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Elliptic curves with rank one and nontrivial 2-part of Tate Shafarevich groups over the $\mathbb{Z}_2$ -extension of $\mathbb{Q}$

This paper constructs an elliptic curve and a family of its quadratic twists that simultaneously possess analytic and algebraic rank one over the cyclotomic $\mathbb{Z}_2$ -extension of $\mathbb{Q}$ while having an infinite Tate-Shafarevich group, achieved by analyzing the 2-adic properties of Mazur-Tate modular elements via Heegner point congruences and an equivariant Coates-Wiles theorem.

Original authors: Li-Tong Deng, Yong-Xiong Li

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Li-Tong Deng, Yong-Xiong Li

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

The Big Picture: A Mathematical Treasure Hunt

Imagine you are a detective trying to solve a mystery about a very special kind of shape called an Elliptic Curve. In the world of math, these aren't just drawings; they are complex equations that act like hidden treasure maps.

The paper by Li-Tong Deng and Yong-Xiong Li is about finding a specific type of "treasure" (mathematical solutions) on these maps, but with a twist: they are looking for these treasures not just on one map, but on an infinite family of maps that get bigger and more complex every time you zoom in.

Here is the breakdown of their adventure:

1. The Characters: The Curve and the Twist

The Main Character (The Curve):
Think of the elliptic curve $E$ as a unique, rigid rollercoaster track defined by a specific equation ( $y^2 + y = x^3 + 2$ ). This track exists on a flat plain (the rational numbers, $\mathbb{Q}$ ).

The "Rank": This is like asking, "How many independent loops can a rollercoaster car make on this track before it gets stuck?"
- If the rank is 0, the car is stuck; there are no new loops.
- If the rank is 1, there is exactly one "master loop" that generates all other possible paths.
- If the rank is 2 or more, there are multiple master loops.

The "Twist":
The authors take this rollercoaster and apply a "magic filter" called a quadratic twist. Imagine taking the track and stretching or flipping it based on a specific number (let's call it $m$ ). This creates a new version of the track, $E(m)$ .

The goal? To find a specific number $m$ (made of two special prime numbers) such that when you twist the track, it has exactly one master loop (Rank 1).

2. The Setting: The Infinite Tower

Usually, mathematicians study these tracks on a single flat plain. But this paper looks at a tower of worlds.

Imagine a tower where the ground floor is our normal world ( $\mathbb{Q}$ ).
The next floor up is a slightly bigger world ( $\mathbb{Q}_1$ ), then a bigger one ( $\mathbb{Q}_2$ ), and so on, forever.
As you go up the tower, the world gets more complex, and the "rollercoaster" might gain new loops or lose them.

The authors wanted to know: Can we find a twisted track that has exactly one master loop on every single floor of this infinite tower?

3. The Mystery: The "Ghost" Solutions (Tate-Shafarevich Group)

This is the most exciting part. In math, sometimes you can prove a solution should exist based on the shape of the curve (the "analytic" side), but you can't actually find the solution (the "algebraic" side).

The Analogy: Imagine you have a map that says, "There is a treasure chest here." You dig and dig, but you find nothing. You know the map is right (the math checks out), but the chest is invisible.
The Tate-Shafarevich Group: This is the mathematical name for these "invisible" or "ghost" solutions.
- If this group is trivial (empty), it means: "If the map says there's a solution, we can find it."
- If this group is nontrivial (infinite), it means: "There are infinitely many ghost solutions that we can't find, even though we know they exist."

The Paper's Big Discovery:
The authors found a specific twisted track ( $E(m)$ ) where:

Rank 1: There is exactly one real, findable master loop on every floor of the tower.
Infinite Ghosts: There are also infinitely many ghost solutions hiding in the shadows on the top floors of the tower.

This is a huge deal because it proves that you can have a "perfectly simple" curve (Rank 1) that is simultaneously haunted by an infinite number of invisible ghosts.

4. How They Solved It: The Detective's Toolkit

How did they prove this? They used three main tools, which we can think of as:

The "Mazur-Tate Elements" (The Magic Compass):
These are special mathematical numbers that act like a compass. They point toward the "ghosts." The authors had to calibrate this compass very carefully. They used a technique involving "congruences" (like checking if numbers leave the same remainder when divided by 2) to ensure the compass pointed exactly where they needed it to, even when the numbers got huge.
The "Heegner Points" (The Real Treasure):
These are specific points on the curve that are guaranteed to exist under certain conditions. The authors used a famous theorem (Gross-Zagier) to show that for their specific twisted curve, there is at least one real, non-ghost point. This proved the Rank is 1.
The "Coates-Wiles Theorem" (The Logic Bridge):
This is a rule that connects the "ghosts" to the "real points." The authors used an "equivariant" version of this rule (a version that works perfectly across the whole infinite tower) to prove that if the real points behave a certain way, the ghosts must be infinite.

5. Why Does This Matter?

Before this paper, mathematicians knew examples of curves with Rank 1, and they knew examples of curves with infinite ghosts. But finding a curve that has both simultaneously, across an infinite tower of number fields, was a major open question.

The Takeaway:
Deng and Li built a mathematical "rollercoaster" that:

Has exactly one real path (Rank 1).
Is haunted by an infinite number of invisible paths (Infinite Tate-Shafarevich group).
Does this consistently as you travel up an infinite tower of worlds.

They solved a puzzle that had stumped mathematicians for a long time, showing that the universe of elliptic curves is even stranger and more complex than we thought: you can have a simple structure that is simultaneously infinitely complex in its hidden depths.

1. Problem Statement and Motivation

The paper addresses a specific question in arithmetic geometry regarding the behavior of elliptic curves over infinite towers of number fields, specifically the cyclotomic $\mathbb{Z}_2$ -extension of $\mathbb{Q}$ , denoted as $Q_\infty$ .

Core Question:
Does there exist an elliptic curve $E/\mathbb{Q}$ and an infinite family of quadratic twists $E(m)$ such that:

The analytic and algebraic ranks of $E(m)$ over the intermediate fields $Q_n$ (where $[Q_n:\mathbb{Q}]=2^n$ ) are exactly 1.
The 2-primary part of the Tate-Shafarevich group, $\text{X}(E(m)/Q_n)(2)$ , is nontrivial (and in the limit over $Q_\infty$ , infinite).

Context:
Previous results by Kurihara and others had established similar phenomena for rank 0 or for specific primes $p$ (often odd primes or under the assumption of the finiteness of the Tate-Shafarevich group). This paper aims to resolve the case for rank 1 and $p=2$ without assuming the Birch and Swinnerton-Dyer (BSD) conjecture or the finiteness of the Tate-Shafarevich group.

2. Methodology

The authors employ a sophisticated combination of Iwasawa theory, modular symbols, and the theory of Heegner points. The methodology is structured around three main pillars:

A. Mazur-Tate Modular Elements and Congruences

The central tool is the study of Mazur-Tate elements associated with the elliptic curve and its quadratic twists.

Construction: They define modular elements $\xi_{Q_n}$ in the group ring $\mathbb{Z}_2[\text{Gal}(Q_n/\mathbb{Q})]$ .
Specialization: They analyze the 2-adic valuations of these elements when specialized to finite-order characters $\rho$ of the Galois group.
The Challenge: Standard methods (Kurihara/Pollack) work well for the base curve $E$ but fail for the quadratic twist $E(m)$ .
The Solution: The authors introduce a new modular element $\xi_E(\chi)$ associated with the pair $(E, \chi)$ , where $\chi$ is the quadratic character of $\mathbb{Q}(\sqrt{m})$ . They prove that the specializations of $\xi_E(\chi)$ are congruent to the Mazur-Tate elements of $E(m)$ modulo 2-adic units.
Higher Congruences: For the exceptional case of primitive characters of $\text{Gal}(Q_3/\mathbb{Q})$ , they derive higher congruence relations (modulo 4) using Hecke actions on modular symbols to determine the precise 2-adic valuation.

B. Iwasawa Theory and Equivariant Coates-Wiles Theorem

They utilize an equivariant version of the Coates-Wiles theorem for elliptic curves with Complex Multiplication (CM).
This allows them to relate the non-vanishing of $L$ -values (analytic rank) to the structure of the Selmer group (algebraic rank) over the $\mathbb{Z}_2$ -extension.
They establish that the 2-power Selmer group of $E(m)$ over $Q_\infty$ is free of rank one over the Iwasawa algebra $\Lambda = \mathbb{Z}_2[[\Gamma]]$ .

C. Heegner Points and Non-Torsionness

To prove the existence of a point of infinite order (rank $\ge 1$ ), they use the Gross-Zagier theorem and Kolyvagin's theorem.
They construct a specific Heegner point associated with $E(-3q)$ and the field $\mathbb{Q}(\sqrt{-2p})$ .
Using the $p$ -adic Waldspurger formula (Bertolini-Darmon-Prasanna) and congruence properties of Heegner points, they prove that this point is non-torsion under specific conditions on the primes $p$ and $q$ .

3. Key Contributions

Resolution of the Rank 1, $p=2$ Case: The paper provides the first unconditional construction of an elliptic curve and a family of quadratic twists with rank 1 over the cyclotomic $\mathbb{Z}_2$ -extension where the Tate-Shafarevich group is infinite.
New Modular Element Technique: The authors successfully extend the method of Kurihara to handle quadratic twists at the prime 2 by introducing the auxiliary modular element $\xi_E(\chi)$ and establishing congruence relations modulo 2 and 4. This overcomes the failure of direct application of existing methods to twisted curves.
Generalization: The result generalizes previous findings from a fixed curve or fixed field to an infinite family of number fields ( $Q_n$ ) with degrees tending to infinity, and to a family of quadratic twists.

4. Main Results

Theorem 1.3 (Main Theorem):
Let $E$ be the elliptic curve defined by $y^2 + y = x^3 + 2$ (conductor 243, CM by $\mathbb{Q}(\sqrt{-3})$ ). Let $p$ and $q$ be distinct primes satisfying:

$p \equiv q \equiv 7 \pmod{12}$ .
The Fourier coefficients $a_p$ and $a_q$ are odd.
$\left(\frac{-2p}{q}\right) = 1$ .
$3 \nmid h(-q)h(-6pq)$ (where $h$ is the class number).

Let $m = pq$. Then:

(i) Over $\mathbb{Q}$ , the rank of $E(m)(\mathbb{Q})$ is 0, and the analytic rank is 0.
(ii) For every $n \ge 1$ $n \geq 1$ , over the field $Q_n$ $Q_{n}$ :
- The analytic rank is 1: $\text{ord}_{s=1} L(E(m)/Q_n, s) = 1$ .
- The algebraic rank is 1: $\text{rank}(E(m)(Q_n)) = 1$ .
- The Tate-Shafarevich group $\text{X}(E(m)/Q_\infty)(2)$ is infinite (both its order and 2-rank are infinite).

Numerical Examples:
The authors provide numerical data (Appendix C) showing infinitely many such pairs $(p, q)$ exist (e.g., $p=31, q=7$ ), confirming the theoretical conditions are not vacuous.

5. Significance

Unconditional Proof: The result holds without assuming the BSD conjecture or the finiteness of the Tate-Shafarevich group, which are major open problems in number theory.
Understanding $p=2$ Phenomena: The prime 2 is notoriously difficult in Iwasawa theory due to supersingular reduction and the lack of a simple control theorem. This paper demonstrates a robust method to handle 2-adic properties in the context of quadratic twists.
Infinite Families: By showing that the rank remains 1 and the Sha group becomes infinite as the degree of the field extension grows, the paper provides deep insight into the asymptotic behavior of arithmetic invariants in $\mathbb{Z}_p$ -extensions.
Methodological Advancement: The technique of using "twisted" modular elements to bypass the limitations of standard Kurihara methods opens new avenues for studying other arithmetic problems involving twists and Iwasawa invariants.

In summary, this paper successfully constructs a concrete family of elliptic curves that exhibit the "rank 1, infinite Sha" phenomenon over the cyclotomic $\mathbb{Z}_2$ -extension, utilizing a novel blend of modular symbol congruences and Heegner point theory.