Unconditional Iwasawa Main Conjecture for ordinary semistable elliptic curves over global function fields: removing the unramified μ-minimality hypothesis

The Iwasawa Main Conjecture (IMC) predicts that the ideal generated by the $p$ -adic $L$ -function of an elliptic curve over a $\mathbb{Z}_p^d$ -extension equals the characteristic ideal of the Pontryagin dual of its $p^\infty$ -Selmer group. Tan–Trihan–Tsoi (arXiv:2603.11615, 2026) proved this conjecture for ordinary semistable elliptic curves over global function fields of characteristic $p$ , over extensions ramified only at ordinary places, but their theorem is conditional on an unramified μ-minimality hypothesis — that the μ-invariant of the $p$ -adic $L$ -function does not increase under specialization to the unramified $\mathbb{Z}_p$ -direction — which they verify only generically, for $p>3$ , on a Zariski open dense locus of the moduli of semistable curves. The problem is to remove this hypothesis: establish the IMC unconditionally for every such curve, extension, and prime $p$ (including $p\leq 3$ and curves with positive μ-invariant over the unramified direction), or refute the conjectured equality by an explicit counterexample.

Background

Iwasawa theory studies the growth of arithmetic invariants — here the $p$ -primary part of Selmer groups of an elliptic curve — in infinite towers of extensions, and predicts that a $p$ -adic $L$ -function built from special values of Hasse–Weil $L$ -functions encodes the Selmer data through an ideal-theoretic equality, the Iwasawa Main Conjecture (IMC). The present problem lives in the positive-characteristic setting: $K$ is a global function field, i.e. a finite extension of $\mathbb{F}_q(t)$ with $q$ a power of the prime $p$ , and the same prime $p$ serves as the Iwasawa prime.

An elliptic curve $A/K$ has good ordinary reduction at a place if the reduced curve is smooth and the $p$ -torsion group scheme of its special fiber has an étale part of rank $p$ (the opposite of supersingular reduction, where the $p$ -torsion is entirely local), and multiplicative reduction at a place if the special fiber degenerates to a torus; $A$ is semistable if every place is good or multiplicative. A $\mathbb{Z}_p^d$ -extension is a Galois extension $L/K$ whose Galois group $\Gamma$ is isomorphic to the direct product of $d$ copies of the $p$ -adic integers. Every global function field has a distinguished everywhere-unramified $\mathbb{Z}_p$ -extension $L_0=K_\infty^{(p)}$ , obtained by extending the finite constant field, and it plays a special role below.

On the algebraic side one takes the $p^\infty$ -Selmer group $\mathrm{Sel}_{p^\infty}(A/L)$ — the subgroup of $H^1(L,A[p^\infty])$ of classes whose restriction to every place $w$ of $L$ lies in the image of the Kummer map $A(L_w)\otimes\mathbb{Q}_p/\mathbb{Z}_p\to H^1(L_w,A[p^\infty])$ — and its Pontryagin dual $X_L$ (the group of continuous characters into $\mathbb{Q}_p/\mathbb{Z}_p$ , with the compact-open topology). $X_L$ is known to be finitely generated over the Iwasawa algebra $\Lambda_\Gamma=\mathbb{Z}_p[[\Gamma]]$ (Tan–Trihan–Tsoi 2025), and one defines its characteristic ideal $\mathrm{CH}_{\Lambda_\Gamma}(X_L)$ — the product of the prime powers appearing in an elementary-divisor decomposition — with the convention that it is the zero ideal $(0)$ precisely when $X_L$ is non-torsion. On the analytic side, Tan (arXiv:2603.10576) constructed a $p$ -adic $L$ -function $\mathscr{L}_{A/L}\in\mathbb{Q}_p\Lambda_\Gamma$ , uniquely characterized by interpolation of Hasse–Weil $L$ -values twisted by finite-order characters of $\Gamma$ , and proved that it satisfies a functional equation, specialization formulae and restriction formulae parallel to those of $X_L$ , together with $\mu(\mathscr{L}_{A/L})=\mu(X_L)$ ; here the μ-invariant of a nonzero element $f$ of $\mathbb{Q}_p\Lambda_\Gamma$ is the unique integer $\mu(f)$ such that $p^{-\mu(f)}f$ lies in $\Lambda_\Gamma$ and is not divisible by $p$ . The IMC is the predicted equality of ideals

$(\mathscr{L}_{A/L})=\mathrm{CH}_{\Lambda_\Gamma}(X_L)\quad\text{in }\Lambda_\Gamma.$

Before 2026 this equality was known for $L=K$ , for ordinary constant elliptic curves (defined over the finite constant field, with ordinary reduction; more generally constant ordinary abelian varieties, Lai–Longhi–Tan–Trihan 2016), and for semistable curves over the unramified $\mathbb{Z}_p$ -extension $L_0/K$ (Lai–Longhi–Tan–Trihan 2016 for semistable abelian varieties; Tan 2026).

Tan–Trihan–Tsoi (arXiv:2603.11615, March 2026) prove the IMC for a general $\mathbb{Z}_p^d$ -extension $L/K$ ramified only at finitely many places where $A$ has good ordinary or multiplicative reduction, assuming $A$ semistable everywhere and in the source papers' class of ordinary elliptic curves — the curves for which the analytic side is constructed in Tan (arXiv:2603.10576): every non-constant curve (not defined over the constant field), and every ordinary constant curve. The key new input is a χ-formula identifying the χ-isotypic characteristic ideals of Selmer modules with the corresponding specializations of $\mathscr{L}_{A/L}$ ; the reduction from rank $d$ to rank $2$ then proceeds by a Weierstrass-preparation lifting argument — using that a nonzero element of $\mathbb{Z}_p[[T]]$ is, up to a unit, a power of $p$ times a monic polynomial with non-leading coefficients divisible by $p$ (a distinguished polynomial), so that an equality of ideals can be lifted from sufficiently many one-variable specializations. However, the main theorem is conditional on the unramified μ-minimality hypothesis: writing $\tilde L=LL_0$ and $p^{\tilde L L_0}$ for the specialization homomorphism induced by the projection $\mathrm{Gal}(\tilde L/K)\to\mathrm{Gal}(L_0/K)$ , one always has $\mu(p^{\tilde L L_0}\mathscr{L}_{A/\tilde L})\geq\mu(\mathscr{L}_{A/\tilde L})$ , and the hypothesis requires equality — i.e. that μ attain its minimum along the unramified direction. The hypothesis holds for a large class of intermediate $\mathbb{Z}_p$ -directions (Tan 2026, Lemma 6.2.5) and holds whenever $\mu(\mathscr{L}_{A/L_0})=0$ , equivalently $\mu(X_{L_0})=0$ . By Lai–Longhi–Suzuki–Tan–Trihan (2021), $\mu(X_{L_0})$ equals the dimension of the Tate–Shafarevich group of $A$ (verifying a suggestion of Ulmer) and admits an unconditional formula in terms of the Faltings height (a measure of the arithmetic complexity of $A$ ) and Frobenius slopes of the Hasse–Weil $L$ -function; in particular it is not always zero. The appendix of arXiv:2603.11615 proves only that for $p>3$ the locus of semistable curves with $\mu=0$ is Zariski open and dense in the natural moduli spaces (building on Theorem 6.3.1 of that 2021 work). For curves with positive μ, and for the small primes $p\leq 3$ where the minimal-Weierstrass-model parameter-space arguments are unavailable, the hypothesis — and hence the unconditional IMC — remains open.

Problem Statement

Prove, or refute by an explicit counterexample, the Iwasawa Main Conjecture for ordinary semistable elliptic curves over global function fields in full generality, removing the unramified μ-minimality hypothesis under which Tan–Trihan–Tsoi (arXiv:2603.11615) proved it. The instances are the source paper's own, with no restriction on μ-invariants: for every prime $p$ ; for every elliptic curve $A$ over a global function field $K$ of characteristic $p$ (i.e. $K$ a finite extension of $\mathbb{F}_q(t)$ with $q$ a power of $p$ ) that is semistable at every place (every place has good or multiplicative reduction) and that belongs to the source papers' class of ordinary elliptic curves — the curves for which the analytic side is constructed in Tan (arXiv:2603.10576): every non-constant curve (not defined over the finite constant field of $K$ ; these are modular in the sense of Deligne–Weil, which the construction invokes), together with every constant curve (defined over the finite constant field) that has ordinary reduction, constant supersingular curves being the only exclusion; and for every $\mathbb{Z}_p^d$ -extension $L/K$ with $d\geq 1$ ramified only at finitely many places where $A$ has good ordinary or multiplicative reduction — the equality of ideals

$(\mathscr{L}_{A/L})=\mathrm{CH}_{\Lambda_\Gamma}(X_L)\quad\text{in }\Lambda_\Gamma=\mathbb{Z}_p[[\mathrm{Gal}(L/K)]]$

— equation (3) of Tan–Trihan–Tsoi (arXiv:2603.11615) — must hold. Here $\mathscr{L}_{A/L}\in\mathbb{Q}_p\Lambda_\Gamma$ is the $p$ -adic $L$ -function of $A$ over $L$ constructed in Tan (arXiv:2603.10576), $(\mathscr{L}_{A/L})$ the principal ideal it generates, $\mathrm{Sel}_{p^\infty}(A/L)$ the $p^\infty$ -Selmer group of $A$ over $L$ , $X_L$ its Pontryagin dual (a finitely generated $\Lambda_\Gamma$ -module), and $\mathrm{CH}_{\Lambda_\Gamma}(X_L)$ its characteristic ideal, taken to be the zero ideal when $X_L$ is non-torsion (in which case the asserted equality reduces to $\mathscr{L}_{A/L}=0$ via the proven compatibility $\mu(\mathscr{L}_{A/L})=\mu(X_L)$ ).

By Theorem 1 of arXiv:2603.11615 the μ-invariants of $\mathscr{L}_{A/L}$ and of $\mathrm{CH}_{\Lambda_\Gamma}(X_L)$ coincide, so the two ideals can differ only in invariants finer than the μ-invariant.

References

Ki-Seng Tan, Fabien Trihan, Kwok-Wing Tsoi, 'Iwasawa Main Conjecture for ordinary semistable elliptic curves over global function fields', arXiv:2603.11615 (2026), DOI 10.48550/arXiv.2603.11615, https://arxiv.org/abs/2603.11615
Ki-Seng Tan, ' $p$ -adic $L$ -functions for elliptic curves over global function fields', arXiv:2603.10576 (2026), DOI 10.48550/arXiv.2603.10576, https://arxiv.org/abs/2603.10576
King-Fai Lai, Ignazio Longhi, Takashi Suzuki, Ki-Seng Tan, Fabien Trihan, 'On the μ-invariants of abelian varieties over function fields of positive characteristic', Algebra & Number Theory 15 (2021), no. 4, 863–907, DOI 10.2140/ant.2021.15.863, https://doi.org/10.2140/ant.2021.15.863
King Fai Lai, Ignazio Longhi, Ki-Seng Tan, Fabien Trihan, 'The Iwasawa Main Conjecture for semistable abelian varieties over function fields', Mathematische Zeitschrift 282 (2016), no. 1–2, 485–510, DOI 10.1007/s00209-015-1550-4, https://doi.org/10.1007/s00209-015-1550-4
King Fai Lai, Ignazio Longhi, Ki-Seng Tan, Fabien Trihan, 'The Iwasawa main conjecture of constant ordinary abelian varieties over function fields', Proceedings of the London Mathematical Society 112 (2016), no. 6, 1040–1058, DOI 10.1112/plms/pdw019, https://doi.org/10.1112/plms/pdw019
Sohan Ghosh, Jishnu Ray, Takashi Suzuki, 'On μ-invariants and isogenies for abelian varieties over function fields', arXiv:2407.21431 (2024); published in Mathematische Annalen 396 (2026), no. 1, DOI 10.1007/s00208-026-03535-0, https://arxiv.org/abs/2407.21431
Anwesh Ray, 'Greenberg's μ=0 conjecture for lisse sheaves over global function fields', arXiv:2607.10728 (2026), DOI 10.48550/arXiv.2607.10728, https://arxiv.org/abs/2607.10728
Ki-Seng Tan, Fabien Trihan, Kwok-Wing Tsoi, 'The μ-invariant change for abelian varieties over finite p-extensions of global fields', Research in the Mathematical Sciences 12 (2025), no. 4, Article 70, DOI 10.1007/s40687-025-00557-z, https://doi.org/10.1007/s40687-025-00557-z