[Paper Review] Two-dimensional families of hyperelliptic jacobians with big monodromy

Yuri G. Zarhin|arXiv (Cornell University)|Oct 24, 2013

Algebraic Geometry and Number Theory24 references4 citations

TL;DR

This paper constructs explicit two-dimensional families of hyperelliptic jacobians over global fields of characteristic ≠ 2 with big monodromy and no nontrivial endomorphisms. By choosing distinct parameters $ t_1, t_2 \in K $ such that the polynomial $ f(x) = (x - t_1)(x - t_2)u(x) $ has Galois group $ \mathbf{S}_{2g} $ or $ \mathbf{A}_{2g} $, the jacobian $ J(C_f) $ achieves maximal Galois image on $ \ell $-adic Tate modules and trivial endomorphism ring, ensuring big monodromy and simplicity.

ABSTRACT

Let $K$ be a global field of characteristic different from 2 and $u(x)\in K[x]$ be an irreducible polynomial of even degree $2g\ge 6$, whose Galois group over $K$ is either the full symmetric group $S_{2g}$ or the alternating group $A_{2g}$. We describe explicitly how to choose (infinitely many) pairs of distinct elements $t_1, t_2$ of $K$ such that the $g$-dimensional jacobian of a hyperelliptic curve $y^2=(x-t_1)(x-t_2))u(x)$ has no nontrivial endomorphisms over an algebraic closure of $K$ and has big $\ell$-adic monodromy.

Motivation & Objective

• To construct explicit two-dimensional families of hyperelliptic jacobians over global fields with big monodromy and no nontrivial endomorphisms.
• To identify conditions on parameters $ t_1, t_2 \in K $ such that the jacobian of $ y^2 = (x - t_1)(x - t_2)u(x) $ has trivial endomorphism ring and maximal Galois image on $ \ell $-adic Tate modules.
• To prove that such jacobians are not isogenous to each other and satisfy the Tate and Hodge conjectures in all codimensions.
• To establish that the automorphism group of the jacobian is exactly $ \{ \pm 1 \} $, ruling out nontrivial automorphisms beyond the hyperelliptic involution.

Proposed method

• Constructing hyperelliptic curves $ C_f: y^2 = f(x) $ with $ f(x) = (x - t_1)(x - t_2)u(x) $, where $ u(x) $ is irreducible of degree $ 2g \geq 6 $ with Galois group $ \mathbf{S}_{2g} $ or $ \mathbf{A}_{2g} $.
• Using Galois-theoretic arguments to show that any $ \bar{K} $-isomorphism between jacobians must arise from a fractional linear transformation in $ \mathrm{PGL}_2(\bar{K}) $ preserving the branch points.
• Applying Hilbert’s Theorem 90 to show that if such a transformation is not defined over $ K $, it leads to a nontrivial automorphism of the jacobian, contradicting the triviality of $ \mathrm{End}(J(C_f)) $.
• Proving that the only automorphism of the jacobian is $ \pm 1 $, which implies that no nontrivial automorphism exists unless the transformation preserves the root set setwise and acts trivially on it.
• Using the Torelli theorem to reduce isomorphism of jacobians to isomorphism of the underlying hyperelliptic curves via $ \mathrm{PGL}_2 $-action on branch points.
• Analyzing the Galois action on $ \ell $-torsion points and showing that the image of the Galois representation is open in the symplectic similitude group, implying big monodromy.

Experimental results

Research questions

• RQ1 Under what conditions on $ t_1, t_2 \in K $ is the jacobian of $ y^2 = (x - t_1)(x - t_2)u(x) $ simple with trivial endomorphism ring?
• RQ2 When does the Galois representation on the $ \ell $-adic Tate module of such a jacobian have big monodromy?
• RQ3 Can distinct choices of $ t_1, t_2 $ yield non-isomorphic jacobians with trivial endomorphism rings and big monodromy?
• RQ4 What constraints does the triviality of $ \mathrm{End}(J(C_f)) $ impose on the existence of $ \bar{K} $-isomorphisms between such jacobians?
• RQ5 To what extent do the Tate and Hodge conjectures hold for self-products of these jacobians?

Key findings

• For infinitely many pairs $ (t_1, t_2) \in K \times K $ with $ t_1 \neq t_2 $, the jacobian $ J(C_f) $ of $ y^2 = (x - t_1)(x - t_2)u(x) $ has trivial endomorphism ring $ \mathrm{End}(J(C_f)) = \mathbb{Z} $.
• The Galois representation on the $ \ell $-adic Tate module of $ J(C_f) $ has image that is open in the symplectic similitude group, implying big monodromy.
• No nontrivial $ \bar{K} $-automorphism exists on $ J(C_f) $, so $ \mathrm{Aut}(J(C_f)) = \{ \pm 1 \} $.
• The jacobians associated to distinct pairs $ (t_1, t_2) $ are pairwise non-isomorphic over $ \bar{K} $, due to the non-existence of $ \mathrm{PGL}_2(K) $-maps preserving the branch points.
• The Tate conjecture holds for all self-products $ J(C_f)^m $ in all codimensions, as all Tate classes are linear combinations of divisor classes.
• In characteristic zero with a complex embedding, the Hodge conjecture holds for all self-products $ J(C_f)^m $ in all codimensions.