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Determinants of Mahler measures and special values of $L$ -functions

This paper establishes general formulas expressing the Mahler measures of two specific families of bivariate polynomials in terms of $L$ -values of cusp forms when their zero loci define CM elliptic curves, and proves that determinants of matrices formed by the Mahler measures of their Galois conjugates in totally real number fields of degrees 2 and 4 are non-zero rational multiples of the corresponding $n$ -th derivatives of these $L$ -functions at zero.

Original authors: Detchat Samart, Zhengyu Tao

Published 2026-07-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Detchat Samart, Zhengyu Tao

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 detective trying to solve a mystery where two completely different worlds are secretly speaking the same language. On one side, you have the world of geometry and shapes, specifically a special kind of curved surface called an elliptic curve. Think of these as intricate, doughnut-shaped loops that mathematicians love to study because they hold secrets about numbers. On the other side, you have the world of averages and measurements, specifically something called a "Mahler measure." If you take a complex mathematical formula and ask, "What is the average size of this formula if you plug in every possible number on a circle?" the answer you get is its Mahler measure. It's like measuring the average volume of a strange, multi-dimensional balloon.

For a long time, mathematicians suspected that these two worlds—the shape of the doughnut and the average size of the formula—were connected by a hidden bridge. This bridge is built from L-functions, which are like magical scorecards that record the hidden patterns of numbers. The big question was: Can we prove that the average size of the formula (the Mahler measure) is exactly equal to a specific number on the scorecard (a special value of an L-function)? This isn't just a game; solving it helps us understand the deep, fundamental structure of numbers themselves, linking geometry, algebra, and analysis in a way that feels almost like magic.

In this paper, the authors, Detchat Samart and Zhengyu Tao, act as master cartographers, drawing a detailed map of this hidden bridge for two specific families of mathematical formulas. They focus on two famous "families" of equations, let's call them the Plus-Family and the Cube-Family. These families are special because when you tweak a single number (called a parameter $t$ ) inside them, the shape of the resulting curve changes. The authors are particularly interested in the moments when these curves have a special kind of symmetry called Complex Multiplication (CM). You can think of CM as the curve having a secret "super-power" that makes it behave in a very orderly, predictable way, almost like a clockwork mechanism.

The team's first major task was to find every single possible value of the parameter $t$ that gives these curves their super-power, but only for numbers that live in "small" worlds (mathematical fields with a degree of 4 or less). They built a clever computer algorithm to hunt these values down, much like a metal detector scanning a beach for gold. They found a treasure trove: dozens of specific numbers, including some that are just simple integers, some that involve square roots (like $8 + 6\sqrt{2}$ ), and even some complex quartic numbers. They organized these findings into neat lists, essentially creating a "Who's Who" of these special curves.

Once they had their list of special curves, they tackled the main mystery: What is the exact relationship between the Mahler measure and the L-function?

For the simpler cases (where the number system is just a simple extension of the real numbers, like a quadratic field), they proved that if you take the Mahler measures of the curve and its "twins" (curves that look slightly different but are mathematically related), and arrange them into a 2x2 grid, the determinant of that grid (a specific calculation you do with the numbers in the grid) equals a special value of the L-function. It's like saying if you mix the ingredients of the curve's average size in a specific recipe, the result is exactly the secret code on the L-function scorecard.

They took this a step further for the more complex, 4-dimensional number systems. Here, they arranged the Mahler measures of four related curves into a massive 4x4 grid. They proved that the determinant of this giant grid is also a non-zero rational multiple of a special L-value. In other words, even in these complicated, high-dimensional worlds, the average size of the formula and the L-function scorecard are locked together by a precise, unbreakable mathematical chain.

The authors didn't just guess these connections; they proved them. They used a mix of deep theoretical math and heavy-duty computation. They used powerful computer software to calculate the values of these measures and L-functions to hundreds of decimal places to spot the patterns (using a tool called PSLQ, which is like a super-smart pattern recognizer). Once the pattern was spotted, they used rigorous mathematical logic to prove that the pattern holds true forever, not just for the numbers they checked.

However, the paper is also honest about what it doesn't do. The authors explicitly state that they are not trying to translate their results into the language of "K-theory" or "Beilinson's conjectures," which are the grand, high-level theories that predict these connections should exist. They are focusing purely on the computational aspect: finding the exact formulas and proving the determinants are non-zero. They also clarify that their results rely on the curves having that special "super-power" (CM). If the curves don't have CM, the simple formulas they found might not work, and the bridge might look very different.

In summary, this paper is a triumph of calculation and classification. It takes two mysterious mathematical concepts, finds every special case where they align perfectly in small number systems, and writes down the exact equations that link them. It confirms that for these specific, symmetric curves, the average size of the formula is indeed a direct reflection of the hidden number patterns in the L-function, providing a concrete, verified map of a territory that was previously only a theory.

Technical Summary: Determinants of Mahler Measures and Special Values of L-Functions

Problem Statement
The paper investigates the relationship between the Mahler measures of two specific families of bivariate polynomials, $P_t = x + x^{-1} + y + y^{-1} + \sqrt{t}$ and $Q_t = x^3 + y^3 + 1 - \sqrt[3]{t}xy$ , and the special values of $L$ -functions associated with the elliptic curves defined by their zero loci. While numerical evidence (Boyd, Rodriguez Villegas) suggests that for rational $t$ yielding Complex Multiplication (CM) curves, the Mahler measure equals a rational multiple of the derivative of the $L$ -function at $s=0$ (or related special values), rigorous proofs are scarce. The authors address two primary problems:

Classification: Determine all parameters $t$ in number fields $K$ of degree $n \leq 4$ for which the associated elliptic curves ( $F_t$ for $P_t$ and $C_t$ for $Q_t$ ) possess Complex Multiplication.
Determinantal Identities: For totally real number fields $K$ of degree $n=2$ and $n=4$ , establish explicit identities where the determinant of an $n \times n$ matrix (entries being Mahler measures of Galois conjugates of $t$ ) equals a non-zero rational multiple of the $n$ -th derivative of the $L$ -function of the curve at $s=0$ (or the value at $s=2$ ).

Methodology
The authors employ a computational and theoretical framework combining modular forms, CM theory, and lattice reduction algorithms:

CM Classification Algorithm: They adapt algorithms from previous works (Guo et al., Samart et al.) to exhaustively list CM points in the fundamental domains of congruence subgroups $\Gamma_0(4)$ and $\Gamma_0(3)$ . By utilizing modular equations and Hauptmoduls ( $t_P(\tau)$ and $t_Q(\tau)$ ), they map CM points $\tau$ to algebraic parameters $t$ with degree $\leq 4$ . This relies on the fact that the degree of the $j$ -invariant of a CM point corresponds to the class number of the associated imaginary quadratic field.
Modular Form Decomposition: For each identified CM curve, the authors utilize the fact that the $L$ -function of a CM elliptic curve over a number field $K$ factors into a product of $L$ -functions of newforms with CM. They explicitly identify these newforms using the LMFDB (L-functions and Modular Forms Database) and PARI/GP.
Theta Series and Mahler Measures: Using Proposition 5.1, they express the Mahler measures $\mu(t)$ and $\nu(t)$ as linear combinations of $L$ -values of specific theta series (cusp forms) evaluated at CM points. These theta series are constructed from quadratic forms derived from the CM lattice.
PSLQ Algorithm: To discover the specific linear relations between the Mahler measures and the $L$ -values, the authors use the PSLQ algorithm (integer relation detection) with high-precision numerical data. They compute Mahler measures via hypergeometric formulas and $L$ -values via PARI/GP.
Determinant Construction: By arranging the Mahler measures of Galois conjugates into specific matrix structures (often related to the regulator pairing in Beilinson's conjecture), they demonstrate that the determinant of these matrices simplifies to a product of the underlying newform $L$ -values, which corresponds to the $L$ -function of the elliptic curve over $K$ .

Key Contributions and Results

Complete Classification of CM Parameters: The paper provides a complete list of parameters $t$ in number fields of degree up to 4 for which $F_t$ and $C_t$ are CM curves.
- For $F_t$ : 3 rational integers, 12 pairs of real quadratic integers, 4 pairs of imaginary quadratic integers, 24 Galois orbits of totally real quartic integers, and 24 conjugate orbits of non-totally real quartic integers.
- For $C_t$ : 3 rational integers, 15 pairs of real quadratic integers, 2 pairs of imaginary quadratic integers, 4 conjugate orbits of non-totally real cubic integers, 24 Galois orbits of totally real quartic integers, and 25 conjugate orbits of non-totally real quartic integers.
- Data is provided in Tables 1 and 2.
General Formulas for Mahler Measures: The authors derive general formulas (Proposition 5.1) expressing $\mu(t)$ and $\nu(t)$ in terms of $L$ -values of cusp forms for any CM parameter $t$ . This generalizes previous results known only for rational $t$ .
Determinantal Identities for Totally Real Fields:
- Quadratic Case ( $n=2$ ): For 12 pairs of real quadratic parameters for $F_t$ and 15 for $C_t$ , the authors prove identities of the form:
  $\det \begin{pmatrix} a\mu(t) & b\mu(t^\sigma) \\ -\mu(t^\sigma) & \mu(t) \end{pmatrix} = r \pi^{-4} L(F_t, 2) = s L''(F_t, 0)$
  (and analogous forms for $C_t$ ). These are listed in Table 1.
- Quartic Case ( $n=4$ ): For 24 Galois orbits of totally real quartic parameters for both families, the authors establish $4 \times 4$ determinant identities. These determinants, involving Mahler measures of the four Galois conjugates, equal rational multiples of $L^{(4)}(F_t, 0)$ (or $L^{(4)}(C_t, 0)$ ). These are listed in Table 3.
- Notably, for parameters where the cube root of $t$ falls within a specific region $K_Q^\circ$ (specifically $t \in (-1, 27)$ ), the authors utilize a "modified Mahler measure" $\tilde{n}(t)$ to maintain the validity of the identities.
Non-Totally Real Cases: The paper also provides formulas expressing Mahler measures for non-totally real $t$ (cubic and quartic) as linear combinations of $L$ -values of newforms (Theorems 5.6 and 5.8), though these do not yield the same determinant identities as the totally real cases.

Significance and Claims
The paper claims to provide a "complete solution" to the classification of CM parameters for these families up to degree 4 and to establish a comprehensive set of new determinantal identities linking Mahler measures to special $L$ -values.

Computational Focus: The authors explicitly state they do not prove these identities via the regulator approach or by translating them into the language of Beilinson's conjectures (which would require constructing explicit elements in $K_2$ ). Instead, they focus on the "computational aspects," verifying that the determinants of Mahler measures match the expected $L$ -values up to rational constants.
Novelty: The authors assert that none of the $4 \times 4$ determinant identities for the families $F_t$ and $C_t$ listed in their tables were previously known in the literature.
Limitations: The results are restricted to totally real number fields for the determinant identities. The authors note that for non-totally real fields, the Galois conjugates of the Mahler measures are not linearly independent in the same way, preventing the formation of similar determinant identities. They also observe that for non-CM curves, such identities are not guaranteed, though they offer a conjecture for $Q$ -curves in the quadratic case based on numerical evidence.

The work serves as a significant extension of previous results (e.g., by Boyd, Rodriguez Villegas, Rogers, and the authors' own prior work) by moving from rational parameters to higher-degree number fields and providing a systematic computational verification of the relationship between Mahler measures and $L$ -functions in these contexts.

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