at \(p\). (The author also refers to \(p\) as an ordinary prime of \(f\).) In earlier work, D. Niebur [Ill. J. Math. 19, 448-449 (1975; Zbl 0301.10025)] treated the classic discriminant \(\Delta= \sum^\infty_{n=1} \tau(n) q^n\), where \(\tau(n)\) is Ramanujan’s tau function, by using an arithmetical identity expressing \(\tau(n)\) in terms of \(\sigma (n)\), the sum of the divisors of \(n\). Niebur found that the only non-ordinary primes for \(\Delta\) between 2 and 65064 are 2, 3, 5, 7 and 2411.
The present author considers the unique normalized eigenforms \(\Delta_k\) of weight \(k\) where \(k\in \{12, 16, 18, 20, 22, 24\}\), \((\Delta_{12} = \Delta)\) and obtains generalizations of Niebur’s formula that are amenable to computation, and then uses these to determine all the non-ordinary primes \(p\leq 10^6\) for each \(\Delta_k\).


11F03 Modular and automorphic functions
11F33 Congruences for modular and \(p\)-adic modular forms


Zbl 0301.10025
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