[Paper Review] Drinfeld modular curves have many points

Lenny Taelman|ArXiv.org|Feb 8, 2006

Algebraic Geometry and Number Theory5 references3 citations

TL;DR

This paper proves that reductions of Drinfeld modular curves modulo prime ideals achieve asymptotically optimal numbers of rational points over finite fields, extending Ihara's result for classical modular curves. By adapting Ihara's trace argument to the Drinfeld modular setting, it establishes that the genus-g curve $X_0(\mathfrak{n})$ reduced modulo $\mathfrak{p}$ has at least $(q^m - 1)(g - 1)$ rational points over $\mathbb{F}_{q^{2m}}$, where $q^m$ is the size of the residue field of $\mathfrak{p}$, thus constructing explicit towers of curves with many points.

ABSTRACT

Ihara's proof that the reduction of the modular curve $X_0(n)$ at a prime $p$ not dividing $n$ has many points over a quadratic extension is adapted to the drinfeld modular curves $X_0(n)$. In order to do so, some properties of drinfeld modular varieties that have no proof in the literature are proven. The text is more or less self-contained.

Motivation & Objective

• To establish that reductions of Drinfeld modular curves attain the asymptotically optimal number of rational points over finite fields, analogous to Ihara's result for classical modular curves.
• To extend Ihara's trace argument—previously applied to modular curves—to the context of Drinfeld modular curves, which lack a comprehensive foundational theory in the literature.
• To construct explicit towers of curves of increasing genus over finite fields with asymptotically maximal point counts, providing new examples of curves with many points.
• To develop the necessary theory of Drinfeld modular curves, including level structures, moduli functors, and compactifications, to support the main result.

Proposed method

• Adapts Ihara's trace method to Drinfeld modular curves by constructing two maps from $X_0(\mathfrak{n}\mathfrak{p})$ to $X_0(\mathfrak{n})$ via isogenies and Frobenius twists.
• Uses the action of the Frobenius endomorphism $\tau^m$ on the reduction modulo $\mathfrak{p}$ to define a correspondence $\Pi \cup \Pi^t$ on the product surface $X_0(\mathfrak{n}) \times X_0(\mathfrak{n})$ over $\Bbbk(\mathfrak{p})$.
• Identifies the set of $\mathbb{F}_{q^{2m}}$-rational points with the intersection $\Pi \cap \Pi^t$, which corresponds to fixed points under $\tau^m \circ \tau^m$.
• Applies the Hurwitz formula and Euler-Poincaré characteristic computation to relate the genus of the covering curve $\tilde{T}$ to the number of special points in $\Pi \cap \Pi^t$, which are in bijection with rational points.
• Establishes that the number of rational points is bounded below by $(q^m - 1)(g - 1)$, where $g$ is the genus of the curve and $q^m = |\Bbbk(\mathfrak{p})|$, using the genus formula $g_0 - 1 = 2(g - 1) + \#\{\text{special points}\}$.
• Demonstrates that the moduli space $X_0(\mathfrak{n})$ parametrizes Drinfeld $\mathbf{A}$-modules of rank 2 with a $\Gamma_0(\mathfrak{n})$-level structure, and that its reduction modulo $\mathfrak{p}$ is smooth and geometrically irreducible.

Experimental results

Research questions

• RQ1 Can Ihara's method for bounding rational points on modular curves be extended to Drinfeld modular curves?
• RQ2 Do reductions of Drinfeld modular curves modulo prime ideals achieve the asymptotic bound of $(\sqrt{q} - 1 + o(1))g$ for the number of rational points over finite fields?
• RQ3 What is the precise number of $\mathbb{F}_{q^{2m}}$-rational points on the reduction of $X_0(\mathfrak{n})$ modulo a prime $\mathfrak{p}$ of residue field size $q^m$?
• RQ4 How do the geometric and arithmetic properties of Drinfeld modular curves—such as level structures and isogenies—support the construction of curves with many points?
• RQ5 What role do special points in the correspondence $\Pi \cap \Pi^t$ play in counting rational points via the Euler-Poincaré characteristic?

Key findings

• The reduction of $X_0(\mathfrak{n})$ modulo a prime $\mathfrak{p}$ has at least $(q^m - 1)(g - 1)$ rational points over $\mathbb{F}_{q^{2m}}$, where $q^m = |\Bbbk(\mathfrak{p})|$ and $g$ is the genus of the curve.
• The number of rational points is bounded below by $(q^m - 1)(g - 1)$, which matches the asymptotic bound predicted by the Drinfeld–Vladut bound when $q^m$ is a square.
• The key geometric object is the correspondence $\Pi \cup \Pi^t$ on $X_0(\mathfrak{n}) \times X_0(\mathfrak{n})$, whose fixed points under $\tau^m \circ \tau^m$ correspond to $\mathbb{F}_{q^{2m}}$-rational points.
• The genus of the covering curve $\tilde{T}$ satisfies $g_0 - 1 = 2(g - 1) + \#\{\text{special points}\}$, and this equality is used to derive the lower bound on rational points.
• The proof relies on the fact that the two projections from $T$ to $X_0(\mathfrak{n})$ have degree $q^m + 1$, which is essential for the Hurwitz formula application.
• The construction yields a tower of curves of increasing genus over $\mathbb{F}_{q^{2m}}$ with asymptotically maximal point counts, providing new explicit examples of optimal curves.