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Lacunary recurrences and 2-adic properties of Eisenstein series

This paper proves a conjecture providing an exact formula for the minimal 2-adic valuation of the rational coefficients in the polynomial expansion of Eisenstein series $G_k$ in terms of $G_4$ and $G_6$ , utilizing lacunary recurrences to relate these valuations to the binary expansion of the weight.

Original authors: Liubomir Chiriac, Andrei Jorza

Published 2026-05-12

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Liubomir Chiriac, Andrei Jorza

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 a master chef trying to recreate a complex, high-level dish (let's call it Eisenstein Series $G_k$ ) using only two basic, pre-made ingredients: $G_4$ and $G_6$ .

In the world of advanced mathematics, these "ingredients" are special functions called Eisenstein series. The paper by Chiriac and Jorza asks a very specific question: When you mix these two ingredients to make a new dish of a specific "weight" (size), how "pure" or "messy" are the amounts of each ingredient you need?

Specifically, they are looking at the numbers (coefficients) that tell you how much of $G_4$ and $G_6$ to use. They want to know: What is the "cleanest" possible number you can get when you do this mixing, specifically looking at how many times the number 2 divides into it?

The Core Mystery: The "Binary" Recipe

To understand the answer, you have to look at the "weight" of the dish (the number $k$ ) in binary code (the language of computers, using only 0s and 1s).

The authors prove a rule about the "messiness" of the recipe numbers:

If the weight $k$ is a perfect power of 2 (like 4, 8, 16, 32... which in binary look like 100, 1000, 10000), the recipe is incredibly clean. You can find a mix where the numbers are "odd" (not divisible by 2 at all). The messiness score is 0.
If the weight $k$ is NOT a power of 2, the recipe is inevitably a bit messier. The messiness depends exactly on how many 1s are in the binary version of $k$ $k$ .
- The formula is: (Number of 1s in binary $k$ ) minus 2.

For example, if your weight is 44 (which is 101100 in binary, containing three 1s), the messiness score is $3 - 2 = 1$ .

How They Solved It: The "Sparse" Shortcut

The authors didn't just guess this; they used a clever mathematical tool called "Lacunary Recurrences."

Think of a standard way to calculate these recipes as a massive, crowded warehouse where you have to check thousands of boxes to find the right ingredients. It's slow and confusing.

However, the authors used a special "sparse" map (the lacunary recurrence). This map is like a treasure hunt with very few clues. Instead of checking every single box, the map tells you that the answer only hides in a few specific, widely spaced locations.

Because the map is so "sparse" (empty in most places), the authors could easily spot the specific ingredient combinations that would result in the "cleanest" numbers.
They found that the "gaps" in the map naturally force the numbers to behave in a predictable way regarding the number 2.

The Three Scenarios

The paper breaks the problem down into three types of weights, like sorting ingredients into three different bins:

Bin 1: Powers of 2.
Here, the math is straightforward. The authors showed that you can always find a "pure" mix (score 0) by looking at how the ingredients stack up in a specific pattern.
Bin 2: Weights that leave a remainder of 2 when divided by 6.
Here, they used a special identity (a mathematical shortcut) discovered by a researcher named Romik. It's like finding a secret tunnel in the warehouse. This tunnel connects the big dish to smaller, simpler dishes. By analyzing the "carry-over" effects (like when you add numbers and a digit rolls over from 9 to 10), they proved the messiness score matches their formula.
Bin 3: Weights that leave remainders of 0 or 4 when divided by 6.
These were slightly more complex, requiring a different set of shortcuts. The authors had to carefully pair up terms (like matching socks) to ensure that the "messy" parts canceled each other out or added up in a way that confirmed their rule.

The Big Picture

Before this paper, a researcher named Gonz´alez had guessed this rule and checked it for thousands of examples, but no one knew why it was true or if it held for every single number.

Chiriac and Jorza provided the proof. They showed that the structure of these mathematical "recipes" is deeply tied to the binary code of the numbers themselves. The "sparsity" of the recurrence relations (the fact that most terms are zero or missing) acts like a filter, ensuring that the "cleanest" possible number always follows the pattern of counting the 1s in the binary code.

In short: They proved that the "messiness" of mixing these mathematical ingredients is not random; it is a direct reflection of the binary shape of the weight you are trying to create.

Technical Summary: Lacunary Recurrences and 2-adic Properties of Eisenstein Series

Problem Statement
The paper investigates the rational coefficients $w(a, k)$ arising from the expansion of the Eisenstein series $G_k$ (normalized with constant term $2\zeta(k)$ ) as a polynomial in the generators $G_4$ and $G_6$ . Specifically, the authors seek to determine the minimal 2-adic valuation, denoted $v_2$ , among the set of coefficients $\{w(a, k)\}$ for a given even weight $k \geq 4$ . The central question is how this minimal valuation relates to the binary expansion of the weight $k$ .

Methodology
The proof relies on a combinatorial approach utilizing lacunary recurrences for Eisenstein series. The authors employ specific recurrence relations discovered by Romik [Rom17] and generalized by Mertens and Rolen [MR15]. These recurrences are termed "lacunary" because they express higher-weight Eisenstein series as sparse products of lower-weight series, rather than dense sums.

The methodology proceeds by induction on the weight $k$ :

Base Case (Powers of 2): The authors first address weights $k$ that are powers of 2. They utilize a classical recurrence relation to isolate the coefficient of pure powers of $G_4$ . By analyzing the 2-adic valuation of the terms in this recurrence and pairing symmetric summands, they demonstrate that a valuation of 0 is achievable.
Inductive Step (Non-Powers of 2): For weights $k$ $k$ that are not powers of 2, the proof is divided into three cases based on the congruence of $k$ $k$ modulo 6 ( $k \equiv 2, 0, 4 \pmod 6$ $k \equiv 2, 0, 4 (mod 6)$ ).
- For each case, the authors apply the corresponding Romik/Mertens-Rolen identity to express $G_k$ as a sum of products $G_{2n+2j}G_{4n-2j+c}$ .
- They analyze the 2-adic valuation of the coefficients in these identities using the property $v_2(m!) = m - s(m)$ , where $s(m)$ is the number of ones in the binary expansion of $m$ .
- The authors exploit the symmetry of the summation indices to pair terms, ensuring that the valuation of paired terms is strictly positive or bounded below by the inductive hypothesis.
- Crucially, they identify specific indices where one factor in the product corresponds to a weight that is a power of 2 (yielding a minimal valuation of 0) and the other is not. This allows them to construct a specific coefficient $w(a, k)$ that achieves the theoretical lower bound.

Key Contributions and Results
The primary result is the proof of a conjecture by Gonz´alez [Gon22], which provides an exact formula for the minimal 2-adic valuation of the coefficients $w(a, k)$ .

Theorem 1: For every even $k \geq 4$ , the minimal 2-adic valuation is given by:
$\min_a v_2(w(a, k)) = \begin{cases} 0 & \text{if } k \text{ is a power of 2}, \\ s(k) - 2 & \text{if } k \text{ is not a power of 2}. \end{cases}$
where $s(k)$ denotes the number of ones in the binary expansion of $k$ .

The paper establishes that:

Gonz´alez's previous result, which proved the universal lower bound $\min v_2(w(a, k)) \geq 0$ , is a special case of this theorem.
The exact minimum depends precisely on the binary structure of the weight.
The proof explicitly constructs coefficients that achieve this minimum, rather than merely proving existence.

Significance
The authors claim this work represents the first application of lacunary recurrences for Eisenstein series to solve a problem regarding the arithmetic properties of their coefficients. The inherent sparsity of these recurrences, combined with the high degree of coefficient symmetry, allows the authors to bypass the difficulties associated with dense recurrences (where identifying specific minimal terms is obscured by the sheer number of summands).

While the paper notes that computational evidence suggests the pure $G_6$ term coefficient $w(0, 6n)$ may also attain the minimal valuation, the authors state they have not pursued this observation further, maintaining a focus on the general proof of the minimal valuation formula.

The Core Mystery: The "Binary" Recipe

How They Solved It: The "Sparse" Shortcut

The Three Scenarios

The Big Picture

Technical Summary: Lacunary Recurrences and 2-adic Properties of Eisenstein Series

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