Meng Fai Lim
8 min
Abstract
We present certain results on the Iwasawa theory of an abelian variety with potentially good ordinary reduction at all primes above $p$. These are then applied to study Diophantine stability and integally Diophantine extensions. Along the way, we also obtain some results pertaining to Mazur growth conjecture which refine previous results of Gajek-Leonard, Hatley, Kundu and Lei. Finally, we extend our investigation to the case of an elliptic curve with good supersingular reduction at the prime $p$ and make a similar analysis.
Alex: Like distinct factors piling up until they can't fit into a fixed polynomial?
Sam: Precisely. It's like distinct prime-like directions in a lattice: projection caps how many can explode before the structure limits them. This bounds bad lines. For elliptic curves over Q with good ordinary reduction at p—clean modulo p—and F a finite abelian imaginary extension, Kato and Rubin's work confirms the torsion, yielding uncountably many stable cyclic extensions per level.
Alex: The mechanism hinges on coprimality and projection tying back to cyclotomic data. Now, with finitely many bad lines, what about rank growth, like Mazur's conjecture?
Sam: It sharpens rank growth results. For an elliptic curve over Q with good ordinary reduction at p—formal group height one—and conditions like p splitting in imaginary quadratic K with μ-invariant one in cyclotomic dual Selmer, ranks stay bounded in all Z_p-extensions of K except the anti-cyclotomic one.
Alex: One exceptional direction, but capped elsewhere?
Sam: Yes. It drops prior assumptions on module structure. A similar result holds for abelian varieties of GL2-type over totally real fields.
Alex: Now, Diophantine stability—how do good directions help?
Sam: A pair L1 over L2, both degree p^n in a Z_p-extension of base F, is stable if points over L1 match those over L2 exactly. With cyclotomic dual torsion, most lines avoid the finite bad set, so ranks bound there. By Wingberg, only finitely many lines have infinite torsion, leaving finitely generated points. Subfields give stable pairs—uncountably many.
Alex: Finitely generated means points stabilize at levels. Unconditional for elliptic curves?
Sam: Yes—for E over Q with good ordinary reduction at p, F any finite abelian imaginary extension of Q: uncountably many stable pairs per n, via Kato-Rubin.
Alex: What about integrally Diophantine extensions?
Sam: Building on Shlapentokh: if ranks match positively over L1 and L2, L2's integers are Diophantine-definable in L1 via polynomials. With positive base rank preserved in good directions, uncountably many such pairs under ordinary reduction and torsion.
Alex: How does it construct those from the bound?
Sam: Uncountably many Z_p-extensions have bounded rank. Pick L1 over L2 at height p^n with matching positive rank. Shlapentokh applies: ranks match means L2's integers definable in L1 by existential equations.
Alex: Unconditional version?
Sam: Corollary 5.6: F any finite abelian imaginary of Q, E over Q with good ordinary reduction at p, analytic rank one so base rank positive over F, Kato confirms torsion. Uncountable integrally Diophantine p^n-pairs over F for every n ≥ 1.
Alex: Now, broader cases like non-abelian?
Sam: Proposition 5.7: E over F with potential good ordinary at p-primes, cyclotomic dual finitely generated. For any finite Galois p-extension L of F, stable pairs uncountable at every n. If positive rank over F, integrally Diophantine too. Example: p=3, 79A1 curve over Q with cube roots, rank one, torsion as Z_p—uncountable for any Galois 3-L.
Alex: Section six refines the bound conjecturally?
Sam: Under cyclotomic dual structure—pseudo-isomorphic to Z_p powers—and no Z_p /U^e for e≥2, U-valuation equals base Selmer p-corank. Greenberg conjecture links to base corank. Bad lines then bounded by corank; if p-Shafarevich finite over F, at most base Mordell-Weil rank.
Alex: Conjecturally capped by base rank.
Sam: Yes. It ends with signed Selmer for supersingular elliptic curves at p over imaginary quadratics where p splits: plus/minus norms control local conditions, bounding bad sets via similar logic.
Alex: Signed versions broaden it.
Sam: For E over Q supersingular at p, F imaginary quadratic with p split: signed Selmers split by trace parity. Cyclotomic signed duals torsion bounds each bad set; total bad set their union. Unconditionally, uncountable stable pairs if base p-Selmer finite or conductors split.
Alex: What limitations?
Sam: Relies on unproven Mazur conjecture for cyclotomic torsion; suitable F not always guaranteed. Supersingular covers ++ and -- known torsion; cross terms partial.
Alex: Strength in the mechanism—projection, coprimality, valuations—tying settings together. A meaningful step in controlling rank growth and stability. Thanks for breaking it down, Sam. That's our look at this paper on Iwasawa theory over Z_p²-extensions. Thanks for listening to ResearchPod.