New phenomena arising from L-invariants of modular forms

TL;DR

This work develops a practical framework for computing ${\mathcal{L}}$-invariants of $p$-new eigenforms through exceptional zeros of $p$-adic $L$-functions, producing a dataset of over $10^5$ invariants and revealing striking patterns. It connects these invariants to local and global Galois representations via deformation rings and the Emerton–Gee stack, using a locally analytic approach on eigencurves and a ghost-series model to understand slopes. The paper proposes a distribution conjecture for ${\mathcal{L}}$-invariants (analogous to Gouvêa’s slope conjecture) and provides compelling numerical and heuristic evidence, supported by recent proofs in special cases. It also raises questions about horizontal data masses, integrality phenomena, and the precise deformation-theoretic mechanisms underpinning the observed regularities, highlighting the deep interplay between $p$-adic analysis, modular forms, and Galois representations with implications for $p$-adic Langlands program. The results offer a concrete path to organizing large-scale arithmetic data within deformation-space geometries and may inform future investigations into constant-slope phenomena and $p$-adic Birch–Swinnerton-Dyer-type questions.

Abstract

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.

Figures (24)

• Figure 1: $2$-adic slopes in level $\mathop{\mathrm{SL}}\nolimits_2({\mathbb Z})$ for $k\leq 512$. (Total data: 5334 eigenforms.)
• Figure 2: Left: Normalized $2$-adic slopes in level $\mathop{\mathrm{SL}}\nolimits_2({\mathbb Z})$ for $k\leq 512$. Right: Histogram of normalized slopes. (Total data: 5334 eigenforms. Num. bins: 19.)
• Figure 3: $2$-adic valuations of ${\mathcal{L}}$-invariants in level $\Gamma_0(2)$ with respect to weights $10 \leq k \leq 804$. (Total data: 13465 eigenforms.)
• Figure 4: Left: Normalized $\frac{6}{k}v_2({\mathcal{L}}_f)$ in level $\Gamma_0(2)$ for $10\leq k\leq 804$. Right: Histogram of normalized slopes. (Total data: 13465 eigenforms. Num. bins = 31.)
• Figure 5: $3$-adic slopes of ${\mathcal{L}}$-invariant slopes in level $\Gamma_0(3)$ with respect to weights $6 \leq k \leq 508$. (Total data: 10752 eigenforms.)

Theorems & Definitions (34)

• Conjecture 1.1
• Theorem 1.2: LTXZ-Proof
• Conjecture 1.3
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Remark 2.3
• Proposition 2.4
• proof