math.NT — Number Theory

An analogue of the Erdos-Kac theorem for signed Iwasawa invariants of modular forms

Contributed by Anwesh Ray

Erdos and Kac proved that the number of distinct prime factors of an integer has a Gaussian limiting distribution. We prove an analogue for the signed Iwasawa invariants of non-ordinary weight-two modular forms. Under the hypotheses of this paper, we construct infinitely many forms congruent to a given form $g$ that realize every common nonnegative increase of its two signed $\lambda$-invariants. We count the levels realizing each increase. In the resulting family, the common increase is asymptotically Gaussian, with mean and variance $\delta_\Pi\log\log X+O_g(1)$ for a positive constant $\delta_\Pi$.