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

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Contributed by Anwesh Ray ↗

Version 1 / Oct 12, 2026 / CC BY 4.0

Abstract

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 gg that realize every common nonnegative increase of its two signed λ\lambda-invariants. We count the levels realizing each increase. In the resulting family, the common increase is asymptotically Gaussian, with mean and variance δΠlog⁡log⁡X+Og(1)\delta_\Pi\log\log X+O_g(1) for a positive constant δΠ\delta_\Pi.

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Version history

  1. v1Submitted by Anwesh RayInitial depositCurrentOct 12, 2026