Number of $K$-rational points with given $j$-invariant on modular curves

This paper presents methods to compute the number of $K$-rational points with a specific $j$-invariant on arbitrary modular curves, applying these techniques to classify the possible counts of cyclic $n$-isogenies and points on Cartan modular curves for elliptic curves over number fields, while also providing an algorithm to determine the number of rational CM points.

The Gist

Imagine you have a giant, magical library called the Modular Curve Library. Inside this library, every book represents a specific type of elliptic curve (a special kind of shape used in advanced math and cryptography).

Each book has a unique ID number called a $j$-invariant. Think of this like a barcode or a fingerprint. If two books have the same barcode, they are essentially the same shape, just maybe rotated or stretched slightly.

Now, imagine you are a librarian (a mathematician) who wants to answer a very specific question:

"If I pick a specific barcode (a specific $j$-invariant), how many different versions of this book can I find on the shelves that are written in a specific language (a specific number field $K$)?"

This paper, written by Ivan Novak, is essentially a cataloging guide for this library. It tells us exactly how many "rational" (language-specific) versions of a book exist for any given barcode, without needing to know exactly which library or which language we are looking at. It's a universal rulebook.

Here is a breakdown of the paper's main ideas using simple analogies:

1. The Core Problem: Counting "Isogenies"

In the world of elliptic curves, there are special connections between shapes called isogenies. You can think of an isogeny as a "bridge" or a "tunnel" connecting two elliptic curves.

• The Question: If I have a specific curve (with a specific $j$-invariant), how many bridges of a certain size (degree $n$) can I build starting from it that stay within my specific language (number field)?
• The Old Way: For simple cases (like prime numbers), mathematicians knew the rules. If you have a bridge of size 3, you usually have 0, 1, 2, or 4 bridges. It was like a simple game with fixed rules.
• The New Discovery: Novak found that when the bridge size is a composite number (like 4, 6, or 9), the rules get much more complicated. You can have a weird number of bridges, like 5 or 7, depending on how the "Galois group" (the secret code of the language) interacts with the shape.

2. The "Galois Group" as a Security Guard

To understand why the number of bridges varies, imagine the Galois Group as a security guard at the entrance of the library.

• The guard checks every potential bridge.
• If the bridge looks "rational" (fits the language rules), the guard lets it through.
• If the bridge looks "weird" (doesn't fit the language), the guard blocks it.
• Novak's work calculates exactly how many bridges get through the guard for any possible security guard configuration. He didn't just look at one guard; he looked at every possible guard and listed every possible outcome.

3. The Main Results: The "Menu" of Possibilities

The paper provides a "menu" of possible answers.

• For Prime Powers ($p^k$): If you are looking for bridges of size $p^k$ (where $p$ is a prime number), the number of bridges you can find is restricted to a very specific list.
  • Analogy: Imagine you are ordering ice cream. You might think you can get any number of scoops, but the shop owner (the math) says, "You can only get 0, 1, 2, 4, 8, or 16 scoops." Novak figured out the exact list of allowed scoop numbers for every flavor.
• For Composite Numbers: If the bridge size is a mix of primes (like 12), the answer is just the product of the answers for the prime parts. It's like building a Lego tower: if you know how many ways you can build a red block and a blue block, you know how many ways you can build a red-and-blue tower.

4. The "Cartan" Curves: Special VIP Sections

The paper also looks at special sections of the library called Cartan Modular Curves.

• Analogy: Imagine the main library has a VIP lounge. The rules for entering this lounge are different. The "bridges" here behave differently.
• Novak calculated the possible number of points in these VIP lounges for specific types of numbers. This is important because these VIP lounges often appear when studying Complex Multiplication (CM) curves—curves with extra symmetry, like a snowflake.

5. The Algorithm: A Recipe for CM Curves

The most practical part of the paper is an algorithm (a step-by-step recipe).

• The Problem: Finding "CM points" (points on curves with extra symmetry) is notoriously hard. It's like trying to find a needle in a haystack where the haystack keeps changing shape.
• The Solution: Novak combined his counting rules with known facts about CM curves to create a simple computer program.
• How it works: You give the computer a specific barcode ($j$-invariant) and a level ($N$). The program looks up the "security guard" rules for that specific barcode and instantly tells you how many points exist.
• Why it matters: Before this, finding these points required heavy, slow calculations. Now, it's a quick lookup. The author even made the code available online for others to use.

Summary: Why Should You Care?

This paper is like a universal translator for a very complex mathematical language.

1. It solves a puzzle: It tells us exactly how many "bridges" (isogenies) exist between shapes for any given size.
2. It removes guesswork: Instead of checking case-by-case, we now have a complete list of all possible answers.
3. It speeds up research: The new algorithm allows mathematicians to instantly count special points on these curves, which helps in cryptography and understanding the deep structure of numbers.

In short, Novak took a chaotic, confusing library of infinite possibilities and organized it into a neat, predictable catalog, showing us exactly what is possible and what is impossible in the world of elliptic curves.

Technical Summary

1. Problem Statement

The paper addresses the problem of determining the possible cardinalities of the set of $K$-rational points on a modular curve $X_H$ that share a specific $j$-invariant $j_0 \in K$.

• Context: Let $H$ be a subgroup of $GL_2(\mathbb{Z}/N\mathbb{Z})$. The modular curve $X_H$ parametrizes elliptic curves equipped with an $H$-level structure.
• Goal: For a fixed number field $K$ and a fixed $j$-invariant $j_0$, determine the set of possible values for:
  $$\{ x \in X_H(K) \mid j(x) = j_0 \}$$
  The author explicitly states that the work is "purely group-theoretic," meaning it does not attempt to classify the specific number fields or elliptic curves that achieve these counts, but rather determines the list of all possible counts achievable by some field and curve.
• Specific Applications:
  1. Determining the possible number of cyclic $n$-isogenies an elliptic curve over a number field can admit.
  2. Calculating possible point counts on Cartan modular curves ($X_s(p^k)$, $X_{ns}(p^k)$) and their normalizers.
  3. Providing an algorithm to count rational Complex Multiplication (CM) points on arbitrary modular curves.

2. Methodology

The core methodology relies on the interplay between the moduli interpretation of modular curves and the image of Galois representations.

A. Group-Theoretic Formulation

The number of $K$-rational points on $X_H$ with a fixed $j$-invariant $j_0$ is reduced to a counting problem within the group $GL_2(\mathbb{Z}/N\mathbb{Z})$.

• Let $E/K$ be an elliptic curve with $j(E) = j_0$.
• Let $R = \rho_{E,N}(G_K)$ be the image of the mod-$N$ Galois representation in a chosen basis.
• Let $A$ be the image of the automorphism group $\text{Aut}(E)$ in the same basis.
• A point defined by a basis $(P, Q)$ is $K$-rational if and only if $R \subseteq H \cdot A$ (up to conjugation and double cosets).
• Key Formula (Proposition 2.2): The number of rational points is given by:
  $$\sum_{g \in GL_2(\mathbb{Z}/N\mathbb{Z})} \frac{\mathbb{1}_{\{gRg^{-1} \subseteq HgA\}}}{\#(HgA)}$$
  This reduces the geometric problem to counting the number of double cosets $HgA$ such that the conjugate of the Galois image $R$ is contained within the double coset.

B. Analysis of Subgroups

The author analyzes the set $S(H)$, defined as the set of all possible values of the expression above as $R$ ranges over all subgroups of $GL_2(\mathbb{Z}/N\mathbb{Z})$.

• The paper focuses on specific subgroups $H$:
  • $B_0(p^k)$: Upper triangular matrices (corresponding to $X_0(p^k)$).
  • $C_s(p^k)$: Split Cartan subgroups.
  • $C_{ns}(p^k)$: Non-split Cartan subgroups.
  • $N_s(p^k)$ and $N_{ns}(p^k)$: Normalizers of the Cartan subgroups.
• The proofs involve detailed $p$-adic valuation analysis of matrix entries to solve systems of equations derived from the condition $gRg^{-1} \subseteq H$.

C. Algorithm for CM Points

For CM elliptic curves, the Galois image is known to be contained in a specific Cartan subgroup (or its normalizer) based on results by Lozano-Robledo. The author combines this with the counting formulas to create an algorithm that inputs a $j$-invariant and a level $N$ and outputs the number of rational CM points.

3. Key Contributions and Results

A. Cyclic $p^k$-Isogenies (Section 3)

The paper generalizes the classical result for prime $p$ (where the number of isogenies is $0, 1, 2,$ or $p+1$) to prime powers $p^k$.

• Proposition 1.1 (Odd Primes): For an odd prime $p$ and integer $k$, the number of $p^k$-isogenies defined over $K$ belongs to:
  $$\{0\} \cup \{p^j \mid 0 \le j < k\} \cup \{2p^j \mid 0 \le j < k\} \cup \{p^k + p^{k-1}\}$$
• Proposition 1.2 (Power of 2): For $2^k$, the possible numbers are:
  $$\{0\} \cup \{2^m \mid 1 \le m \le k\} \cup \{2^k + 2^{k-1}\}$$
• General $n$: Using the Chinese Remainder Theorem, the possible number of $n$-isogenies is the product of the possible counts for each prime power factor of $n$.
• Refined Results (Proposition 3.8): The paper determines the possible number of $p^{k+1}$-isogenies given a fixed number of $p^k$-isogenies, establishing a dependency structure between levels.
• Geometric Insight (Proposition 3.10): A matrix in $PGL_2(\mathbb{Z}/p^k\mathbb{Z})$ that fixes three points on $\mathbb{P}^1(\mathbb{Z}/p^k\mathbb{Z})$ whose reductions mod $p$ are distinct, must fix all points. This generalizes the classical "3 points fix the line" property to higher powers, though with specific conditions on the reductions.

B. Cartan Modular Curves (Section 4)

The author determines the possible number of points on curves associated with Cartan subgroups and their normalizers.

• Split Cartan ($X_s(p^k)$): Possible counts are $\{0, p^{2k} + p^{2k-1}\} \cup \{2p^{2j} \mid 0 \le j < k\}$.
• Normalizer of Split Cartan ($X_{s}^+(p^k)$): The set of possible counts is a union of several sets involving powers of $p$ and small constants (2 or 3) depending on whether $-1$ is a quadratic residue mod $p$.
• Non-Split Cartan ($X_{ns}(p^k)$): Possible counts are $\{0\} \cup \{2p^{2j}\} \cup \{p^{2k} - p^{2k-1}\}$.
• Normalizer of Non-Split Cartan ($X_{ns}^+(p^k)$): The set of possible values is complex, involving unions of sets $A', B', C', \dots$ which depend on $p \pmod 4$ and $p \pmod 8$.

C. Algorithm for Rational CM Points (Section 5)

• Algorithm: The paper provides a concrete algorithm to compute the number of rational CM points on any modular curve $X_H$.
  1. Identify the quadratic order $\mathcal{O}_{K,f}$ associated with the $j$-invariant.
  2. Construct the specific Cartan subgroup $C_{\delta, \phi}(N)$ and its normalizer $N_{\delta, \phi}(N)$ as described by Lozano-Robledo.
  3. Determine the automorphism group image $A$ (size 2, 4, or 6).
  4. Compute the number of double cosets satisfying the inclusion condition.
• Implementation: The author notes a GitHub implementation of this algorithm, which is significantly faster than previous methods (e.g., computing preimages of the $j$-map) for specific cases like $X_0(88)$.

4. Significance

• Completeness of Classification: The paper provides a complete classification of the possible numbers of isogenies and rational points for a wide class of modular curves, moving beyond specific examples to general structural results.
• Resolution of "3 Points" Intuition: It clarifies the behavior of isogenies for composite levels (specifically prime powers), showing that the "3 points fix all" intuition from $p$ does not hold generally for $p^k$, but a refined version (Proposition 3.10) does hold under specific reduction conditions.
• Computational Efficiency: The proposed algorithm for counting CM points offers a practical, efficient alternative to existing methods, leveraging group-theoretic properties of Galois images rather than heavy geometric computations.
• Foundation for Further Research: The results on Cartan normalizers are directly relevant to Serre's Uniformity Conjecture, as these groups appear as images of Galois representations for CM curves and are the primary obstructions to surjectivity for large primes.

In summary, Novak's work bridges the gap between the abstract group theory of $GL_2(\mathbb{Z}/N\mathbb{Z})$ and the arithmetic geometry of modular curves, providing precise enumerative results and efficient computational tools for studying rational points and isogenies.