On the $v$-adic values of G-functions II

1. Problem Statement

The paper addresses the problem of establishing effective height bounds for points of "special interest" (specifically, singular moduli corresponding to CM elliptic curves) on a 1-parameter family of elliptic curves.

The ultimate goal is to contribute to an effective version of Siegel's Theorem (1935), which provides a lower bound for the class numbers of imaginary quadratic fields. While Siegel's original result is ineffective (the constant depends non-constructively on the discriminant), the "G-functions method," developed by Yves André, aims to produce effective bounds by relating the height of a point to the degree of algebraic relations satisfied by the values of associated G-functions at that point.

The specific gap this paper fills concerns the construction of these algebraic relations (polynomials) for places $v$ where the "central fiber" (a fixed CM elliptic curve $E_0$) has ordinary reduction or where the prime $p$ is ramified in the CM field. Previous work by André and Beukers had only addressed supersingular places (unramified) and archimedean places.

2. Methodology

The author employs the G-functions method within the framework of $p$-adic Hodge theory and the arithmetic of CM elliptic curves.

• Geometric Setup: The study focuses on a family of elliptic curves $f: \mathcal{E} \to S$ over a number field $K$, with a distinguished point $s_0 \in S(K)$ such that the fiber $E_0 = \mathcal{E}_{s_0}$ is a CM elliptic curve. A local parameter $x$ is chosen at $s_0$.
• G-functions: A family of G-functions $Y_G(x)$ is associated with the pair $(\mathcal{E} \to S, s_0)$. These functions arise as solutions to the differential equations governing the de Rham cohomology of the family.
• Comparison Isomorphisms: The core technical tool involves comparing different cohomology theories:
  • Archimedean: de Rham vs. Betti (Grothendieck).
  • Non-Archimedean (Ordinary): de Rham vs. Crystalline (Berthelot-Ogus).
  • Non-Archimedean (Supersingular): de Rham vs. Crystalline, utilizing the action of the endomorphism algebra.
• Canonical Bases: The author constructs "canonical" symplectic bases for the crystalline cohomology $H^1_{crys}$ of the reduction $\tilde{E}_0$ at a place $v$.
  • For ordinary reduction, the basis is formed by eigenvectors of the Frobenius endomorphism.
  • For supersingular reduction, the basis is constructed using the action of the CM field within the quaternion algebra of endomorphisms.
• Isogeny Argument: For points $s$ close to $s_0$ (in the $v$-adic topology) where the fiber $E_s$ is also CM, the author utilizes the existence of an isogeny $\phi: E_s \to E_0$. The functoriality of the comparison isomorphisms allows the translation of the action of $\phi$ on cohomology into algebraic relations among the values of the G-functions.

3. Key Contributions and Results

A. Construction of New Algebraic Relations (Theorem 1.1)

The main result is the construction of non-trivial polynomials $R_{s,v}$ such that $\iota_v(R_{s,v}(Y_G(s))) = 0$ for points $s$ $v$-adically close to $s_0$. The paper extends previous results to two new cases:

1. Ordinary Reduction (The "New" Feature):

   • Result: If $v$ is a place of ordinary reduction for $E_0$, the author constructs a polynomial $R_{s,ord}$ that is independent of the place $v$.
   • Significance: This is a striking structural property. Previously, relations were known to depend on the specific place. Here, a single polynomial works for all ordinary places where $s$ is close to $s_0$. This implies the coefficients of the relation are independent of $v$.
   • Mechanism: The proof relies on the fact that for ordinary reduction, the Frobenius eigenvectors align with the CM eigenvectors, forcing the off-diagonal term of the period matrix to vanish ($Y_{1,2} = 0$).

2. Ramified Primes:

   • Result: The author constructs relations for places $v$ lying over primes $p$ that ramify in the CM field of $E_0$.
   • Mechanism: Even though the standard crystalline comparison isomorphism is more complex at ramified primes, the existence of an isogeny between $E_s$ and $E_0$ (guaranteed by proximity and CM theory) allows the construction of a relation based on the trace of the crystalline endomorphism.

B. Height Bounds (Theorem 6.4)

The paper derives a height bound for singular moduli $j$ (where $E_j$ has CM by a field different from $E_0$):
$$h(j) \leq C_1 ([Q(j):Q] + |P(j)|)^{C_2}$$
where:

• $h(j)$ is the logarithmic Weil height.
• $[Q(j):Q]$ is the degree of the field of definition.
• $|P(j)|$ is the number of supersingular places of proximity (places where $j$ is close to $j_0$ and the reduction is supersingular).

C. Implications for Effective Brauer-Siegel

The paper analyzes the path toward an effective Siegel bound.

• The current bound depends on $|P(j)|$, the number of supersingular places of proximity.
• To achieve a fully effective Siegel bound (where the height is bounded purely by the degree), one must show that $|P(j)|$ is bounded by a polynomial in $[Q(j):Q]$ and the discriminant $D(j)$.
• The author notes that without a bound on $|P(j)|$, the method hits an impasse, as the degree of the constructed polynomials grows with the number of such places.

4. Significance

1. Completing the Picture: The paper completes the classification of $v$-adic relations for G-functions associated with CM elliptic curves, covering the previously missing cases of ordinary reduction and ramified primes.
2. Structural Insight: The discovery that relations at ordinary places are independent of the place $v$ is a profound structural insight into the G-functions method. It suggests a deeper uniformity in the arithmetic of these functions than previously recognized.
3. Progress on Effective Siegel: While the paper does not fully solve the effective Brauer-Siegel problem, it isolates the precise bottleneck: the control of the set $P(j)$ (supersingular places of proximity). It shifts the problem from "constructing relations" to "bounding the number of supersingular places," which is a more tractable arithmetic problem.
4. Methodological Rigor: The paper provides a rigorous, case-by-case proof utilizing modern tools like Berthelot-Ogus isomorphisms and Lauter-Viray generalizations of the Gross-Zagier formula, serving as a comprehensive reference for the arithmetic of G-functions in the context of elliptic curves.

In summary, Papas advances the G-functions method by removing technical restrictions on the reduction type of the central fiber, revealing a new independence phenomenon for ordinary places, and clarifying the specific arithmetic obstacles remaining to achieve an effective version of Siegel's theorem on class numbers.