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$p$ -adic Equidistribution of Special Loci in a Product of Modular Curves

This paper investigates the $p$ -adic equidistribution of intersection loci between a smooth curve in a product of modular curves and modular subvarieties $Y_0(n)$ , demonstrating that these loci converge to a canonical point under specific conditions involving supersingular reduction or increasing $p$ -divisibility, while also proving that the accumulation points of such intersections always form a non-discrete set.

Original authors: Dan Townsend

Published 2026-08-25

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CC BY 4.0

Original authors: Dan Townsend

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of number theory, mathematicians often study shapes that are not made of wood or stone, but of numbers. These shapes, called modular curves, act as maps that organize the behavior of a special class of equations known as elliptic curves. Imagine these elliptic curves as points on a vast, invisible grid. When you look at these points through the lens of a specific type of arithmetic called p-adic analysis, the grid reveals a hidden structure where points can cluster, spread out, or settle into specific patterns. For decades, mathematicians have wondered how these special points distribute themselves across the grid. Do they scatter randomly, or do they follow a strict, predictable law? This question is central to understanding the deep connections between geometry and arithmetic, a field that underpins much of modern cryptography and the theory of numbers.

A recent study by Dan Townsend tackles this question by examining what happens when a smooth, curved path cuts through a grid of these modular points. Specifically, the researcher looked at a single, continuous curve drawn across a space defined by pairs of elliptic curves. The goal was to see where this path intersects with a family of special sub-structures known as modular subvarieties. These subvarieties represent specific relationships between the pairs of curves, such as one curve being a particular kind of transformation of the other. The study asks a simple but profound question: as we look at more and more of these special relationships, do the intersection points spread out evenly across the curve, or do they pile up in certain spots?

The answer depends entirely on the nature of the curve and the specific sequence of relationships being examined. The research reveals that if the curve avoids a particular type of "bad" behavior at a specific prime number, or if the sequence of relationships becomes increasingly divisible by that prime, the intersection points do indeed spread out perfectly evenly. They distribute themselves until they fill the space uniformly, converging on a single, central point that represents the average behavior of the entire system. This result holds true even when the curve is complex and defined over a strange, non-standard number system used in advanced arithmetic.

However, the story changes if the curve passes through regions where the behavior is more chaotic. The study shows that if the curve does not avoid these difficult regions, and if the sequence of relationships does not grow in a specific way, the points do not spread out evenly. Instead, they can cluster in unexpected ways. The author provides a family of examples where this failure to distribute evenly is not just a possibility, but the expected outcome. In these cases, the points refuse to settle into a uniform pattern, defying the neat order seen in the other scenarios.

Beyond the question of distribution, the paper also investigates where these points gather over time. It turns out that for any such curve, the set of intersection points is never truly scattered or isolated; there is always a place where the points accumulate. The study identifies exactly where these clusters form. If the curve passes through a point where the underlying mathematical objects are "ordinary" in a technical sense, the points will gather there if and only if the curve touches a specific type of smooth, repeating path known as a torus. If the curve crosses this path at an angle, the points will not gather. If the curve runs parallel to it, touching it gently, the points will accumulate. This precise characterization allows mathematicians to predict exactly where the density of these special points will be highest.

The research also connects these findings to earlier work by other mathematicians who studied similar patterns. While previous studies showed that these special points are sparse in a broad sense, this new work expands the view to show how they behave in the long run. It confirms that while the points might be rare in any single snapshot, their long-term behavior is governed by strict geometric rules. The study uses powerful tools from a branch of mathematics that treats numbers as if they were distances on a map, allowing the researcher to visualize the curve and the points in a way that makes the patterns clear. By translating the problem into this geometric language, the author proves that the distribution is either perfectly uniform or fails in a very specific, predictable manner.

Ultimately, the paper provides a complete picture of how these special points behave on a curve. It proves that under the right conditions, the points distribute themselves with perfect fairness, filling the space until no area is more crowded than another. When those conditions are not met, the points behave differently, clustering in ways that reveal the underlying geometry of the curve. This work does not just solve a single puzzle; it offers a new way to understand how arithmetic and geometry interact, showing that even in the most abstract realms of mathematics, there are laws that dictate how things spread out and where they gather. The findings suggest that the universe of these numbers is not chaotic, but follows a rigid, elegant structure that can be mapped and understood, provided one knows where to look.

Technical Summary: p-adic Equidistribution of Special Loci in a Product of Modular Curves

Problem Statement
The paper investigates the asymptotic distribution of intersection points between a smooth, irreducible curve $C \subset X(1) \times X(1)$ defined over $\mathbb{C}_p$ and the family of modular subvarieties $Y_0(n)$ . The space $X(1) \times X(1)$ represents the product of two $j$ -lines, serving as the moduli space for pairs of elliptic curves. The study focuses on the behavior of the probability measures $\bar{\delta}_n(C)$ , which are supported on the intersection $C \cap Y_0(n)$ (counting multiplicity), as $n \to \infty$ .

The central question is whether these measures converge weakly to the unique canonical point $x_{can}$ in the Berkovich analytification $C^{Berk}$ , a phenomenon known as equidistribution. The paper distinguishes between two regimes where equidistribution is proven:

The Ordinary Case: Where the curve $C$ avoids points where both coordinates have supersingular reduction.
The Supersingular Case: Where the sequence of indices $n$ (or a subsequence $a_n$ ) exhibits increasing divisibility by $p$ ( $v_p(a_n) \to \infty$ ).

The paper also addresses the structure of the set of accumulation points of the union of these intersections, $L = C \cap \bigcup_n Y_0(n)$ , specifically determining when this set is non-discrete. It notes that if neither of the above conditions is satisfied (i.e., $C$ passes through a point where both coordinates are supersingular and the sequence has bounded $p$ -divisibility), equidistribution is expected to fail, and the author provides a family of examples exhibiting this behavior.

Methodology
The proof strategy relies on translating the problem of weak convergence of measures on the Berkovich space into a problem of bounding the number of intersection points within specific $p$ -adic balls. This reduction is formalized via Lemma 1.4, which states that equidistribution to $x_{can}$ occurs if and only if the proportion of intersection points falling into any ball of radius $r < 1$ (or specific balls at infinity) tends to zero relative to the total degree of the intersection.

The methodology is bifurcated based on the reduction type of the elliptic curves involved:

Ordinary Locus (Serre-Tate Coordinates):
- The analysis utilizes Serre-Tate coordinates to linearize the local geometry of the modular curves $Y_0(n)$ . In these coordinates, the locus $Y_0(n)$ is described as a union of translates of formal tori by $p$ -power roots of unity.
- The intersection of $C$ with these tori translates is reduced to finding zeros of $p$ -adic power series.
- Strassman's Theorem is the primary tool used to bound the number of zeros of these power series, providing uniform bounds on the number of intersections within a fixed ball.
- To characterize accumulation points, the paper employs the density of "modular tori" (tori arising from isogenies) in the space of all possible tori (Lemma 4.3), allowing for the construction of sequences of intersection points converging to specific limits.
Supersingular Locus (Period Maps and Hecke Orbits):
- For supersingular reductions, the paper utilizes the period map defined on deformation discs. This map collapses $p$ -power Hecke orbits.
- By analyzing the properties of Hecke orbits (specifically Theorem 6.1 regarding the independence of the period image for $Y_0(n_0 p^m)$ ), the author demonstrates that for sufficiently large $m$ , the modular curves $Y_0(n_0 p^m)$ eventually miss any fixed ball of radius $R < 1$ in the supersingular locus.
- This leads to a stronger result than mere equidistribution: the intersection becomes empty for large enough $p$ -divisibility.
Infinity (Tate Coordinates):
- The behavior near the boundary (infinity) is analyzed using Tate coordinates derived from $p$ -adic uniformization.
- The modular curves $Y_0(n)$ are shown to be unions of branches of the form $q_2^a = q_1^b$ .
- Similar to the ordinary case, the number of intersections with a fixed curve $C$ is bounded using Strassman's Theorem applied to power series in Tate coordinates, showing the count grows slower than the total degree.

Key Contributions and Results

Theorem 1.1 (Equidistribution):
- Case (i): If $C$ avoids points with supersingular reduction in both coordinates, and $a_n$ is a sequence with $v_p(a_n) \to \infty$ , then $\bar{\delta}_{a_n}(C) \to \delta_{x_{can}}$ .
- Case (ii): If $C$ avoids points where both coordinates are supersingular, then $\bar{\delta}_n(C) \to \delta_{x_{can}}$ for the full sequence $n$ .
- The paper explicitly states that if neither condition is satisfied (i.e., $C$ passes through a point with both coordinates supersingular and the sequence has bounded $p$ -divisibility), equidistribution is expected to fail. The author provides a family of examples where the limiting measure is non-zero on specific balls, contrasting with the complex case where equidistribution holds for all sequences with discriminant tending to $-\infty$ .
Theorem 1.2 (Characterization of Accumulation Points):
The paper provides a precise characterization of which points $P \in C$ are accumulation points of the locus $L = C \cap \bigcup_n Y_0(n)$ , assuming $P$ corresponds to elliptic curves with good ordinary reduction:
- Neither CM: $P$ is an accumulation point if and only if $P$ lies on a torus (in Serre-Tate coordinates).
- Exactly One CM: $P$ is never an accumulation point.
- Both CM: $P$ is an accumulation point if and only if $C$ is tangent at $P$ to a torus passing through $P$ .
- Corollary 4.10: For any curve $C$ , the set of intersection points with the special locus is never discrete; there is always at least one accumulation point.
Relation to Previous Work:
The results are connected to the work of Maulik and Poonen ([10]) on jumping loci. The paper shows that the set of points where coordinates are CM or isogenous is $p$ -adically nowhere dense in $C$ (Proposition 5.2), recovering a specific case of their result using the equidistribution techniques developed here.

Significance and Claims
The paper provides a comprehensive analysis of the equidistribution of special loci in products of modular curves over $\mathbb{C}_p$ , extending previous 1-dimensional results by Herrero, Menares, and Rivera-Letelier.

Contrast with Complex Geometry: The paper highlights a fundamental difference between the $p$ -adic and complex settings. While all sequences of CM points with discriminant tending to $-\infty$ equidistribute over $\mathbb{C}$ , the $p$ -adic setting exhibits a dichotomy: equidistribution holds only under specific conditions regarding supersingular reduction and $p$ -divisibility. In cases where these conditions are not met, the author expects failure and demonstrates it with examples.
Non-Discreteness: A significant finding is that the set of special points on any curve $C$ is never discrete. This is established by showing that accumulation points exist on any curve, a result derived from the density of modular tori and the properties of Serre-Tate coordinates.
Methodological Contribution: The work demonstrates the efficacy of combining Berkovich geometry (to define the limit measure), Serre-Tate coordinates (to linearize the problem in the ordinary case), and period maps (to control the supersingular case) to study arithmetic intersection problems.

The paper does not claim to resolve the equidistribution problem for curves passing through supersingular points with bounded $p$ -divisibility; rather, it identifies this as a regime where equidistribution fails and provides examples of such behavior. The results are presented as a complete characterization of the equidistribution phenomenon under the stated hypotheses.

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