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On refined nonvanishing conjectures by Kurihara and Kolyvagin

This paper extends Kurihara's and Kolyvagin's refined nonvanishing conjectures to cases involving arbitrary reduction types and inert primes, respectively, by introducing a novel method to compute $p$ -divisibility indices of special Galois cohomology elements through a reformulation of the Iwasawa Main Conjectures using determinants of Selmer complexes.

Original authors: Francesc Castella, Takamichi Sano

Published 2026-01-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Francesc Castella, Takamichi Sano

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a massive, ancient puzzle involving numbers, shapes, and hidden patterns. In the world of mathematics, specifically a field called Number Theory, there is a famous object called an Elliptic Curve. Think of an elliptic curve not as a smooth oval, but as a complex, multi-dimensional lattice of points that follows very strict rules.

Mathematicians have long suspected that these curves hold a secret code connecting their geometric shape to a specific type of infinite series called an L-function. This connection is known as the Birch and Swinnerton-Dyer (BSD) conjecture. It's like saying the shape of a mountain (the curve) perfectly predicts the flow of a river (the L-function) running through it.

However, proving this connection is incredibly hard. For decades, mathematicians have tried to build "bridges" to cross the gap between the geometry and the numbers. Two of the most famous bridge-builders were Kurihara and Kolyvagin. They proposed refined versions of the bridge, suggesting that if you look at specific "special numbers" (called invariants) derived from the curve, they should match up perfectly with other "special numbers" derived from the curve's local behavior.

The Problem: The Bridge Was Too Fragile

In a previous paper (referenced as [BCGS26]), the authors of this study managed to prove that these bridges worked, but only under very strict conditions.

The Kurihara Bridge: Only worked if the curve behaved "nicely" at a specific prime number (a prime is a number like 2, 3, 5, 7...). If the curve acted "badly" or "strangely" at that number, the bridge collapsed.
The Kolyvagin Bridge: Only worked if the prime number split in a specific way within a related imaginary world (an imaginary quadratic field). If the prime stayed "inert" (refused to split), the bridge failed.

It was like having a bridge that only works on sunny days or only when the wind blows from the north. The mathematicians wanted to know: Does the bridge work in the rain? Does it work when the wind is from the south?

The Solution: A New Blueprint

In this paper, Francesc Castella and Takamichi Sano have rebuilt these bridges. They have proven that the refined conjectures by Kurihara and Kolyvagin hold true even in those "bad weather" conditions:

For Kurihara: The bridge now works even if the elliptic curve has "bad reduction" (acts strangely) at the prime number.
For Kolyvagin: The bridge now works even if the prime number is "inert" (doesn't split) in the imaginary field.

How Did They Do It? The "Determinant" Trick

To understand their method, imagine you are trying to measure the weight of a ghost. You can't put it on a scale directly.

The Old Way: Previous attempts tried to measure the ghost by looking at its shadow (using something called p-adic L-functions). This worked well in good weather but got distorted in bad weather.
The New Way: Castella and Sano used a new tool called Determinants of Selmer Complexes.
- Think of a Selmer Complex as a giant, multi-layered net designed to catch specific mathematical "fish" (cohomology classes).
- Instead of looking at the shadow, they calculated the determinant of this net. In math, a determinant is like a single number that summarizes the entire volume or "size" of a shape.
- By reformulating the problem in terms of these determinants, they bypassed the need for the fragile "shadow" measurements. They showed that the "size" of the net perfectly matches the "size" of the special numbers predicted by the conjectures, regardless of the weather (the reduction type of the prime).

The Core Discovery: Counting the "Divisibility"

The heart of their proof involves a concept called the p-divisibility index.

Imagine you have a special number, $\delta$ . You want to know how many times you can divide it by a prime number $p$ before it stops being a whole number.
Kurihara and Kolyvagin predicted that this "count" (the index) should be exactly equal to a specific count of "Tamagawa factors" (which are like local correction factors for the curve's behavior at different points).
The authors proved that this count is indeed correct. They showed that the "depth" of the special numbers matches the "depth" of the local corrections perfectly.

The Big Picture: Why This Matters

This paper doesn't just say "we fixed the bridge." It says, "The bridge was never broken; we just needed a better way to measure it."

By proving these refined conjectures in such broad generality, the authors have:

Unified the Theory: They showed that the deep connections between geometry and numbers are robust. They don't break just because a prime number behaves oddly.
Validated the "Main Conjecture": They proved that these refined non-vanishing conjectures are equivalent to the famous Iwasawa Main Conjecture. This is a huge deal because the Iwasawa Main Conjecture is a cornerstone of modern number theory. If you believe the Main Conjecture (which is known to be true in many cases), you automatically believe these refined Kurihara and Kolyvagin conjectures.

Summary in a Nutshell

Think of the elliptic curve as a complex machine.

Kurihara and Kolyvagin built a manual predicting exactly how many gears (special numbers) would turn based on the machine's settings.
Previous proofs could only verify this manual when the machine was running smoothly.
Castella and Sano developed a new diagnostic tool (determinants of Selmer complexes) that works even when the machine is sputtering or running hot.
The Result: They confirmed that the manual is correct in all scenarios, proving that the deep mathematical laws governing these curves are universal and unbreakable.

Technical Summary: On Refined Nonvanishing Conjectures by Kurihara and Kolyvagin

Problem Statement
This paper addresses the refinement of nonvanishing conjectures for elliptic curves $E/\mathbb{Q}$ at an odd prime $p$ . Specifically, it extends the results of [BCGS26] regarding two major conjectures:

Kurihara's Conjectures: Relating the structure of the $p$ -primary Selmer group to analytic quantities $\delta_n$ derived from modular symbols. The refinement (Conjecture B) predicts an exact formula for the $p$ -adic valuation of the order of the Tate–Shafarevich group (or its Bloch–Kato variant) in terms of Tamagawa numbers and the "order of vanishing" of the collection $\{\delta_n\}$ .
Kolyvagin's Conjectures: Relating the nonvanishing of Kolyvagin systems derived from Heegner points to the structure of the Selmer group. The refined version (Conjecture C, by W. Zhang) predicts an exact formula for the $p$ -adic valuation of the order of the Tate–Shafarevich group in terms of Tamagawa numbers and the "order of vanishing" of the Kolyvagin system.

The primary limitation of previous work (specifically [BCGS26]) was the restriction on the reduction type of $E$ at $p$ (requiring good ordinary or specific supersingular conditions) and the splitting behavior of $p$ in the underlying imaginary quadratic field $K$ (requiring $p$ to split). This paper aims to remove these restrictions, allowing $E$ to have any reduction type at $p$ for Kurihara's conjectures, and allowing $p$ to be inert in $K$ for Kolyvagin's conjectures.

Methodology
The proofs of the main theorems (Theorem A and Theorem C) follow a descent strategy similar to [BCGS26], utilizing the relationship between Euler systems and Selmer groups. However, the paper introduces a significant methodological innovation in the computation of the $p$ -divisibility index of specialized cohomology classes.

Reformulation via Determinants: Instead of relying on $p$ -adic $L$ -functions and control theorems in the style of [Gre99], the authors utilize a reformulation of the Iwasawa Main Conjecture in terms of determinants of Selmer complexes, as introduced in [KS24].
$\Lambda$ -adic Kolyvagin Systems: The authors study $\Lambda$ -adic Kolyvagin systems derived from Kato's Euler system (for Kurihara) and Heegner points (for Kolyvagin), where $\Lambda$ is the Iwasawa algebra of the cyclotomic (or anticyclotomic) $\mathbb{Z}_p$ -extension.
Specialization and Descent: They consider the specialization of these systems at non-trivial characters $\alpha$ $α$ close to the trivial character. By combining:
1. An exact formula for the divisibility index of the specialized class derived from the Main Conjecture (involving Tamagawa factors of the Cartier dual).
2. An exact formula for the length of the strict Selmer group derived from Mazur–Rubin's structure theorem.
  The authors establish a link between the analytic invariants (Tamagawa numbers) and the algebraic invariants (divisibility of the Kolyvagin system).
Rigidity: A key technical component is a "rigidity" argument (Proposition 2.1.5 and its analogue) showing that the order of vanishing of the system is stable under restriction to subsets of primes, allowing the comparison between the $\Lambda$ -adic system and the classical system.

Key Contributions

Generalization of Reduction Types: The paper proves the refined Kurihara conjecture for elliptic curves with any reduction type at $p$ (good ordinary, good supersingular, or bad reduction), provided the surjectivity of the mod $p$ Galois representation and non-divisibility of the Manin constant by $p$ .
Inert Primes in Kolyvagin's Context: The paper proves the refined Kolyvagin conjecture for the case where $p$ is inert in the imaginary quadratic field $K$ , a case previously inaccessible via the $p$ -adic $L$ -function methods used in [BCGS26].
New Descent Framework: The paper establishes a framework for computing divisibility indices using determinants of Selmer complexes, bypassing the need for explicit $p$ -adic $L$ -functions in the descent step. This allows for the treatment of cases where $p$ -adic $L$ -functions are not yet fully developed or where the standard control theorems are difficult to apply.

Main Results

Theorem A (Kurihara): For an elliptic curve $E/\mathbb{Q}$ and a prime $p > 3$ satisfying the surjectivity and Manin constant conditions, the following are equivalent:
(i) The refined Kurihara conjecture holds (i.e., $M_\infty(\delta) = \text{ord}_p(\text{Tam}_E)$ ).
(ii) The Iwasawa Main Conjecture for the cyclotomic $\mathbb{Z}_p$ -extension (formulated in terms of determinants of arithmetic complexes) holds.
Corollary: If $p \nmid \text{Tam}_E$ , the nonvanishing of some $\bar{\delta}_n$ is equivalent to the Iwasawa Main Conjecture.
Theorem B: Consequently, the refined Kurihara conjecture is proven for $p$ good ordinary, and for $p$ good supersingular with squarefree conductor $N$ .
Theorem C (Kolyvagin): For an elliptic curve $E/\mathbb{Q}$ and a prime $p > 3$ satisfying the surjectivity and Manin constant conditions, assuming $E$ has good ordinary reduction at $p$ and $p$ is unramified in $K$ , the following are equivalent:
(i) The refined Kolyvagin conjecture holds (i.e., $M_\infty(\kappa^{\text{Heeg}}) = \text{ord}_p(\text{Tam}_E)$ ).
(ii) The anticyclotomic Iwasawa Main Conjecture for Heegner points (formulated in terms of determinants) holds.
Corollary: If $p$ splits in $K$ , Conjecture C holds. The result extends to the inert case under the assumption of the Main Conjecture.

Significance and Claims
The authors claim that their results provide a proof of Kurihara's conjecture and its refinement in many new cases, specifically removing the "non-anomalous" and "good reduction" hypotheses that previously limited the scope of these theorems. For Kolyvagin's conjecture, the paper provides the first proof of the refined conjecture in the case where $p$ is inert in $K$ , a scenario where previous approaches relying on Bertolini–Darmon–Prasanna $p$ -adic $L$ -functions were not applicable.

The paper emphasizes that the core innovation is the "new approach to the descent computations" based on the determinant formulation of the Iwasawa Main Conjecture. This approach allows the authors to control the $p$ -divisibility of special elements in Galois cohomology without relying on the specific analytic properties of $p$ -adic $L$ -functions, thereby broadening the applicability of the descent method to inert primes and various reduction types. The results are presented as conditional on the validity of the respective Iwasawa Main Conjectures, which are known to hold in the ordinary and certain supersingular cases due to prior work by Wan, BCS, CLW, and BSTW.