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Extreme values of class group $L$ -functions

The paper establishes a lower bound for the maximum value of class group $L$ -functions at the central point for imaginary quadratic fields $\mathbb{Q}(\sqrt{-D})$ , demonstrating that this maximum is on average at least $\exp\left(\delta\sqrt{\frac{\log D \log \log \log D}{\log \log D}}\right)$ for any $\delta < \frac{1}{4}$ .

Original authors: João C. C. Vargas

Published 2026-08-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: João C. C. Vargas

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 the number system as a vast, invisible city where every building is a number. Some numbers are simple, standing alone like solitary houses, while others are complex structures built from smaller parts. Deep within this city, mathematicians study "L-functions," which are like secret maps or weather reports that predict how these buildings are arranged. These maps are crucial because they hold the keys to understanding the fundamental rules of arithmetic, much like how a weather map helps us predict a storm. One of the most famous mysteries in this city is the "Lindelöf hypothesis," a prediction that these maps should never show values that get too wildly huge. However, just as a calm sea can hide a massive tsunami, mathematicians suspect that under the right conditions, these values might actually spike to enormous heights. The question is: how high can they really go, and can we prove it?

This paper, written by João Campos-Vargas, dives into a specific neighborhood of this mathematical city: the "imaginary quadratic fields." Think of these as special districts where the rules of geometry are slightly twisted, involving numbers with a square root of negative one. In these districts, the "class group" acts like a unique fingerprint for the area, describing how the buildings (ideals) can be rearranged. The author investigates the "L-functions" attached to these fingerprints, specifically looking for the biggest possible value they can reach at a critical point called the "central point."

The paper's main discovery is a lower bound for the maximum value of these L-functions. In simpler terms, the author proves that if you look at a large collection of these special districts, the biggest "spike" you will find is at least as large as a very specific, incredibly fast-growing number. This number is roughly the exponential of the square root of the logarithm of the district size, multiplied by a few other logarithmic factors. The author shows that, on average, these values are at least as big as:

$\exp \left( \delta \sqrt{\frac{\log D \log \log \log D}{\log \log D}} \right)$

where $D$ represents the size of the district and $\delta$ is any number smaller than $1/4$ .

This finding is significant because it supports a popular belief among mathematicians that these values can indeed get much larger than the conservative "Lindelöf hypothesis" predicts. While the famous "Riemann Hypothesis" (a giant rulebook for the city) suggests these values shouldn't be too crazy, this paper shows that if you look at the average behavior across many districts, you can definitely find values that break the "safe" limits. Crucially, this result is proved unconditionally, meaning it stands on its own without needing to assume the Riemann Hypothesis is true. The author uses a clever technique called the "resonance method," which is like tuning a radio to a specific frequency to amplify a faint signal. By carefully choosing which "primes" (the basic building blocks of the districts) to focus on, the author amplifies the L-function values to reveal their true, massive potential.

However, the paper also draws a careful line. It does not prove that every single district has these huge spikes. Instead, it proves that if you take the average of the maximums across many districts, the result is huge. This is because the method relies on the presence of "split primes"—primes that break apart in a specific way within the district. The author notes that while we expect about half of the primes to split, we cannot guarantee this for every single district without assuming the Riemann Hypothesis, which remains unproven. In fact, the paper suggests that for some specific districts, all the necessary primes might be "inert" (refusing to split), which would make the spike disappear. So, while the paper confirms that these massive values exist in the average, it leaves open the mystery of whether they exist in every single case.

The journey to this result involves a lot of mathematical machinery. The author starts with an "approximate functional equation," which is a tool that lets them rewrite the L-function in a way that is easier to handle. They then construct a "resonator," a mathematical device made of specific prime numbers, designed to resonate with the L-functions and boost their values. The tricky part is ensuring that this resonator doesn't accidentally boost the "trivial" values (the background noise) too much. The author solves this by carefully limiting the size of the set of primes used, ensuring that the signal (the maximum value) is much louder than the noise.

Ultimately, the paper confirms that the "true order of magnitude" for these values is likely much larger than previously thought, aligning with the conjectures of Farmer, Gonek, and Hughes. It suggests that the probabilistic nature of these numbers allows for extreme values that are far more dramatic than the strict upper bounds predicted by other hypotheses. The work stands as a rigorous, unconditional proof: it shows that if you look at the average, you will find these extreme values, and it does so without relying on unproven hypotheses. The paper doesn't just guess; it provides a step-by-step demonstration that these mathematical giants are real, even if they are hiding in the shadows of some specific districts.

Technical Summary: Extreme Values of Class Group L-Functions

Problem Statement
The paper investigates the magnitude of the maximum value of $L(1/2, \chi)$ where $\chi$ ranges over the characters of the class group of an imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$ . While the Lindelöf hypothesis predicts $L(1/2, \chi) \ll_\epsilon D^\epsilon$ , and the Generalized Riemann Hypothesis (GRH) implies a bound of $\exp((\frac{\log 2}{2} + o(1)) \frac{\log D}{\log \log D})$ , probabilistic models by Farmer, Gonek, and Hughes suggest the true order of magnitude for the maximum is significantly larger:
$\max_{\chi} |L(1/2, \chi)| = \exp((c + o(1)) \sqrt{\log D \log \log D})$
for some positive constant $c$ . Previous work by Bondarenko and Seip established this order of magnitude for the Riemann zeta function and Dirichlet $L$ -functions using the resonance method. This paper aims to determine if this same order of magnitude holds for class group $L$ -functions, specifically addressing the challenges posed by the symplectic nature of the family and the distribution of split primes.

Methodology
The author employs the resonance method, originally developed by Soundararajan and adapted by Bondarenko and Seip. The core strategy involves constructing a "resonator" function $R_\chi$ to maximize the ratio $V/W$ , where $V$ is a weighted sum of $L$ -values and $W$ is the sum of the weights.

Approximate Functional Equation: The $L$ -values are expressed via an approximate functional equation involving a weight function $W(x)$ , allowing the $L$ -values to be written as sums over integral ideals.
Resonator Construction: The resonator coefficients $R_\chi$ are defined as sums over a set of ideals $\mathcal{M}$ with a multiplicative function $f$ . The set $\mathcal{M}$ is carefully constructed to be a subset of squarefree ideals generated by prime ideals lying above rational primes in a specific range: $(e \log M \log_2 M, \exp(\lfloor (\log_2 M)^\gamma \rfloor) \log M \log_2 M]$ .
Trivial Character Constraint: A critical technical hurdle is ensuring the contribution of the trivial character does not dominate the sum. The paper establishes that if the size of the set $\mathcal{M}$ satisfies $|\mathcal{M}| \leq h_D / (3D^{1/4} \log D)$ , the trivial character's contribution is negligible compared to the main term.
Handling Split Primes: Unlike previous applications of the resonance method to Dirichlet $L$ -functions, the lower bound here depends on the sum of $1/\sqrt{N\mathfrak{p}}$ over prime ideals $\mathfrak{p}$ that split in $\mathbb{Q}(\sqrt{-D})$ . If primes in the chosen range are inert, the bound becomes trivial.

Key Contributions and Results

Theorem 1 (Average Case): The paper proves that the conjectured order of magnitude holds on average over fundamental discriminants $-D \in [-2X, -X]$ . Specifically, for any $\delta < 1/4$ :
$\left( \prod_{X \leq D \leq 2X}^{\flat} M_D \right)^{1/N_X} \geq \exp\left( \delta \sqrt{\frac{\log X \log_3 X}{\log_2 X}} \right)$
where $M_D$ is the maximum $L$ -value for a given $D$ , and the product excludes at most one exceptional discriminant. This is the first unconditional proof of this order of magnitude in a symplectic family (excluding a function field example).
Theorem 2 (Pointwise Technical Bound): The paper provides a precise lower bound for a specific discriminant $-D$ that depends explicitly on the splitting behavior of primes in a specific range. If $P$ is the set of prime ideals above primes in the range defined by parameters $M$ and $\gamma$ , then:
$\max_{\chi} L(1/2, \chi) \geq \exp\left( (1+o(1)) \sqrt{\frac{\log M \log_2 M}{\log_3 M}} \sum_{\mathfrak{p} \in P} \frac{1}{\sqrt{N\mathfrak{p}}} \frac{1}{\sqrt{p}(\log p - \log_2 M - \log_3 M)} \right)$
This result highlights that the magnitude of the bound is directly tied to the number of split primes in the interval $(\log D)^{1+\epsilon}$ .

Significance and Limitations
The paper claims that the averaging feature in Theorem 1 is necessary because the method requires primes of size $(\log D)^{1+\epsilon}$ to split to achieve the desired bound. The author notes that even under GRH, one cannot guarantee that sufficiently many such primes split for every discriminant; GRH only ensures splitting for primes larger than $\log^2 D$ .

Consequently, the paper does not prove that the conjectured maximum holds for every fundamental discriminant unconditionally. Instead, it demonstrates that the bound holds on average, leveraging the probabilistic expectation that half of the relevant primes split. The author raises an open question: whether the existence of extreme values for every discriminant (as predicted by the Farmer-Gonek-Hughes conjecture) would imply stronger results about prime splitting than GRH currently provides, or if the order of magnitude can only be attained using primes larger than $\log^2 D$ .

The work is presented as an adaptation of the resonance method to the symplectic family of class group characters, overcoming specific obstacles related to the positivity of coefficients and the distribution of split primes in imaginary quadratic fields.

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