Products of point counts of higher genus curves over finite fields

Abstract: Let $E/\mathbb Q$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the conjecture of Birch and Swinnerton-Dyer asserts that $$\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x) ^{{\text{rank}(E(\mathbb} Q))}$$ as $x \to \infty$. In this paper, we formulate a similar conjectural asymptotic for smooth projective curves of genus at least 2, in which the contributions to the conjectured asymptotic come not only from the rank of the Jacobian but also from the Sato--Tate group of the curve. The key analytic input in formulating our conjecture is a conjecture due to Kurokawa (2012) on the convergence of Euler products of entire $L$-functions on the critical line. We also provide some numerical evidence for our conjecture in various cases.

• The paper formulates a BSD-type conjecture predicting that the product of normalized point counts behaves like C(log x)^{rk(Jac(X))−rk(NS(Jac(X)))+1} for genus-at-least-two curves.
• The derivation combines conjectural Euler-product convergence, automorphy, BSD for Jacobians, and a Tate-type relation between the exterior-square L-function and the Jacobian’s Néron–Severi rank.
• Numerical tests on four genus-2 curves support the predicted exponents, including negative logarithmic powers for curves with exceptional Sato–Tate groups and extra Jacobian endomorphisms.

Overview

This paper, by Bucur, Kedlaya, and Sheth, formulates a conjectural analogue of the original Birch and Swinnerton-Dyer (BSD) conjecture for smooth projective curves of genus $g \geq 2$ over $\mathbb{Q}$. The original version of BSD asserts that for an elliptic curve $E/\mathbb{Q}$ with point counts $N_p$, the partial Euler product satisfies $$\prod_{p \leq x} N_p/p \sim C(\log x)^{rk(E(\mathbb{Q}))}$$. The authors conjecture that for a curve $X$ of genus at least 2, the corresponding product obeys

$$\prod_{\substack{p \leq x \ p \notin S_X}} \frac{N_p}{p} \sim C(\log x)^{rk(Jac(X)) - rk(NS(Jac(X))) + 1},$$

where $S_X$ is the finite set of bad primes together with primes for which $N_p = 0$. The distinguishing feature relative to the elliptic case is the correction term $-rk(NS(Jac(X))) + 1$, governed by the Sato–Tate group of the curve. Notably, this exponent can be negative, a phenomenon impossible for elliptic curves, where $\mathbb{Q}$0 always.

The conjectural framework

The derivation rests on three conjectural pillars. First, the authors invoke the Kaneko–Koyama–Kurokawa conjecture (a form of the "Deep Riemann Hypothesis") on the convergence of Euler products of entire $\mathbb{Q}$1-functions on the critical line. In the isobaric setting, it predicts that the partial Euler product of an automorphic $\mathbb{Q}$2-function at its central point converges to a constant multiple of the $\mathbb{Q}$3-value, with a rate of $\mathbb{Q}$4 where $\mathbb{Q}$5 is the order of vanishing, and an explicit factor $\mathbb{Q}$6 sourced from the second-moment $\mathbb{Q}$7-function $\mathbb{Q}$8. The exponent $\mathbb{Q}$9 enters via the estimate $E/\mathbb{Q}$0.

Second, they assume modularity: that $E/\mathbb{Q}$1 for an abelian variety $E/\mathbb{Q}$2 coincides with an automorphic $E/\mathbb{Q}$3-function, admitting meromorphic continuation and a functional equation. This is known for CM abelian varieties and, for $E/\mathbb{Q}$4, by the modularity theorem.

Third, they use a conjecture relating the exterior square $E/\mathbb{Q}$5-function to the Néron–Severi rank. Writing the motivic factorization $E/\mathbb{Q}$6, they conjecture that $E/\mathbb{Q}$7 is holomorphic and nonzero at $E/\mathbb{Q}$8. Combined with the theorem of Costa–Fité–Sutherland that $E/\mathbb{Q}$9 — itself a consequence of Faltings' isogeny theorem — this yields $N_p$0, a special case of Tate's conjecture.

Derivation of the curve conjecture

The mechanism is as follows. Applying the Kaneko–Koyama–Kurokawa conjecture to $N_p$1 at the central point $N_p$2, together with Tate's BSD conjecture $N_p$3, gives $N_p$4 for the reciprocal partial Euler product. On the other hand, the trace formula gives $N_p$5, and expanding the product $N_p$6 in terms of $N_p$7 produces a correction factor

$N_p$8

Mertens' estimate applied to the normalized exterior square $N_p$9-function yields $$\prod_{p \leq x} N_p/p \sim C(\log x)^{rk(E(\mathbb{Q}))}$$0 with $$\prod_{p \leq x} N_p/p \sim C(\log x)^{rk(E(\mathbb{Q}))}$$1; the classical Mertens estimate handles the $$\prod_{p \leq x} N_p/p \sim C(\log x)^{rk(E(\mathbb{Q}))}$$2 term. Exponentiating, the correction factor is $$\prod_{p \leq x} N_p/p \sim C(\log x)^{rk(E(\mathbb{Q}))}$$3, and combining the two asymptotics produces the conjectured exponent $$\prod_{p \leq x} N_p/p \sim C(\log x)^{rk(E(\mathbb{Q}))}$$4. The authors emphasize that the entire discrepancy from the naive guess $$\prod_{p \leq x} N_p/p \sim C(\log x)^{rk(E(\mathbb{Q}))}$$5 stems from the trace formula for curves; for abelian varieties, where $$\prod_{p \leq x} N_p/p \sim C(\log x)^{rk(E(\mathbb{Q}))}$$6, no such correction appears, and the analogous conjecture reads $$\prod_{p \leq x} N_p/p \sim C(\log x)^{rk(E(\mathbb{Q}))}$$7.

Relation to the Kurokawa–Tanaka hypothesis

The paper situates its conjectures within a hypothesis of Kurokawa and Tanaka asserting that for any algebraic variety $$\prod_{p \leq x} N_p/p \sim C(\log x)^{rk(E(\mathbb{Q}))}$$8, the product $$\prod_{p \leq x} N_p/p \sim C(\log x)^{rk(E(\mathbb{Q}))}$$9 should be asymptotic to $X$0 for integers $X$1 and positive reals $X$2. Kurokawa–Tanaka verified this unconditionally for projective space, Grassmannians, and certain matrix groups. The present conjectures can be read as refinements that explicitly identify $X$3 in the two most direct higher-genus generalizations of BSD. The authors note that the hypothesis requires interpretation: primes with $X$4 (finite in number by Lang–Weil) and primes of bad reduction must be excluded, and $X$5 may depend on the chosen integral model at bad primes.

Numerical evidence

The authors present numerical evidence for four genus-2 curves from the LMFDB, plotting $X$6 against $X$7 for primes up to 5693 and comparing the slope of the best-fit line with the conjectured exponent.

Curve (LMFDB)	Sato–Tate group	$X$8	$X$9	Conjectured exponent	Fitted slope
440509.a.440509.1	$$\prod_{\substack{p \leq x \ p \notin S_X}} \frac{N_p}{p} \sim C(\log x)^{rk(Jac(X)) - rk(NS(Jac(X))) + 1},$$0	4	1	4	≈ 3.77
277.a.277.1	$$\prod_{\substack{p \leq x \ p \notin S_X}} \frac{N_p}{p} \sim C(\log x)^{rk(Jac(X)) - rk(NS(Jac(X))) + 1},$$1	0	1	0	≈ 0.05
504.a.27216.1	$$\prod_{\substack{p \leq x \ p \notin S_X}} \frac{N_p}{p} \sim C(\log x)^{rk(Jac(X)) - rk(NS(Jac(X))) + 1},$$2	0	2	−1	≈ −1.06
400.a.409600.1	$$\prod_{\substack{p \leq x \ p \notin S_X}} \frac{N_p}{p} \sim C(\log x)^{rk(Jac(X)) - rk(NS(Jac(X))) + 1},$$3	0	3	−2	≈ −2.003

The last two examples are of particular interest because the conjectured exponent is negative, so the product tends to zero like a negative power of $$\prod_{\substack{p \leq x \ p \notin S_X}} \frac{N_p}{p} \sim C(\log x)^{rk(Jac(X)) - rk(NS(Jac(X))) + 1},$$4 — behavior with no analogue in the elliptic setting. These are precisely the cases with non-generic Sato–Tate groups, where $$\prod_{\substack{p \leq x \ p \notin S_X}} \frac{N_p}{p} \sim C(\log x)^{rk(Jac(X)) - rk(NS(Jac(X))) + 1},$$5 reflects extra endomorphisms of the Jacobian. The fitted slopes agree closely with the conjectured exponents, though the data range is modest.

Limitations and open questions

The paper is explicitly conjectural: no unconditional asymptotic is proved for any higher-genus curve, and the derivation depends on the Kaneko–Koyama–Kurokawa conjecture, the automorphy of $$\prod_{\substack{p \leq x \ p \notin S_X}} \frac{N_p}{p} \sim C(\log x)^{rk(Jac(X)) - rk(NS(Jac(X))) + 1},$$6 for general abelian varieties, and the holomorphy/nonvanishing of $$\prod_{\substack{p \leq x \ p \notin S_X}} \frac{N_p}{p} \sim C(\log x)^{rk(Jac(X)) - rk(NS(Jac(X))) + 1},$$7. The relation between the original and modern formulations of BSD remains unresolved even for elliptic curves: it is not known whether the modern formulation plus the Riemann Hypothesis for $$\prod_{\substack{p \leq x \ p \notin S_X}} \frac{N_p}{p} \sim C(\log x)^{rk(Jac(X)) - rk(NS(Jac(X))) + 1},$$8 implies the product asymptotic unconditionally, though it implies it outside a set of finite logarithmic measure. The numerical evidence covers only genus-2 curves over a limited prime range, and the behavior of the constant $$\prod_{\substack{p \leq x \ p \notin S_X}} \frac{N_p}{p} \sim C(\log x)^{rk(Jac(X)) - rk(NS(Jac(X))) + 1},$$9 — for which Goldfeld gave an explicit formula in the elliptic case — is not addressed here for higher genus. Whether the exponent $S_X$0 admits an unconditional interpretation, or whether the conjecture extends to varieties beyond curves and abelian varieties with an identifiable $S_X$1, remains open.

Conclusion

The paper proposes a precise, Sato–Tate-sensitive refinement of the original BSD product asymptotic for curves of genus at least 2 and for abelian varieties, deriving both from a coherent framework built on Euler product convergence at the central point and Tate-type conjectures for the exterior square $S_X$2-function. The conjecture subsumes the elliptic case, predicts genuinely new phenomena (negative exponents) for curves with exceptional Sato–Tate groups, and is supported by numerical evidence whose fitted slopes match the predicted exponents. Its verification in any nontrivial higher-genus case would require substantial progress on the underlying analytic and automorphic conjectures.