Congruences for the ratios of Rankin--Selberg $L$-functions

This paper computationally investigates the principle that congruences between holomorphic cuspforms induce congruences between the special values of their associated Rankin–Selberg $L$-functions, leading to the formulation of a precise general conjecture.

The Gist

Imagine you are a detective trying to solve a mystery about two very different-looking twins. One twin, let's call him F, is a complex, high-energy mathematician. The other, F-prime, looks almost identical to F, but if you look closely at their fingerprints (their mathematical "coefficients"), you notice they are slightly different. However, if you view their fingerprints through a special pair of "modular glasses" (a specific mathematical filter called a prime ideal), they look exactly the same.

This paper by Narayanan and Raghuram asks a fascinating question: If these two twins look the same through our special glasses, do their "life stories" (specifically, the special values of their Rankin-Selberg L-functions) also tell the same story when viewed through those same glasses?

Here is a breakdown of the paper's journey, using everyday analogies:

1. The Core Principle: "If the Source is the Same, the Result Should Match"

In the world of numbers, there's a famous rule (a principle from Iwasawa theory) that says: Congruence in, Congruence out.

• The Input: If two mathematical objects (like our twins F and F-prime) are "congruent" (basically, they are the same modulo a specific number), they should behave similarly.
• The Output: The paper investigates whether this similarity holds true for the Rankin-Selberg L-functions. Think of an L-function as a complex "scorecard" or a "fingerprint" that summarizes the behavior of a number pattern.
• The Twist: Instead of comparing the whole scorecard, the authors compare the ratios of specific points on the scorecard (like comparing the score at level 20 to the score at level 21). They want to know: If F and F-prime are twins, is the ratio of their scores at these levels also the same?

2. The Tools: The Mathematical "Kitchen"

To test this, the authors had to cook up some very specific recipes. They couldn't just guess; they needed to calculate the exact values of these L-functions.

• Algorithm 1 (The Projection Chef): They used a method developed by Shimura and Hida. Imagine you have a messy, non-holomorphic (wobbly) cake. This algorithm acts like a special press that flattens the wobbly parts into a perfect, smooth, "holomorphic" cake. This allows them to extract the exact "algebraic" value of the L-function.
• Algorithm 2 (The Modular Symbol Map): This is like using a GPS map to navigate the space of these numbers. It translates the abstract problem of calculating infinite sums into a concrete problem of counting paths on a grid (modular symbols).
• The Computer: They ran these recipes on SAGE, a powerful open-source mathematics software, acting as their high-speed calculator.

3. The Experiments: Testing the Twins

The authors didn't just talk about theory; they ran five specific experiments (Examples 3.1 to 3.5) to see if the rule held up.

• Experiment 1 & 2 (The Galois Twins): They took two forms that were "Galois conjugates" (mathematical twins that are mirror images of each other in a complex number field). They checked if their L-function ratios matched modulo a large prime. Result: Yes! The ratios were congruent.
• Experiment 3 (The Non-Twin Twins): They took two forms that weren't even mirror images but still shared a congruence. Result: Yes! The rule still held.
• Experiment 4 (The Exception): Here, they froze one form and changed the other form (the lower weight one). They found that usually, the rule works, BUT there was one specific case where it failed.
  • Why? It turned out that the "background music" (the abelian L-functions) was playing a different tune that drowned out the congruence. It was like two singers singing the same song, but one had a microphone that was slightly out of tune, ruining the harmony.
• Experiment 5 (The Ramanujan Connection): They compared a famous cusp form (Ramanujan's Delta function) with an Eisenstein series (a different type of number pattern). These are famous "congruent" partners (the Ramanujan congruence). Result: The rule held perfectly, confirming a deep historical connection.

4. The Big Conclusion: The Conjecture

After running these tests, the authors formulated a Conjecture (a strong mathematical guess).

They propose that:

If two forms are congruent, their L-function ratios will be congruent UNLESS there is a specific "interference" from the background numbers (the abelian part) that cancels out the match.

They added a safety clause (Condition 16) to their conjecture to account for that one exception they found in Experiment 4. It's like saying, "The rule works, provided the background noise isn't too loud."

5. Why Does This Matter?

This isn't just about solving a puzzle for fun.

• The Bridge: It builds a bridge between the "shape" of a number pattern (the form) and the "value" it produces (the L-function).
• The Future: The authors mention that a companion paper proves a variation of this using heavy-duty machinery called "Eisenstein Cohomology." This paper provides the computational evidence that says, "Hey, the heavy machinery is likely working correctly because our computer experiments show the pattern holds up."

Summary in One Sentence

The authors used powerful computer algorithms to verify that when two complex number patterns look identical through a specific mathematical lens, their resulting "scores" (L-function ratios) also match, with only a few rare exceptions caused by background noise.

Technical Summary

Here is a detailed technical summary of the paper "Congruences for the Ratios of Rankin–Selberg L-functions" by P. Narayanan and A. Raghuram.

1. Problem Statement and Motivation

The paper investigates a fundamental principle in number theory, rooted in Iwasawa theory: congruences between modular forms should induce congruences between the special values of their associated $L$-functions.

Specifically, the authors focus on Rankin–Selberg $L$-functions ($L(s, f \times g)$) attached to pairs of holomorphic cusp forms $f$ and $g$. The core question is:
If two primitive cusp forms $f$ and $f'$ are congruent modulo a prime ideal $\mathfrak{P}$ (i.e., their Fourier coefficients satisfy $a(n, f) \equiv a(n, f') \pmod{\mathfrak{P}}$), does this imply a congruence between the ratios of their critical values?

The authors formulate the specific conjecture:
$$f \equiv f' \pmod{\mathfrak{P}} \implies \frac{L(m, f \times g)}{L(m+1, f \times g)} \equiv \frac{L(m, f' \times g)}{L(m+1, f' \times g)} \pmod{\mathfrak{P}}$$
where $m$ is a critical integer. The paper addresses the technical difficulty that $L$-values themselves are often transcendental; however, their ratios (after removing archimedean factors and periods) lie in number fields, making congruences well-defined.

2. Methodology

The authors employ a computational approach supported by theoretical algorithms to verify this principle in various cases. The methodology relies on two primary algorithms implemented in SAGE:

A. Algorithm for Rankin–Selberg L-values (Shimura–Hida)

To compute the algebraic part of $D(m, f \times g)$ (the Dirichlet series part of the $L$-function), the authors utilize the interpretation of $L$-values as Petersson inner products.

• Holomorphic Projection: They express the $L$-value as $\langle f^\rho, h_0 \rangle$, where $h_0$ is the holomorphic projection of a nearly holomorphic modular form constructed from $g$ and an Eisenstein series $E^*_{\lambda, N}$.
• Recursive Calculation: Using Maaß–Shimura differential operators ($\delta^{(r)}_\lambda$), they compute the projection $h_0$ recursively via Lemma 2.6.
• Basis Expansion: The resulting form $h_0$ is expanded in a basis of eigenforms. The coefficient corresponding to $f$ yields the desired algebraic value.

B. Algorithm for Standard L-values (Modular Symbols)

To compute the algebraic parts of standard $L$-functions (needed when one form is an Eisenstein series or for normalization), they use the modular symbols method.

• Hecke Equivariance: They utilize the pairing between cusp forms and modular symbols, which is Hecke-equivariant.
• Linear Algebra: By constructing matrices of Hecke operators acting on the space of modular symbols and solving for the eigenvector corresponding to the Fourier coefficients of the form, they extract the values of critical $L$-ratios.

C. Sturm's Criterion

To establish the initial congruence $f \equiv f' \pmod{\mathfrak{P}}$, the authors use Sturm's bound (Theorem 2.11). This allows them to verify congruence by checking only a finite number of Fourier coefficients (up to a specific bound depending on the weight and level).

3. Key Contributions and Results

A. Computational Verification

The paper provides rigorous computational verification of the congruence principle in five distinct scenarios:

1. Galois Conjugates: $f, f' \in S_{24}(SL_2(\mathbb{Z}))$ and $S_{30}(SL_2(\mathbb{Z}))$ with $g \in S_{12}(SL_2(\mathbb{Z}))$. Here $f'$ is a Galois conjugate of $f$.
2. Non-Conjugate Forms: $f, f' \in S_{13}(\Gamma_0(3), \chi)$ where $f'$ is not a Galois conjugate of $f$, but they are congruent modulo 13.
3. Fixed High Weight: $f \in S_{26}(SL_2(\mathbb{Z}))$ is fixed, while $g, g' \in S_{13}(\Gamma_0(3))$ vary.
4. Cusp vs. Eisenstein: The famous Ramanujan congruence where $g$ is a cusp form ($\Delta$) and $g'$ is an Eisenstein series ($E_{12}$), both congruent modulo 691.

B. Discovery of an Exceptional Case

A critical finding is the identification of a condition where the congruence fails for the completed $L$-function ratios, even when the forms are congruent.

• Case: In Section 3.4, for $f \in S_{26}$ and $g, g' \in S_{13}$, the congruence holds for the Dirichlet series ratios $D(m, f \times g)/D(m+1, f \times g)$ but fails for the completed $L$-function ratios $L(m, f \times g)/L(m+1, f \times g)$ at the extreme critical point $m=24$.
• Reason: The failure is attributed to the ratio of the abelian $L$-factors (involving the Dirichlet $L$-function of the character $\psi$). Specifically, the denominator of the ratio of Bernoulli numbers introduces a factor of the prime $\mathfrak{P}$ (valuation $v_{\mathfrak{P}} = -3$), which cancels out the congruence.

C. Formulation of a Precise Conjecture

Based on the examples and the identified exception, the authors formulate Conjecture 4.1.
The conjecture states that if $f \equiv f' \pmod{\mathfrak{P}}$, then the ratio congruence holds provided a specific condition on the archimedean and finite Euler factors is met:
$$v_{\mathfrak{P}} \left( \frac{L_\infty(m, f \times g) L_N(2m+2-k-l, \chi\psi)}{L_\infty(m+1, f \times g) L_N(2m+4-k-l, \chi\psi)} \right) \geq 0$$
This condition ensures that the "non-algebraic" parts of the $L$-function do not introduce negative valuations that would destroy the congruence.

4. Significance

• Bridging Computation and Theory: The paper demonstrates how high-precision computational number theory can be used to test deep conjectures in the Langlands program and Iwasawa theory.
• Refinement of Principles: It moves beyond the vague notion that "congruences imply congruences" to a precise, conditional statement involving the valuation of archimedean and finite factors.
• Algorithmic Framework: The paper provides explicit, implementable algorithms for computing algebraic parts of Rankin–Selberg $L$-values, which are essential for future numerical investigations in automorphic forms.
• Foundation for Proof: The authors note that a companion paper ([9]) uses these computational insights and the machinery of Eisenstein cohomology (Harder–Raghuram) to prove a variation of this conjecture, validating the computational findings theoretically.

In summary, this work rigorously tests the relationship between modular form congruences and $L$-value congruences, successfully verifying the principle in most cases while pinpointing the exact arithmetic obstruction (the valuation of the $L$-factors) that causes it to fail in specific boundary cases.