[Paper Review] Manin's conjecture vs. Malle's conjecture

Takehiko Yasuda|arXiv (Cornell University)|May 18, 2015

Algebraic Geometry and Number Theory19 references3 citations

TL;DR

This paper establishes a heuristic correspondence between Manin’s conjecture on rational points of singular Fano varieties and Malle’s conjecture on the distribution of number field extensions, using quotient varieties of projective space by finite group actions and Dirichlet series analysis. The key result is that the asymptotic behavior of rational points on such quotients matches the distribution of Galois extensions when the age invariant of the group action aligns with the geometric invariants of the variety.

ABSTRACT

By a heuristic argument, we relate two conjectures. One is a version of Manin's conjecture about the distribution of rational points on a Fano variety. We concern specific singular Fano varieties, namely quotients of projective spaces by finite group actions, and their singularities play a key role. The other conjecture is a generalization of Malle's conjecture about the distribution of extensions of a number field. Main tools are several Dirichlet series and previously obtained techniques, especially the untwisting, for the counterpart over a local field.

Motivation & Objective

• To explore the connection between the distribution of rational points on singular Fano varieties and the distribution of Galois extensions of number fields.
• To investigate how the geometry of quotient varieties $\overline{X} = \mathbb{P}^d/G$ relates to arithmetic counting functions in number field counting problems.
• To unify two conjectures—Manin’s (on rational points) and Malle’s (on extensions)—via the McKay correspondence and height zeta functions.
• To analyze the role of singularities and crepant resolutions in both arithmetic and geometric counting problems.
• To propose a refined version of Malle’s conjecture using the age function and conjugacy class structure of finite groups acting on representations.

Proposed method

• Constructs a Fano variety $\overline{X} = \mathbb{P}^d/G$ as a quotient of projective space by a finite group $G$-action, with log terminal singularities.
• Defines a height function on rational points of $\overline{X}$ using adelic metrics on the anti-canonical bundle.
• Introduces primitive $K$-points as those not coming from intermediate covers, to exclude accumulating thin subsets.
• Applies untwisting techniques from local fields to relate global height zeta functions to Dirichlet series.
• Uses the $V$-discriminant and extended $V$-discriminant to count $G$-fields over a number field $K$.
• Analyzes the abscissa of convergence of Dirichlet series $Z^\mathrm{disc}(s)$ and relates it to the asymptotic growth of rational points and extensions.

Experimental results

Research questions

• RQ1 How do the asymptotic counts of rational points on quotient Fano varieties $\overline{X} = \mathbb{P}^d/G$ relate to the distribution of $G$-extensions of a number field?
• RQ2 To what extent does the age invariant $\mathrm{age}(G)$ of a finite group action determine the growth rate of rational points and number field extensions?
• RQ3 Can the structure of $K$-conjugacy classes with minimal age explain the logarithmic terms in both Manin’s and Malle’s conjectures?
• RQ4 What is the role of primitive rational points in reconciling the two conjectures, especially in the presence of accumulating thin subsets?
• RQ5 How do the poles of height zeta functions and $V$-discriminant Dirichlet series reflect the geometric and arithmetic invariants of the quotient variety?

Key findings

• For $\mathrm{age}(G) > 1$, the Dirichlet series $Z^\mathrm{disc}(s)$ converges at $s=1$, implying that the rational point count on $X_{\mathrm{prim}}(K)$ has a simple pole at $s=1$, consistent with Conjecture 5.6.
• When $\mathrm{age}(G) = 1$, the right-most pole of $Z^\mathrm{disc}(s)$ is at $s=1$ with order $\upsilon(G)$, matching the logarithmic term $\gamma(X)$ in Manin’s conjecture.
• For $\mathrm{age}(G) < 1$, the abscissa of convergence of $Z^\mathrm{disc}(s)$ is $s = 1/\mathrm{age}(G) > 1$, which aligns with the growth rate $B^{1/\mathrm{age}(G)}$ in Conjecture 12.10.
• The asymptotic formula $N_{G,V,K}(B) \sim CB^{1/\mathrm{age}(G)}(\log B)^{\upsilon(G)}$ generalizes Malle’s conjecture when $G \subset S_n$ and $V = K^{2n}$.
• The primitive rational points $X_{\mathrm{prim}}(K)$ provide a natural subset for which Conjecture 5.6 holds, and their height zeta function matches the $V$-discriminant series under the assumptions.
• The correspondence is most precise when $\mathrm{age}(G) = 1$, where both conjectures predict the same pole structure and growth order.