[Paper Review] A rational map between two threefolds

Kenichiro Kimura|ArXiv.org|Oct 11, 2004

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper constructs an explicit dominant rational map of degree 3 from the product of three elliptic curves $E^3$ to the threefold $V_{33} o \mathbb{P}^5$ defined by two cubic equations. The map is induced by a ring homomorphism between their coordinate rings, and it realizes a geometric correspondence predicted by Tate's conjecture, linking the $L$-series of $H^3(\widetilde{V}_{33})$ to a modular form of weight 4 on $\Gamma_0(9)$ and a Hecke character over $\mathbb{Q}(\sqrt{-3})$. The key contribution is the explicit realization of this correspondence via birational geometry and field extensions.

ABSTRACT

A rational map between certain specific threefolds is given in an explicit manner.

Motivation & Objective

• To construct an explicit rational correspondence between the threefold $V_{33}$ and the product $E^3$ of three elliptic curves, as predicted by Tate's conjecture.
• To verify that the $L$-series of $H^3(\widetilde{V}_{33})$, associated with a modular form of weight 4 on $\Gamma_0(9)$, matches a piece of the cohomology of $E^3$.
• To provide a geometric realization of the Galois representation in $H^3$ of $V_{33}$ via a dominant rational map of degree 3 from $E^3$.
• To extend the known web of correspondences among threefolds with modular $L$-series, particularly those related to $\Gamma(3)$ and $\Gamma_0(9)$-modular forms.

Proposed method

• The construction uses affine patches: the affine piece of $V_{33}$ where $X_3 \neq 0$, defined by two cubic equations in five variables.
• The affine open subset $(E \setminus \{X_2 = 0\})^3$ is described by three cubic equations in six variables, corresponding to the product of three copies of the curve $E: x^3 + y^3 + 1 = 0$.
• A ring homomorphism is defined from $\mathbb{Q}[X_0, X_1, X_2, X_4, X_5]$ to the coordinate ring of $(E \setminus \{X_2 = 0\})^3$, mapping $X_0 \mapsto -x_1 y_3$, $X_1 \mapsto -y_1 y_3$, $X_2 \mapsto x_3$, $X_4 \mapsto -x_2 y_3$, $X_5 \mapsto -y_2 y_3$.
• This homomorphism induces a dominant rational map from $E^3$ to $V_{33}$, with the function field $\mathbb{Q}(E^3)$ generated over $\mathbb{Q}(V_{33})$ by $y_3$ satisfying $y_3^3 + X_2^3 + 1 = 0$, confirming the degree 3 extension.
• The rational map is shown to be dominant by verifying that the image is dense and that the field extension has degree 3, using the algebraic independence and relations in the coordinate rings.

Experimental results

Research questions

• RQ1 Is there a geometric correspondence between $V_{33}$ and $E^3$ that explains the matching $L$-series of their $H^3$ cohomology groups?
• RQ2 Can the rational map from $E^3$ to $V_{33}$ be explicitly constructed, and what is its degree?
• RQ3 Does the field extension $\mathbb{Q}(E^3)/\mathbb{Q}(V_{33})$ induced by the map have degree 3, as required for a degree 3 rational map?
• RQ4 How does this correspondence relate to known modular forms and Hecke characters, particularly those of weight 4 on $\Gamma_0(9)$?
• RQ5 Can this construction be generalized or linked to other threefolds with modular Galois representations in $H^3$?

Key findings

• A dominant rational map of degree 3 is explicitly constructed from $E^3$ to $V_{33}$, where $E$ is the elliptic curve $X^3 + Y^3 + Z^3 = 0$.
• The map is induced by a ring homomorphism sending $X_0 \mapsto -x_1 y_3$, $X_1 \mapsto -y_1 y_3$, $X_2 \mapsto x_3$, $X_4 \mapsto -x_2 y_3$, $X_5 \mapsto -y_2 y_3$ on the affine patches.
• The function field $\mathbb{Q}(E^3)$ is generated over $\mathbb{Q}(V_{33})$ by $y_3$, with $y_3^3 + X_2^3 + 1 = 0$, confirming the degree of the map is 3.
• The existence of this map realizes the correspondence predicted by Tate's conjecture for the $L$-series $L(H^3(\widetilde{V}_{33}), s) = L(f, s)$ with $f$ a newform of weight 4 on $\Gamma_0(9)$.
• The $L$-series of $H^3(\widetilde{V}_{33})$ matches a piece of the cohomology of $E^3$, as noted in Remark 4.5 of [4], and this map provides a geometric realization of that match.
• The construction confirms that $V_{33}$ and $E^3$ are linked via a rational correspondence that aligns their modular $L$-functions and Galois representations in $H^3$.