2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves I

1. Problem Statement

The paper addresses the fundamental problem of determining the set of rational points $X(\mathbb{Q})$ on hyperelliptic curves $X$ of genus $g > 1$. While Baker's method (for integral points) and the Chabauty–Coleman method (for rational points) are effective in specific cases, they face significant limitations:

• Baker's Method: Relies on height bounds which are often computationally intractable.
• Chabauty–Coleman Method: Requires the Mordell–Weil rank $r$ of the Jacobian $J$ to satisfy $r < g$.
• Quadratic Chabauty: Extends Chabauty to cases where $r = g$, but fails when $r > g$ or when the endomorphism algebra is small (e.g., $\mathbb{Z}$).

The specific motivation for this work is the curve $X: y^2 - y = x^5 - x$ (the "BMSST curve"), which has genus 2 and a Jacobian of Mordell–Weil rank 3 with endomorphism algebra $\mathbb{Z}$. Standard methods cannot determine $X(\mathbb{Q})$ for this curve. The paper aims to develop a method to compute $X(\mathbb{Q})$ for curves where the rank exceeds the genus, specifically by bounding the dimension of Bloch–Kato Selmer groups associated with the second étale cohomology of the Jacobian.

2. Methodology

The author introduces a "nonabelian 2-descent" framework that generalizes classical 2-descent on the Jacobian. The methodology proceeds through several key stages:

A. Galois Cohomology of the Wedge Square

Instead of studying $H^1(K, J[2])$ (which approximates the Mordell–Weil group), the paper studies $H^1(K, \wedge^2 J[2])$.

• Theoretical Basis: For a hyperelliptic curve with a rational Weierstrass point, the paper establishes an explicit isomorphism between $H^1(K, \wedge^2 J[2])$ and the kernel of a norm map between units of specific étale algebras:
  $$H^1(K, \wedge^2 J[2]) \simeq \text{Ker}\left( (K_f^{(2)})^\times \otimes \mathbb{F}_2 \xrightarrow{Nm} K_f^\times \otimes \mathbb{F}_2 \right)$$
  where $K_f = K[x]/(f(x))$ and $K_f^{(2)}$ is a specific quadratic extension of $K_f$ related to pairs of roots of $f$.

B. The Nonabelian $(x-T)$ Map

The core innovation is a "nonabelian generalization" of the classical $(x-T)$ map used in 2-descent.

• Construction: Using the étale fundamental group $\pi_1^{\text{ét}}$, the author constructs a quotient $\Gamma$ which is a central extension of $J[4]$ by $\wedge^2 J[2]$.
• Explicit Formula: For points $b, z \in X(K)$, the paper derives an explicit formula for the cohomology class $\kappa_{\Gamma, b}(z) \in H^1(K, \Gamma)$. If $[z] - [b] \in 4J(K)$, this class lifts to an element in $H^1(K, \wedge^2 J[2])$.
• The Formula: The lift is given explicitly in terms of the coordinates of $z$ and $b$ and the roots of $f$:
  $$\text{Lift} \propto 2 \left( 1 + \frac{c_\alpha u_\alpha - c_\beta u_\beta}{\beta - \alpha} \right)$$
  where $c_\alpha = x(b) - \alpha$ and $u_\alpha$ is a square root related to the difference in $x$-coordinates.

C. Local Conditions and Crystalline Lifts

To bound the global Selmer group, the paper analyzes local conditions at primes $v$ (specifically $v=2$ and $v \neq 2$).

• Away from 2: The paper uses the Picard–Lefschetz formula and monodromy operators to show that for curves with specific reduction types, the local Bloch–Kato subspace $H^1_f$ coincides with the unramified subspace $H^1_{ur}$.
• At 2: This is the most technical part. The author uses mixed extensions and the theory of integral crystalline lifts. By analyzing the structure of the unipotent fundamental group and using the nonabelian $(x-T)$ map, the paper provides criteria to determine when a cohomology class in $H^1(\mathbb{Q}_2, \wedge^2 J[2])$ admits a crystalline lift. This allows for the explicit computation of the dimension of the local Bloch–Kato Selmer group.

D. The Chabauty–Coleman–Kim Method

The paper integrates these descent results into the Chabauty–Coleman–Kim framework.

• Finiteness Criterion: The set of rational points is finite if the dimension of the global Bloch–Kato Selmer group is strictly less than the dimension of the local Selmer group (adjusted for the rank of the Jacobian).
• Generalized Heights: The paper constructs a "generalized height" function $h_2: X(\mathbb{Q}_2) \to W$ which is quadratic on rational points. By computing the image of known rational points under this map, one can define a subspace $W_0$. The set $X(\mathbb{Q}_2)_2$ is then the preimage of $W_0$, effectively isolating the rational points.

3. Key Contributions

1. Explicit 2-Descent for $\wedge^2 J[2]$: The paper provides the first explicit field-theoretic description and computational algorithm for the cohomology group $H^1(K, \wedge^2 J[2])$ for hyperelliptic curves, moving beyond the standard $J[2]$ descent.
2. Nonabelian $(x-T)$ Map: It introduces a concrete formula for lifting Kummer classes from $J[4]$ to $\wedge^2 J[2]$ using the geometry of the curve and its fundamental group.
3. Local Analysis at $p=2$: It develops a systematic method to compute the crystalline subspace of the Bloch–Kato Selmer group at the prime 2, a notoriously difficult case for $p$-adic methods.
4. Algorithm for High-Rank Curves: It demonstrates a practical algorithm to compute $X(\mathbb{Q})$ for curves where the Mordell–Weil rank $r \ge g$, a regime previously inaccessible to Chabauty-type methods.

4. Results

The paper applies these methods to the BMSST curve $X: y^2 - y = x^5 - x$:

• Rank Analysis: The Jacobian has rank 3 and genus 2. Standard Chabauty fails ($3 \not< 2$).
• Selmer Dimension: The author proves that $\dim H^1_f(\mathbb{Q}, \wedge^2 J[2]) = 3$.
• Finiteness: Using the dimension bounds derived from the local analysis at 2, the paper verifies that the dimension of the relevant Selmer scheme is small enough to ensure the finiteness of $X(\mathbb{Q}_2)_2$.
• Rational Points: By combining the finiteness result with the Mordell–Weil sieve (using pre-existing data from Bugeaud et al.) and the generalized height function, the paper determines the complete set of rational points.
  $$X(\mathbb{Q}) = \{ \infty, (0,0), (0,1), (1,0), (1,1), (2,-5), (2,6), (3,-15), (3,16), (-1,0), (-1,1), (30, -4929), (30, 4930), (1/4, 15/32), (1/4, 17/32), (-15/16, -185/1024), (-15/16, 1209/1024) \}$$
  This answers an open question posed by Bugeaud, Mignotte, Siksek, Stoll, and Tengely.

5. Significance

• Breaking the Rank Barrier: This work represents a major step forward in the arithmetic of curves, providing a viable path to determine rational points on hyperelliptic curves with Mordell–Weil rank greater than the genus.
• Nonabelian Arithmetic Geometry: It successfully translates deep nonabelian cohomological concepts (fundamental groups, mixed extensions) into explicit, computable algebraic formulas.
• Foundational for Future Work: The techniques developed here (specifically the handling of $p=2$ and the explicit formulas for $\wedge^2$) lay the groundwork for the sequel [Dog23], which applies these methods to thousands of curves, and for extending the theory to higher genus and even-degree hyperelliptic curves (as noted in the work of Berry).
• Resolution of Open Problems: It provides a definitive solution to a specific, long-standing problem in Diophantine geometry, demonstrating the power of the Bloch–Kato conjecture framework in practical computation.