New phenomena arising from L-invariants of modular forms

This article explains how to practically compute L-invariants of p-new eigenforms using p-adic L-series and exceptional zero phenomena. As proof of the utility, we compiled a data set consisting of over 150,000 L-invariants. We analyze qualitative and quantitative features found in the data. This includes conjecturing a statistical law for the distribution of the valuations of L-invariants in a fixed level as the weights of eigenforms approach infinity. One novel point of our investigation is that the algorithm is sensitive to compiling data for fixed Galois representations modulo p. Therefore, we explain new perspectives on L-invariants that are related to Galois representations. We propose understanding the structures in our data through the lens of deformation rings and moduli stacks of Galois representations.

💡 Research Summary

The paper presents a concrete algorithm for computing the L‑invariants of p‑new eigenforms by exploiting the exceptional‑zero phenomenon of p‑adic L‑functions, and then uses a massive data set (over 150,000 L‑invariants) to formulate and support a new statistical law governing their p‑adic valuations. After a brief historical overview of Archimedean distribution results (Ramanujan–Petersson bounds, Sato–Tate, vertical Sato–Tate) the authors turn to the p‑adic side, recalling Gouvêa’s conjecture on the distribution of p‑adic slopes as the weight tends to infinity. They explain that Mazur–Tate–Teitelbaum’s exceptional‑zero formula identifies the L‑invariant L_f with the ratio –2 a′_p(k)/a_p(f), where a′_p(k) is the derivative of the p‑th Hecke eigenvalue along the Coleman–Mazur eigencurve.

The computational core builds on Lauder‑Von­k’s logarithmic‑time algorithms for Hecke operators, allowing the authors to evaluate a′_p(k) efficiently even for very large weights. The special shape of a_p(f) for forms of level N·p (with p∤N), namely a_p(f)=±p^{k/2−1}, simplifies the input data and makes the algorithm practical. The implementation details (precision handling, choice of p‑adic logarithm branch, data storage) are described, and the authors compare their method with earlier approaches by Gräf and by Anni‑Böckle‑Gräf‑Troja, emphasizing the substantial speed‑up and the ability to attach each L‑invariant to a specific eigenform.

With the algorithm in hand, the authors compute L‑invariants for a wide range of levels and weights, assembling a data set of more than 150,000 entries. They first present raw visualisations (scatter plots, histograms) showing that, for a fixed tame level N, the p‑adic valuation v_p(L_f) appears to become uniformly distributed on a fixed interval as the weight grows. To make this observation precise they introduce, for each mod p Galois representation ρ arising from eigenforms, the normalized statistic x_T(ρ)=((p+1)/k)·v_p(a_p(f)) where the average runs over all forms of weight ≤T with ρ_f≅ρ. Empirically, x_T(ρ) tends to the Lebesgue measure on