[Paper Review] Arithmetic on Elliptic Threefolds

Rania Wazir|ArXiv.org|Dec 23, 2001

Algebraic Geometry and Number Theory10 references16 citations

TL;DR

This paper establishes an analytic formula linking the rank of rational points on an elliptic threefold over a number field to the residue of a generating series involving Frobenius trace averages over reductions modulo primes. Using étale cohomology, the Shioda-Tate formula, and Tate's Conjecture, it proves that the residue at s=1 of a sum over primes of the average Frobenius trace times log(q_p)/q_p^s equals the Mordell-Weil rank of the threefold's rational sections.

ABSTRACT

In a recent paper, Rosen and Silverman showed that Tate's conjecture on the order of vanishing of L(E,s) implies Nagao's formula, which gives the rank of an elliptic surface in terms of a weighted average of fibral Frobenius trace values. The aim of this article is to extend their result to the case of elliptic threefolds, and deduce, from Tate's conjecture, a Nagao-type formula for the rank of an elliptic threefold E. This will require a two-pronged approach: on the one hand, we need some cohomological results in order to derive a Shioda-Tate-like formula for elliptic threefolds; on the other, we compute an "average" number of rational points on the singular fibers and relate this to the action of Galois on those fibers.

Motivation & Objective

• To extend Nagao’s analytic rank formula from elliptic surfaces to elliptic threefolds.
• To establish a connection between the arithmetic of rational points on elliptic threefolds and geometric invariants via Frobenius traces.
• To prove that the residue of a generating series involving average Frobenius traces equals the Mordell-Weil rank of the threefold.

Proposed method

• Establish an isomorphism between étale cohomology groups of the base surface and the threefold as Galois modules.
• Apply a generalized Shioda-Tate formula for elliptic threefolds to relate Néron-Severi rank to the Mordell-Weil rank.
• Use the Lefshetz Fixed Point Theorem to equate two counts of rational points over finite fields: globally and fiberwise.
• Reinterpret the resulting fiberwise average trace formula in terms of L-functions and their logarithmic derivatives.
• Apply Tate’s Conjecture to relate the order of vanishing of L-functions to the rank of Galois-invariant cohomology and Néron-Severi groups.
• Combine all components to derive the analytic rank formula at s=1.

Experimental results

Research questions

• RQ1 Does Nagao’s analytic rank formula for elliptic surfaces extend to elliptic threefolds under Tate’s Conjecture?
• RQ2 How does the average Frobenius trace over fibers relate to the Mordell-Weil rank of rational sections on an elliptic threefold?
• RQ3 Can the rank of the Mordell-Weil group be recovered from the residue of a zeta-like series built from Frobenius traces?

Key findings

• The residue at s=1 of the Dirichlet series ∑_p -A_p(ℰ) log(q_p)/q_p^s equals the Mordell-Weil rank of ℰ(S/k).
• The isomorphism H^1_ét(S/k̄, Q_l) ≅ H^1_ét(ℰ/k̄, Q_l) as Galois modules holds for elliptic n-folds, including threefolds.
• The Shioda-Tate formula for elliptic threefolds decomposes the Mordell-Weil rank into contributions from the Néron-Severi group of the base and fiber components.
• The Frobenius trace average A_p(ℰ) is linked to the logarithmic derivative of L-functions via the Lefshetz Fixed Point Theorem.
• Under Tate’s Conjecture, the order of vanishing of L_2(ℰ,s) at s=2 equals the Néron-Severi rank of ℰ.
• The final formula confirms that the analytic rank formula for elliptic surfaces generalizes to threefolds when Tate’s Conjecture holds.