Analytic rank-one elliptic curves over function fields and their rank over certain ring class fields

Abstract: Let $E/k$ be a non-isotrivial elliptic curve over a global function field $k$ of characteristic $p>3$, and $G\subset \mathrm{Gal}(k^{\mathrm{sep}}/k)$ be a topologically finitely generated subgroup. We prove that if $E/k$ has analytic rank $1$, then its rank over the fixed subfield $L^G$ is infinite, where $L$ is the infinite ring class extension of some finite separable extension $K/k$. If $E/k$ has analytic rank $0$, then we prove that the same holds provided there exists an imaginary quadratic extension $K/k$ such that $E/K$ has analytic rank $1$ and satisfies the Heegner hypothesis.

• The paper establishes that non-isotrivial analytic rank-one elliptic curves over function fields exhibit infinite Mordell-Weil rank in fixed subfields of ring class field towers.
• The paper employs techniques like Heegner point constructions and the function field Gross–Zagier–Zhang formula to derive explicit growth estimates and recursion relations.
• The paper verifies a case of Larsen’s conjecture by analyzing the rich pro‑p structure of Galois groups in infinite ring class field extensions.

Analytic Rank-One Elliptic Curves Over Function Fields and Their Rank Over Certain Ring Class Fields

Introduction and Motivation

The paper addresses deep arithmetic properties of non-isotrivial elliptic curves $E$ defined over global function fields $k$ of characteristic $p > 3$, specifically focusing on the rank of $E$ over infinite towers of ring class fields and their Galois fixed subfields. The study is situated at the intersection of Diophantine geometry, the arithmetic of elliptic curves, and Iwasawa theory in the function field context. The motivating conjecture is Larsen's conjecture, which asserts infinitude of rank for abelian varieties over fields fixed by topologically finitely generated subgroups of the Galois group. The work builds on the Heegner point method and functional field analogues of the Gross-Zagier formula to establish unconditional results in the function field setting, paralleling but also diverging significantly from the number field case.

Main Results

Let $E/k$ be a non-isotrivial elliptic curve with analytic rank one. Let $L$ be the union of all ring class field extensions of finite separable extensions $K/k$, and let $G$ be a topologically finitely generated subgroup of $\text{Gal}(k^{\text{sep}}/k)$. The main theorem of the paper states:

• If $E/k$ has analytic rank 1, then the rank of $k$0 over $k$1 (the fixed field under $k$2 inside $k$3) is infinite.
• If $k$4 has analytic rank 0, the same holds provided there exists an imaginary quadratic extension $k$5 such that $k$6 has analytic rank 1 and satisfies the Heegner hypothesis.

These results represent the first complete proof of Larsen's conjecture for rank one elliptic curves over function fields, extending Breuer’s construction of infinite rank in the full ring class field tower [Breuer 2004].

Methodology and Technical Themes

The proof technique is based on a combination of arithmetic geometric methods and analytic tools:

• Heegner Points in Function Field Settings: The authors construct towers of Heegner points via Drinfeld modules and modular parametrizations, exploiting the ring class field tower structure intrinsic to function fields. The norm-compatibility and recurrence relations for Heegner systems are established and exploited to generate infinitely many independent points.
• Function Field Gross–Zagier–Zhang Formula: Essential use is made of the function field analogue of the Gross-Zagier-Zhang formula (as proved by Qiu), connecting the non-torsionness of Heegner points with non-vanishing of $k$7-function derivatives, specifically $k$8. The automorphic Langlands correspondence over function fields is also invoked to transfer between geometric and automorphic data.
• Detailed Galois Module Analysis: Leveraging class field theory, the group $k$9 is shown to have a rich pro-$p > 3$0 structure, allowing powerful descent arguments beyond the capabilities of the number field case, notably because the vertical ring class tower in function fields has infinitely many generators.

For analytic rank zero curves, the argument requires the additional existence of an auxiliary imaginary quadratic extension $p > 3$1 such that the analytic rank over $p > 3$2 is 1 and the Heegner hypothesis holds. The existence of such $p > 3$3 is guaranteed by Ulmer’s geometric non-vanishing results only after possibly a finite separable base change.

Strong Numerical and Theoretical Claims

The main strong claim is the unconditional infinitude of the Mordell-Weil rank of $p > 3$4 over $p > 3$5 under stated hypotheses. This is shown to hold for all choices of topologically finitely generated $p > 3$6, considerably generalizing earlier results which required $p > 3$7 to be procyclic or imposed torsion assumptions. The paper demonstrates explicit, asymptotically precise lower bounds for the $p > 3$8-part of extension degrees in the ring class field tower and establishes exponential growth in the relevant recurrences for Heegner trace relations, precluding the possibility of a finitely generated Mordell-Weil group in the fixed subfields.

Implications and Future Directions

Practical Implications:

• The results describe the behavior of Mordell-Weil ranks in infinite, non-abelian extensions of global function fields, which is significant for explicit point construction, possible advances in algorithmic arithmetic geometry over function fields, and for the conceptual foundations of Iwasawa theory beyond the cyclotomic case.

Theoretical Implications:

• The methods close an analogue of Larsen’s conjecture in rank one for function fields, paralleling progress in number fields and reinforcing the central role of Heegner points and the Gross-Zagier framework.
• The paper’s approach reveals stark differences between number fields and function fields, because the pro-$p > 3$9 part of the Galois group in the function field ring class tower is much richer, drastically simplifying some recursion and field-descent arguments.

Future Developments:

• The techniques suggest possible attacks on Larsen’s conjecture for abelian varieties of higher dimensions or for higher analytic rank, contingent on generalizations of the Gross-Zagier formula and the construction of suitable Euler systems in the function field setting.
• Further exploration of the interaction between infinite field extension towers and the arithmetic of elliptic curves (possibly with non-constant $E$0-invariants or in positive characteristic with small $E$1) would benefit from this analytic framework.

Conclusion

This work unconditionally establishes that non-isotrivial analytic rank-one elliptic curves over function fields of characteristic $E$2 have infinite Mordell-Weil rank in fixed fields of certain infinite ring class field extensions, for any topologically finitely generated Galois subgroup. The methodological innovation is the precise control of Heegner systems in the function field ring class tower, supported by the function field Gross-Zagier-Zhang formula and explicit Galois descent. The results answer a substantial case of Larsen’s conjecture in the function field setting, and the techniques open prospects for analogous results in broader contexts.