Modular Polynomials F_N(x,j): Theory & Computation

• Modular polynomials F_N(x,j) are algebraic relations connecting the modular parameter x and the j-invariant, encapsulating key geometric and arithmetic properties of modular curves and elliptic curves.
• They enable efficient computation of modular parametrizations, detailed analysis of fiber structures, and precise evaluation of poles, cusps, and CM points.
• Their construction via q-expansion coefficient matching provides actionable algorithms to link analytic modular forms with deep arithmetic investigations, including aspects of the Birch and Swinnerton-Dyer conjecture.

A modular polynomial $F_N(x, j)$ is a two-variable integer polynomial that algebraically relates two modular functions—typically the parameter $x$ on a modular curve $X_0(N)$ (often interpreted as the image of the coordinate function $x$ under a modular parametrization $\varphi: X_0(N) \rightarrow E$ for an elliptic curve $E$) and the classical $j$-invariant. Modular polynomials of this type encode the geometric and arithmetic relations between points on modular curves and elliptic curves, including isogenies, fiber structure, action on CM/heegner points, poles, cusps, and ramifications. They are essential for effective calculation of modular parametrizations, explicit evaluation of modular forms, and arithmetic investigations—especially those connected to the Birch and Swinnerton-Dyer conjecture (BSD) and rational points on $E$.

1. Algebraic Construction and Defining Properties

Let $E$ be a complex elliptic curve of conductor $N$ with modular invariant $j(E) \in \mathbb{Q}$, and let $\varphi: X_0(N) \to E$ be a holomorphic modular parametrization. The composition of the first coordinate function $x$ on $E$ with $\varphi$ defines a meromorphic function $x: X_0(N) \to \mathbb{C}\cup \{\infty\}$. Together with the pull-back of the $j$-invariant to $X_0(N)$, one obtains an algebraic relation: $$F_N(x, j) = \sum_{k=0}^{K} \sum_{l=0}^{L} c_{k,l}\, x^k\, j^l = 0$$ where the coefficients $c_{k,l}$ are integers determined by the $q$-expansion data of $x$ and $j$; $K$ and $L$ are degree bounds reflecting the geometry of $X_0(N)$ and the degree of $\varphi$ (Wang, 18 Sep 2025). Typically, $K$ is bounded by the index $\mu = [\mathrm{SL}_2(\mathbb{Z}): \Gamma_0(N)]$ and $L$ by $2d$, with $d=\deg(\varphi)$.

The construction is algorithmic:

• Compute enough terms in the $q$-expansions of $x(\tau)$ and $j(\tau)$.
• Set up the linear system for the $c_{k,l}$ from the expansion

$$\sum_{n=0}^M \left( \sum_{k, l} c_{k,l}\, c(k,l;n) \right) q^n = 0$$

and solve for the coefficients.

The resulting polynomial relation encapsulates the entire fiber structure of $\varphi$ and the algebraic dependency between $x$ and $j$, serving as the bridge between analytic modular forms and the arithmetic of $E$.

2. Applications: Evaluation, Fibers, Poles, Cusps, and Ramification

Once $F_N(x,j)$ is constructed, it yields algorithms for a spectrum of arithmetic tasks:

• Poles of $\varphi$: Write $F_N(x, j)$ as $A_K(j)x^K + \dots$; points with $x(\tau)\to \infty$ (i.e., poles) must satisfy $A_K(j(\tau))=0$. Zeros of $A_K$ correspond to non-cuspidal poles. At a pole, the leading $x^K$ term dominates.
• Cusps: Express $F_N(x,j)$ as $B_L(x) j^L + \dots$; at cusps where $j(\tau)\to\infty$, the leading $j^L$ term gives $B_L(x(\tau))=0$, allowing exact evaluation of $\varphi$ at cusps.
• Fibers: For $P=(\alpha,\beta) \in E(\mathbb{C})$, solve $F_N(\alpha, j)=0$ for $j$ to obtain preimages on $X_0(N)$. To recover the corresponding $\tau$ in $\mathbb{H}$, use auxiliary relations, e.g., $f_N(x, J)$, linking $x$ and $J(\tau)=j(N\tau)$.
• Ramification: Points with nontrivial stabilizer under $\varphi$ have coalescing $j$-values (multiple roots). Thus $F_N(x(τ_0), j)=0$ together with $\partial_j F_N(x(τ_0), j)=0$ signals ramification. Resultant computations and fiber evaluations solidify detection.
• Minimal Polynomial of CM/Heegner Points: Given a CM point $[\tau]$ on $X_0(N)$, $F_N(x, j(\tau))=0$ produces the minimal polynomial for $x(\tau)$ and encodes all Galois conjugates.

3. Heegner Points, Galois Structure, and Explicit Trace Formulas

Modular polynomials $F_N(x,j)$ encode the Galois structure of Heegner and CM points:

• For a CM-point $[\tau]$ on $X_0(N)$, both $j(\tau)$ and $j(N\tau)$ are algebraic. The image $\varphi(\tau)$ in $E$ is then an algebraic point whose minimal polynomial is computable via factors of $F_N(x, j(\tau))$.
• The paper provides a total trace formula for Heegner points. For example, on $X_0(389)$, one computes the semi-trace $P=\sum_{\sigma\in H} \varphi(\tau)^{\sigma}$ in $E(L)$, where $H$ is a subgroup of the Galois group of the splitting field $L=\mathbb{Q}(\sqrt{-3}, j(389\tau))$ (Wang, 18 Sep 2025). This links modular parametrization fibers, Galois theory, and complex multiplication.
• The method applies the Artin isomorphism, relating ideal classes to conjugates of $j(\tau)$, so $F_N(x,j)$ bridges analytic and Galois-theoretic perspectives.

4. Explicit Connection to the BSD Conjecture

Via the modular polynomial $F_N(x,j)$ and modular parametrization $\varphi$, one can construct sequences of algebraic numbers linked to rational points of infinite order in $E(\mathbb{Q})$:

• For $P\in E(\mathbb{Q})$ of infinite order, the paper shows that there exists an infinite sequence $\{(j(\tau_n), j(N\tau_n))\}$, each of bounded degree (by $\deg\varphi$), associated to $P$.
• These sequences provide regulator data, potentially giving new angles for computational approaches to the BSD conjecture for elliptic curves of rank $\geq 2$ (Wang, 18 Sep 2025).

5. Examples and Case Studies

Numerical and symbolic examples demonstrate practical implementation:

• For small levels $N$ (e.g., $N=11$), $F_{11}(x,j)$ and $f_{11}(x, J)$ can be written down explicitly, simplifying calculations compared to classical modular equations $\Phi_N(X,Y)$.
• For $N=389$, the data file for $F_{389}(x,j)$ is about 20 MB, whereas the classical $\Phi_{389}$ equation would be nearly 900 MB. This efficiency enables computations of fibers, ramification, and minimal polynomials for CM images, even at high levels.

6. Computational Aspects and Algorithmic Efficiency

The construction of $F_N(x,j)$ proceeds by matching $q$-expansion coefficients, solving for $c_{k,l}$ as the coefficients of sufficiently many terms are known a priori to belong to $\mathbb{Z}$ (Wang, 18 Sep 2025). Bounded degrees in $x$ and $j$ permit real-time computation of fibers, poles, and minimal polynomials. The method adapts to the calculation of rational functions expressing $\varphi$ as $x = P_1(j, J)/Q_1(j, J)$, generalizing approaches from Kolyvagin and enabling compact storage.

In high-level arithmetic tasks (fibers, traces, ramification), one leverages the algebraic properties of $F_N(x, j)$, systematic coefficient matching, and resultant computation, outperforming the direct use of classical modular equations at elevated levels.

7. Theoretical Significance and Future Directions

Modular polynomials $F_N(x, j)$ unify analytic, algebraic, and arithmetic aspects of modular forms, modular curves, and elliptic curves. They serve as "total formulas" interconnecting fiber structure, ramification, Galois actions, Heegner/CM points, and rank data for $E$. The approach supports computational and theoretical investigations into rational point structure and regulator data—thus providing actionable methodology for questions such as BSD.

This framework, as illustrated by detailed examples and explicit algorithms in (Wang, 18 Sep 2025), demonstrates that $F_N(x, j)$ is central to modern arithmetic geometry, particularly modular parametrizations, explicit calculations of special points, and the effective arithmetic of elliptic curves over $\mathbb{Q}$.