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Unexpected primes of good reduction in quotients of modular and Shimura curves

This paper classifies all zero-dimensional spaces of weight 2 newforms with squarefree level and fixed Atkin-Lehner signs, using this classification to identify unexpected primes of good reduction for Atkin-Lehner quotients of modular and Shimura curves.

Original authors: Oana Padurariu, Sun Woo Park, John Voight

Published 2026-07-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Oana Padurariu, Sun Woo Park, John Voight

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 world of numbers as a massive, bustling city called Modular City. In this city, there are special buildings called Modular Curves. These aren't ordinary buildings; they are shaped by deep, hidden rules of arithmetic. Some of these buildings are huge and complex, while others are tiny, simple, and have a "genus" of zero (which is like saying they are just a single point or a simple loop with no holes).

For a long time, mathematicians have been fascinated by these zero-genus buildings because they are rare and special. But the real mystery in this paper isn't just about the buildings themselves; it's about what happens when you take a big, complex building and smash it down into a smaller, simpler version using a specific set of rules called Atkin–Lehner quotients.

Think of the Atkin–Lehner group as a team of demolition experts. They look at a building (defined by a number $N$ ) and decide which parts to keep and which to toss. They do this based on a "sign pattern," which is like a code of plus and minus signs assigned to the building's prime number ingredients.

The Great Hunt for Empty Rooms

The authors, Oana Padurariu, Sun Woo Park, and John Voight, asked a very specific question: Are there any combinations of building size ( $N$ ) and demolition code (sign pattern) that result in a building with absolutely no "new" rooms left?

In math-speak, they were looking for "zero-dimensional spaces of newforms." If you imagine the building's rooms as a library of new stories (newforms), they wanted to find the specific blueprints where the library ends up completely empty.

They knew that for most huge buildings, the library is always full. The bigger the building, the more rooms it has. But for smaller, specific sizes, it's possible that the demolition code wipes out every single new story, leaving the library empty.

The Main Finding:
The team didn't just guess; they went out and found every single one of these "empty library" blueprints. They proved that there are only a finite number of them. They created a master list (Tables 2, 3, 4, and 5 in the paper) that acts like a "Wanted Poster" for these specific, empty configurations. If your building size and sign pattern aren't on that list, your library is guaranteed to have at least one new story.

The Surprise: Good Reduction at Bad Places

Here is where the story gets really cool. Usually, if a building has a "bad" ingredient (a prime number $p$ that causes trouble), the whole structure is expected to crumble or have "bad reduction" at that spot. It's like a house built on a swamp; you expect the floor to be wobbly.

However, the authors discovered something unexpected. Sometimes, when you take a modular curve and smash it down into its Atkin–Lehner quotient, the resulting smaller building suddenly becomes stable at a prime number where the original building was wobbly.

They call these "unexpected primes of good reduction."

The Analogy:
Imagine a giant, rickety bridge (the original curve) that is known to collapse if you step on a specific rusty bolt (the prime $p$ ). You'd expect any smaller bridge built from its parts to also collapse on that bolt. But the authors found cases where, after the demolition crew does their work, the new, smaller bridge is perfectly solid on that very same rusty bolt. The "badness" of the bolt has been magically neutralized by the way the building was smashed down.

What They Proved:
They didn't just find a few examples; they classified all of these surprising cases for a specific type of building (squarefree levels). They proved that:

If the new, smaller building has any size at all (genus > 0), it will always be wobbly at the primes that were part of the "discriminant" (the $D$ part of the blueprint).
But for the other primes, they found the exact list of cases where the building becomes stable. This list is found in Tables 6 and 7.

What They Ruled Out

The paper is very clear about what doesn't happen.

They proved that you cannot have an infinite number of these "empty library" blueprints. If you keep making the building bigger and bigger, you will eventually run out of empty libraries; the rooms will always appear.
They ruled out the idea that these "good reduction" surprises can happen at any prime. They specifically showed that if the prime is part of the discriminant $D$ , the building will always be bad there. The magic only works on the other primes.

How Sure Are They?

This isn't a guess or a simulation. The authors used a powerful combination of math formulas (trace formulas) and computer power to prove their results.

They used a formula to estimate how many rooms a building should have.
They calculated the "error terms" (the messy parts of the math) to be sure the estimate was accurate enough.
They wrote a computer program (using a tool called Magma) to check every single candidate building up to a certain size.
They proved that for any building larger than a specific huge number (like $N > 5 \times 10^{14}$ for some cases), it is mathematically impossible for the library to be empty.

So, when they say "we found all of them," they mean it. They have the complete, verified list.

The Takeaway for a Curious Teen

Think of this paper as a treasure map. The treasure isn't gold, but knowledge.

The Map: A list of specific numbers and sign patterns.
The X Marks the Spot: These are the rare, magical moments where a complex mathematical structure simplifies so perfectly that it loses all its "new" complexity (becoming zero-dimensional) or gains a stability it shouldn't have (good reduction at a bad prime).
The Result: The authors have drawn the entire map. There are no hidden islands left to discover in this specific territory. If you try to build a modular curve with a squarefree level and a specific sign pattern, and it's not on their list, you can be 100% sure it will have new stories to tell and won't have these specific "miracle" properties.

They even found a specific example mentioned in the intro: A curve called $X_0(194)$ is wobbly at the prime 2, but its smashed-down version, $X^*_0(194)$ , is perfectly solid at 2. That's the kind of "unexpected" magic they hunted down and cataloged.

Technical Summary: Unexpected Primes of Good Reduction in Quotients of Modular and Shimura Curves

Problem Statement
The paper addresses two interconnected problems in arithmetic geometry and the theory of modular forms. First, it seeks to classify all pairs $(N, \varepsilon_N)$ , where $N$ is a squarefree integer and $\varepsilon_N$ is a sign pattern for the Atkin–Lehner group $W_N$ , such that the space of newforms $S_2^{\text{new}, \varepsilon_N}(N)$ is zero-dimensional. Second, it applies this classification to identify "unexpected" (or "gained") primes of good reduction for Atkin–Lehner quotients of modular and Shimura curves. Specifically, the authors investigate cases where a quotient curve $X^*_0(D; N)$ or its Jacobian possesses good reduction at a prime $p$ dividing the level (or discriminant), despite the curve typically having bad reduction at such primes.

Methodology
The authors employ a combination of analytic trace formulas and explicit computational enumeration.

Trace Formula Analysis: The dimension of the Atkin–Lehner eigenspace $S_2^{\text{new}, \varepsilon_N}(N)$ is estimated using a trace formula adapted from Martin [Mar18]. The dimension formula consists of a main term, which grows linearly with $N$ (specifically $N/2^{\omega(N)}$ ), and error terms controlled by class numbers of imaginary quadratic fields (roughly $O(\sqrt{N} \log N)$ ).
Positivity Criteria: By isolating the main term and bounding the error terms using the Louboutin bound for class numbers, the authors derive explicit inequalities (Lemma 2.5, Corollary 2.18). These inequalities demonstrate that for a fixed number of prime factors $\omega(N)$ , the dimension is strictly positive once $N$ exceeds a certain computable bound. This reduces the problem of finding zero-dimensional spaces to a finite search.
Computational Enumeration: For the finite range of $N$ below the derived bounds, the authors perform an exhaustive search. They generate squarefree integers $N$ with specific congruence classes and prime factor counts, apply filters based on necessary inequalities for vanishing dimensions, and finally compute the exact dimensions using a Magma implementation of Martin's trace formula.
Application to Reduction: To determine primes of good reduction for the quotients $X^*_0(D; N)$ , the authors utilize the Jacquet–Langlands correspondence and the decomposition of the Jacobian into modular abelian varieties. They establish that the Jacobian has good reduction at $p$ if and only if all newform spaces in the isotypic decomposition with levels divisible by $p$ are zero-dimensional (Lemma 3.7).

Key Contributions and Results

Classification of Zero-Dimensional Spaces (Theorem 1.1): The paper provides a complete list of all squarefree levels $N$ and Atkin–Lehner sign patterns $\varepsilon_N$ for which $S_2^{\text{new}, \varepsilon_N}(N) = \{0\}$ . These pairs are enumerated in Tables 2 through 5. The authors prove that for sufficiently large $N$ (depending on $\omega(N)$ and the congruence class of $N$ ), no such zero-dimensional spaces exist.
Classification of Gained Primes (Theorem 1.2): The authors classify all triples $(D, N, p)$ $(D, N, p)$ where $X^*_0(D; N)$ $X_{0}^{*} (D; N)$ has genus $>0$ $> 0$ and possesses good reduction at a prime $p$ $p$ dividing $DN$.
- Part (a) confirms that for these curves, the curve has good reduction at $p$ if and only if its Jacobian does.
- Part (b) establishes that if $p$ divides the discriminant $D$ , the curve always has bad reduction.
- Part (c) lists the specific cases where good reduction occurs at primes dividing the level $N$ (or $DN$), detailed in Tables 6 and 7.
Corollary on Root Numbers: As an immediate consequence, the authors identify exactly 45 squarefree composite integers $N$ for which no weight 2 newform has root number $-1$ (Corollary 2.31).

Significance
The paper's primary significance lies in its effective determination of the "lacking" pairs $(N, \varepsilon_N)$ , a problem previously known to have only finitely many solutions but lacking a complete explicit list. This classification serves as a foundational tool for understanding the reduction properties of quotients of modular and Shimura curves.

The authors highlight the phenomenon of "gained" primes of good reduction, where a quotient curve $X^*_0(194)$ (genus 3) exhibits good reduction at $p=2$ , whereas the parent curve $X_0(194)$ has bad reduction. This result clarifies the behavior of Jacobians of Atkin–Lehner quotients, showing that their reduction properties are strictly governed by the vanishing of specific newform spaces. The work is restricted to the "cleanest" case of squarefree levels, weight 2, and trivial character, with the authors noting that extensions to non-squarefree levels, arbitrary nebentypus, or higher weights are natural but technically distinct directions. All computations are verified using Magma and data from the LMFDB.

The Great Hunt for Empty Rooms

The Surprise: Good Reduction at Bad Places

What They Ruled Out

How Sure Are They?

The Takeaway for a Curious Teen

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