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The anticyclotomic main conjectures for elliptic curves

This paper proves the anticyclotomic Iwasawa main conjectures for rational elliptic curves over imaginary quadratic fields under mild arithmetic assumptions, covering both good ordinary and supersingular cases for the prime $p$ .

Original authors: Massimo Bertolini, Matteo Longo, Rodolfo Venerucci

Published 2026-02-06

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CC BY 4.0

Original authors: Massimo Bertolini, Matteo Longo, Rodolfo Venerucci

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, cosmic puzzle involving a specific type of number pattern called an elliptic curve. Mathematicians have long suspected that the "shape" of this curve (how many solutions it has) is secretly connected to a mysterious "sound" or "frequency" generated by a complex mathematical formula called an L-function. This connection is known as the Birch and Swinnerton-Dyer (BSD) conjecture.

This paper, written by Bertolini, Longo, and Venerucci, is about proving a specific, high-stakes version of this connection. They are looking at the problem through a special lens called Iwasawa theory, which studies how these patterns behave when you zoom out infinitely, layer by layer, like looking at a fractal.

Here is a simple breakdown of their journey and what they found:

1. The Two Worlds: Definite vs. Indefinite

The authors split their investigation into two different "worlds" or scenarios, depending on how the elliptic curve behaves with a specific prime number (let's call it $p$ ).

The Indefinite World (The Open Road): Here, the mathematical "road" is open. The authors use special points on a geometric shape called a Shimura curve (think of these as "landmarks" or "beacons") to build a bridge. They create a sequence of these landmarks that fit together perfectly across the infinite layers of their puzzle. They prove that the "sound" (the $p$ -adic L-function) generated by these landmarks perfectly matches the "shape" (the Selmer group, which counts the solutions) of the curve.
The Definite World (The Closed Room): Here, the road is closed off. You can't just walk along it; you have to use a different trick. The authors use a technique called "level raising." Imagine you have a musical instrument playing a specific tune. They find a way to add extra strings to the instrument (raising the level) to create a new, slightly different tune that is congruent (very similar) to the original one modulo a prime number. This new tune allows them to access the "landmarks" (Heegner points) they couldn't reach before. They then use these to prove the connection in this closed world too.

2. The "Supersingular" Twist

Usually, mathematicians assume the prime number $p$ behaves nicely (it's "ordinary"). But sometimes, $p$ behaves wildly (it's "supersingular").

In this wild scenario, the rules change. The authors had to invent two different versions of their "sound" and "shape" (labeled with a plus or minus sign, $\varepsilon = \pm$ ) to handle the chaos.
They successfully proved that even in this wild, supersingular case, the connection holds true, except for one very specific, tricky situation they call the "exceptional case." In this one rare scenario, the "sound" has a glitch (a zero) that makes it hard to prove the full connection, so they could only prove half of it.

3. The Main Achievement: The "Main Conjecture"

The central goal of the paper is to prove the Anticyclotomic Main Conjecture.

The Analogy: Imagine you have a lock (the algebraic structure of the curve's solutions) and a key (the analytic L-function). For decades, mathematicians have suspected the key fits the lock perfectly.
The Result: This paper proves that the key does fit the lock. Specifically, they show that the "key" (the $p$ -adic L-function) generates the "lock" (the characteristic ideal of the Selmer group).
The Catch: In almost all cases, the key fits perfectly (equality). In that one "exceptional" case mentioned earlier, they can only prove the key turns the lock (divisibility), but they can't yet prove it fits perfectly.

4. The "Reciprocity Laws"

To build their proof, the authors discovered two "laws of exchange" (Reciprocity Laws).

Think of these as a secret handshake between two different mathematical worlds.
Law 1: If you take a special point from the "indefinite" world and look at it from the "definite" world, its value tells you exactly what the "sound" (L-function) should be.
Law 2: Conversely, if you start in the "definite" world, you can translate that information back to the "indefinite" world to find the "sound."
These laws act as the glue holding their entire argument together, allowing them to move information back and forth between the two different scenarios to prove the main conjecture.

5. The Final Verdict: BSD Formulas

Once they proved the Main Conjecture, they could derive the Birch and Swinnerton-Dyer formulas for these specific layers of the puzzle.

What this means: They showed that if the "sound" (L-function) is loud (non-zero), then the number of solutions to the curve is finite and predictable. If the "sound" is silent (zero), the number of solutions grows in a specific, predictable way.
They provided a precise formula linking the "volume" of the sound to the "size" of the solution group, confirming a deep prediction about how numbers behave.

Summary

In short, Bertolini, Longo, and Venerucci have built a massive, unified bridge between two different ways of looking at elliptic curves. They proved that the analytic "music" of the curve perfectly predicts its algebraic "structure," even in difficult and chaotic scenarios, with only one tiny, mysterious exception left to solve. They did this by using clever tricks like "level raising" (adding strings to an instrument) and "reciprocity laws" (secret handshakes between worlds) to navigate the complex landscape of modern number theory.

Technical Summary: The Anticyclotomic Main Conjectures for Elliptic Curves

Problem Statement
This paper addresses the Iwasawa Main Conjectures for a modular elliptic curve $E/\mathbb{Q}$ over the anticyclotomic $\mathbb{Z}_p$ -extension $K_\infty$ of an imaginary quadratic field $K$ . The authors aim to prove the equality between the algebraic side (the characteristic ideal of the Pontryagin dual of the $p$ -primary Selmer group) and the analytic side (the anticyclotomic $p$ -adic $L$ -function) in two distinct settings:

The Definite Case: Where the functional equation of the complex $L$ -function has sign $-1$.
The Indefinite Case: Where the functional equation has sign $+1$ .

The work covers both the ordinary case (where $p$ is a good ordinary prime for $E$ ) and the supersingular case (where $p$ is a supersingular prime). In the supersingular setting, the authors distinguish between "non-exceptional" and "exceptional" cases (the latter occurring when $p$ is inert in $K$ and the sign $\varepsilon = +$ ), noting that the exceptional case involves an extra zero of local nature that complicates the full proof of the conjecture.

Methodology
The proof relies on a unified inductive approach based on the theory of bipartite Euler systems, extending the framework established by Bertolini–Darmon [BD05] and Howard [How04, How06]. The methodology proceeds through the following key technical steps:

Level Raising and Admissible Primes: The authors utilize the level-raising technique at "admissible primes" (primes $\ell$ satisfying specific congruence and splitting conditions relative to the modular form $f$ and the field $K$ ). By raising the level of the modular form at these primes, they construct congruent eigenforms that allow the transition between definite and indefinite settings.
Special Points and Heegner Classes:
- In the indefinite case, the authors use Heegner points on Shimura curves to define a norm-compatible sequence of cohomology classes.
- In the definite case, they utilize Gross points on Gross curves (associated with definite quaternion algebras).
- These points are used to construct global cohomology classes (Kolyvagin systems) in the Selmer groups.
Explicit Reciprocity Laws: The core of the argument involves two explicit reciprocity laws relating the cohomology classes constructed from the special points to the $p$ $p$ -adic $L$ $L$ -functions:
- The First Reciprocity Law relates the residue of a cohomology class at an admissible prime to the value of the $p$ -adic $L$ -function.
- The Second Reciprocity Law relates the finite part of a cohomology class to the $L$ -function of a level-raised form.
Control Theorems and Local Analysis: The paper establishes rigorous control theorems for Selmer groups over the anticyclotomic tower. A significant portion of the work (Section 5) is dedicated to analyzing the structure of $\varepsilon$ -Selmer groups in the supersingular case, relying on recent results regarding local points in $p$ -adic towers (specifically the work of Burungale–Kobayashi–Ota [BKO21] for the inert case).
Inductive Argument: The authors employ an inductive argument on the length of the Selmer groups. By comparing the Selmer groups of the original form $f$ with those of level-raised forms, they establish divisibility relations between the characteristic ideals and the $p$ -adic $L$ -functions.

Key Contributions and Results

Proof of the Main Conjectures (Theorem A):
The authors prove that the ideal generated by the anticyclotomic $p$ -adic $L$ -function $L_p^\varepsilon(f)$ is contained in the characteristic ideal of the Selmer group, with equality holding in the non-exceptional cases.
- Definite Case (DAMC): Proven for both ordinary and supersingular primes (non-exceptional).
- Indefinite Case (IAMC): Proven for both ordinary and supersingular primes (non-exceptional).
- Exceptional Case: In the case where $p$ is inert in $K$ and $\varepsilon = +$ , the authors establish one divisibility but note that the presence of an exceptional zero prevents the proof of the full equality with current methods.
$\varepsilon$ -BSD Formulae (Theorems B and C):
As a consequence of the Main Conjectures, the paper derives explicit Birch and Swinnerton-Dyer (BSD) formulae for the Selmer groups over finite layers of the anticyclotomic extension.
- Definite Case (Theorem B): Relates the length of the Selmer group to the order of vanishing of the complex $L$ -function $L(E/K, \chi, 1)$ .
- Indefinite Case (Theorem C): Relates the length of the Shafarevich–Tate group to the derivative $L'(E/K, \chi, 1)$ and a regulator term.
  These formulae hold with equality in the non-exceptional cases.
Refinement of the Inductive Procedure:
The paper refines the induction argument of Bertolini–Darmon [BD05] to handle the supersingular case uniformly. This includes a detailed analysis of the $\varepsilon$ -local points and the structure of the Selmer groups in the inert supersingular setting, incorporating the proof of Rubin's conjecture by Burungale–Kobayashi–Ota [BKO21].

Significance and Claims
The paper claims to provide a unified proof of the anticyclotomic Main Conjectures for elliptic curves, covering both ordinary and supersingular primes, and both definite and indefinite settings, under a standard set of hypotheses (e.g., $p \geq 5$ , surjectivity of the mod $p$ representation, and specific conditions on the conductor).

Unification: The authors emphasize that the DAMC and IAMC are derived from the same inductive process based on bipartite Euler systems, rather than treating them as separate problems.
Supersingular Extension: The work extends the theory of anticyclotomic Main Conjectures to the supersingular case, a setting that had previously been treated partially (e.g., by Darmon–Iovita [DI08] for split $p$ ) or required different techniques.
Modesty on the Exceptional Case: The authors explicitly state that in the "exceptional" supersingular case (inert $p$ , $\varepsilon=+$ ), the presence of a local zero limits the result to one divisibility. They do not claim to have resolved the full conjecture in this specific subcase, noting that it requires further investigation and potentially new ideas.
Reliance on Deep Theorems: The proof of the non-vanishing of $L$ -values modulo $p$ (crucial for the inductive step in the definite case) relies on the $p$ -converse theorem of Skinner–Urban [SU14] and related results by Fouquet–Wan.

The paper does not propose new experimental applications or future directions beyond the mathematical resolution of the conjectures and the derivation of the BSD formulae. Its primary contribution is the rigorous establishment of the link between the analytic $p$ -adic $L$ -functions and the algebraic Selmer groups in the anticyclotomic setting.