Toroidal families and averages of $L$-functions, II: cubic moments

This paper generalizes previous work on toroidal averages by studying the cubic moments of special values of $L$-functions for Dirichlet characters of prime modulus, while highlighting connections to bilinear forms of trace functions and the number of solutions to monoidal equations in finite fields.

The Gist

Imagine you are a detective trying to solve a massive mystery involving numbers. Specifically, you are looking at a special family of functions called L-functions. These functions are like musical instruments; when you play them at a specific note (the "center" of their range, called $s=1/2$), they produce a value. Sometimes this value is zero (a silent note), and sometimes it's a loud, non-zero number.

The big question in number theory is: How often do these instruments play a loud note? If they are silent too often, it breaks the harmony of the mathematical universe.

This paper, written by four mathematicians, is the sequel to a previous investigation. Here, they are looking at a "cubic moment." In plain English, instead of listening to one instrument, or a pair of instruments, they are listening to three instruments playing together and multiplying their sounds.

$$\text{Sound} = L(1/2, \chi^a) \times L(1/2, \chi^b) \times L(1/2, \chi^c)$$

Here, $\chi$ is a "character," which you can think of as a specific tuning or setting for the instrument. The authors want to know: If we try every possible tuning (for a large prime number $q$), what is the average volume of this triple sound?

The Main Discovery: The "Galant" vs. The "Oxozonic"

The authors found that the answer depends entirely on the relationship between the numbers $a$, $b$, and $c$. They invented some whimsical names for these relationships, borrowing from French and perhaps a bit of whimsy:

1. The "Galant" Trio: If the numbers $a, b, c$ are in a "good" relationship (which they call galant), the average volume is a steady, predictable, non-zero number. It's like a choir singing a perfect chord. The math proves that for these triplets, the instruments almost never all go silent at the same time.
2. The "Oxozonic" Trio: These are special, slightly more complex relationships (like the numbers 1, 1, and 2). Even here, the average volume is non-zero, though the math is trickier.
3. The "Sulfatic" and "Induced" Cases: These are the "bad" or "degenerate" relationships (like 1, 1, -2). The paper admits they haven't fully solved the mystery for these yet; they are the "unsolved cases" left for future detectives.

The Big Takeaway: For almost all combinations of these three numbers, the average product is positive. This implies that there are many tunings where the product is not zero. In fact, they prove that a huge number of these "tunings" result in a non-zero sound.

How Did They Solve It? (The Detective's Toolkit)

To find this average, the authors had to use some very high-tech mathematical tools. Here is how they did it, using analogies:

1. The "Approximate Functional Equation" (The Recipe)

Calculating the exact value of an L-function is like trying to weigh a ghost; it's hard to catch. The authors used a "recipe" (called an Approximate Functional Equation) that turns the ghostly L-function into a sum of many small, manageable numbers. It's like turning a complex song into a list of individual notes that are easier to count.

2. The "Trace Functions" and "Sheaves" (The Hidden Patterns)

When they added up all these notes, they ran into a problem: the numbers were dancing around in a chaotic way. To tame them, they used a branch of math called $\ell$-adic cohomology.

• The Analogy: Imagine the numbers are dancers on a stage. The authors realized these dancers weren't just moving randomly; they were following the choreography of a hidden geometric shape (a "sheaf").
• By understanding the shape of this hidden geometry, they could predict how the dancers would move. If the shape was "Galant" (a simple, strong shape), the dancers moved in a way that cancelled out the chaos, leaving a clean, predictable average.

3. Counting Solutions (The Puzzle)

A huge part of the proof involved counting how many ways you can solve a specific puzzle:
$$l^a \cdot m^b \cdot n^c \equiv \text{something} \pmod q$$
Think of this as trying to find how many ways you can combine three different types of Lego blocks to build a tower of a specific height, but with a twist: the blocks have to fit a specific modular pattern.

• The authors had to prove that for "Galant" triplets, the number of solutions is exactly what you'd expect (random distribution).
• For some specific triplets, they had to assume a famous conjecture (a guess that hasn't been proven yet) to finish the job. They showed that if this guess is true, their formula works perfectly.

Why Does This Matter?

In the world of mathematics, non-vanishing (not being zero) is a golden ticket.

• If $L(1/2, \chi) = 0$, it often means a deep symmetry or a hidden structure in the universe of numbers.
• If the product of three L-functions is never zero for a significant number of characters, it tells us that the universe of numbers is robust and full of "life" (non-zero values).

The authors also proved a "Corollary" (a side result): There are at least $q / (\log q)^{12}$ different tunings where the triple product is not zero.
Since $q$ is a huge prime number, this means there are thousands, millions, or billions of cases where the music is loud. They didn't just find one note; they found a whole orchestra playing.

Summary in One Sentence

This paper proves that when you mix three specific types of number-theoretic "sounds" together, they almost always create a loud, non-zero harmony, provided the ingredients are mixed in the right "Galant" or "Oxozonic" proportions, using a blend of geometry, algebra, and clever counting to prove it.

Technical Summary

Here is a detailed technical summary of the paper "Toroidal Families and Averages of L-Functions, II: Cubic Moments" by Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, and Will Sawin.

1. Problem Statement

The paper investigates the asymptotic behavior of the cubic mixed moments of central values of Dirichlet $L$-functions. Specifically, for a large prime modulus $q$ and fixed non-zero integers $a, b, c$, the authors aim to evaluate the average:
$$M_{a,b,c}(q) := \frac{1}{q-1} \sum_{\chi \pmod q} L(1/2, \chi^a) L(1/2, \chi^b) L(1/2, \chi^c)$$
This work generalizes their previous study of second moments (cubic moments were previously only known for specific cases like $M_{1,1,1}$ or $M_{1,1,-1}$). The central challenge is to determine whether this average converges to a specific constant as $q \to \infty$ and to quantify the error term.

The authors distinguish between different types of triples $(a, b, c)$ based on the geometric properties of associated $\ell$-adic sheaves:

• Galant triples: The generic case where the geometric monodromy group is "large" (simple or a central extension of a simple group).
• Oxozonic triples: Cases where the monodromy group is the orthogonal group $O_4$.
• Induced/Solvable/Sulfatic cases: Special degenerate cases (e.g., $a+b=c$ or specific small integer combinations) which are not fully resolved in this paper but are identified for future work.

2. Methodology

The proof employs a sophisticated synthesis of analytic number theory, algebraic geometry, and the theory of exponential sums. The methodology proceeds through several key stages:

A. Reduction and Decomposition

1. Fourier Analysis: The authors reduce the problem from general integers $a, b, c$ to the case where they are setwise coprime, introducing a "twisted" moment $M_{a,b,c}(\xi; q)$ involving a root of unity $\xi$.
2. Parity Decomposition: The sum is split into contributions from even and odd characters ($\chi(-1) = \pm 1$).
3. Approximate Functional Equations (AFE): The $L$-function products are replaced by rapidly converging Dirichlet series using AFEs. This transforms the problem into evaluating two types of sums:
   • $M_1$ (Main Term): A sum over integers $l, m, n$ satisfying a congruence $l^a m^b n^{\pm c} \equiv \xi \pmod q$.
   • $M_2$ (Error Term): A sum involving normalized Gauss sums and a "monomial Kloosterman-like" exponential sum $K_{a,b,\pm c}$.

B. Bounding the Error Terms ($M_2$)

The core innovation lies in bounding the trilinear sums involving the exponential sums $K_{a,b,\pm c}(u; q)$.

• Sheaf Theory: The authors identify $K_{a,b,\pm c}$ as the trace function of a specific hypergeometric $\ell$-adic sheaf constructed via Katz's theory.
• Monodromy Analysis: They prove that for "galant" and "oxozonic" triples, these sheaves have large geometric monodromy groups (specifically, they are "galant" in the sense of Definition 2.1).
• Bilinear/Trilinear Bounds: Using deep results from $\ell$-adic cohomology (specifically bounds for bilinear and trilinear sums of trace functions from their companion paper [14]), they establish square-root cancellation for these sums. This proves that the $M_2$ terms are negligible ($O(q^{-\eta})$).

C. Analyzing the Main Terms ($M_1$)

The $M_1$ terms count solutions to the congruence $l^a m^b n^{\pm c} \equiv \xi \pmod q$ in dyadic boxes.

• Main Term Extraction: When $\xi = 1$, the solutions to the exact equality $l^a m^b n^{\pm c} = 1$ (in integers) contribute the main term. This leads to a convergent Dirichlet series $D_{a,b,\pm c}(s)$ evaluated at $s=1/2$.
• Error Term Estimation: The remaining solutions (where the congruence holds but the integer equality does not) are bounded using a conjecture on the number of solutions to monomial congruences in small boxes (Conjecture P).
  • The authors prove this conjecture unconditionally when $a=b$.
  • For the general case, they assume Conjecture P, which is supported by average results over primes and known bounds for specific ranges.

3. Key Contributions

1. General Asymptotic Formula: The paper provides an asymptotic formula for the cubic moment $M_{a,b,c}(q)$ for a vast class of triples $(a, b, c)$.

   • Theorem 1.2: For "galant" or "oxozonic" triples, the average is bounded below by a constant $D_{a,b,\pm c} \ge 1$.
   • Theorem 1.3: Assuming Conjecture P, the formula becomes an equality: $M_{a,b,c}(q) = D_{a,b,\pm c} + O(q^{-\eta})$.
   • Theorem 1.5: An unconditional asymptotic formula is proven when averaging over the modulus $q$ in a short interval.

2. Classification of Triples: The authors introduce a rigorous classification (Galant, Oxozonic, Induced, Solvable, Sulfatic) based on the geometry of the underlying sheaves. This clarifies why certain cases are easier (induced) and why others are difficult (sulfatic/oxozonic).

3. Non-Vanishing Results:

   • Corollary 1.4: They prove that for any non-zero $a, b, c$, there exists a positive proportion of characters $\chi \pmod q$ such that $L(1/2, \chi^a)L(1/2, \chi^b)L(1/2, \chi^c) \neq 0$. Specifically, the number of such characters is $\gg q / (\log q)^{12}$.
   • This is a significant step toward understanding the non-vanishing of $L$-functions in families.

4. Technical Advances in Exponential Sums: The paper extends the application of $\ell$-adic sheaf theory to trilinear sums of trace functions, handling the specific complexities of monomial Kloosterman sums $K_{a,b,\pm c}$.

4. Main Results

• Asymptotic Formula: For setwise coprime $a, b, c$ and a prime $q$:
  $$\frac{1}{q-1} \sum_{\chi} L(1/2, \chi^a) L(1/2, \chi^b) L(1/2, \chi^c) = D_{a,b,\pm c} + O(q^{-\eta})$$
  where $D_{a,b,\pm c}$ is a constant derived from a Dirichlet series (equal to 1 for $a=b=c$ or specific cases, and $>1$ otherwise).
• Independence of $d$: The main term $D_{a,b,\pm c}$ is independent of the scaling factor $d$ in the exponents $ad, bd, cd$, provided the triple remains coprime after dividing by $d$.
• Non-Vanishing: The product of three $L$-functions does not vanish for a positive proportion of characters (unconditionally for the average, and for a positive proportion under the conjecture).

5. Significance

• Advancement of the Cubic Moment Problem: This is one of the first papers to treat the cubic moment for general exponents $(a, b, c)$ in Dirichlet families, moving beyond the classical $M_{1,1,1}$ case.
• Geometric Insight: The paper successfully bridges the gap between analytic number theory (moments of $L$-functions) and algebraic geometry (monodromy groups of sheaves). The classification of triples based on the "galant" property of the associated sheaf provides a new structural understanding of why certain moments behave well.
• Non-Vanishing Applications: The results contribute to the broader program of proving that $L$-functions do not vanish at the central point, a crucial step in applications like the non-vanishing of derivatives (related to the Birch and Swinnerton-Dyer conjecture) and subconvexity bounds.
• Methodological Framework: The techniques developed for bounding trilinear sums of trace functions with specific monomial weights are likely to be applicable to other problems involving higher moments or mixed moments of automorphic $L$-functions.

In summary, this paper represents a major theoretical leap in the study of moments of $L$-functions, utilizing the full power of modern $\ell$-adic cohomology to solve a problem that was previously intractable for general parameters.