Newform Eisenstein Congruences of Local Origin

This paper proposes and partially proves a general conjecture regarding the existence of Eisenstein congruences between weight $k \geq 3$ newforms of square-free level $NM$ and weight $k$ new Eisenstein series of square-free level $N$ with arbitrary character, while also exploring connections to the Bloch-Kato conjecture and providing computational examples.

The Gist

Imagine you are a master chef working in a very high-end, mathematical kitchen. In this kitchen, you have two very different types of ingredients:

1. The "Cusp" Ingredients: These are rare, complex, and mysterious spices (called Newforms). They are hard to find, and when you use them, they create unique, unpredictable flavors.
2. The "Eisenstein" Ingredients: These are the basic, predictable staples of the kitchen (called Eisenstein Series). They are easy to make, and their flavor follows a strict, simple recipe.

Usually, these two ingredients are worlds apart. A cusp form is like a secret family recipe, while an Eisenstein series is like a box of instant noodles. They shouldn't taste the same.

The Big Discovery: The "Taste Test"
For over a century, mathematicians have been fascinated by a strange phenomenon: sometimes, if you taste these two ingredients under a specific "magic microscope" (a prime number modulus), they taste identical.

This is called a congruence. It's like saying, "If you look at these two complex dishes through a red filter, they look exactly the same."

The most famous example was found by Ramanujan in the 1920s. He noticed that a very complex dish (the discriminant function $\Delta$) tasted exactly like a simple dish (the Eisenstein series $E_{12}$) when viewed through the filter of the number 691.

What This Paper Does
Dan Fretwell and Jenny Roberts are asking a deeper question: "Can we find these 'taste-alike' matches in more complex kitchens?"

Specifically, they are looking at kitchens with "square-free levels" (a way of describing the complexity of the kitchen's layout). They want to know:

• If we have a complex, rare spice (a Newform) in a big, complicated kitchen.
• And we have a simple, basic spice (an Eisenstein series) in a smaller, simpler kitchen.
• Can we prove they will taste the same under a specific filter?

The "Local Origin" Mystery
The title mentions "Local Origin." Imagine you are trying to match two recipes. Sometimes, the match happens because of a global ingredient (like the whole recipe). But sometimes, the match happens because of just one specific spice in the pantry.

The authors are studying cases where the "taste match" is caused by a single local factor (a specific prime number in the level of the form). They call this "Local Origin."

The Main Conjecture (The Chef's Guess)
The authors propose a general rule (a conjecture) to predict when these matches will happen. They say:

"A rare, complex spice (Newform) will taste exactly like a simple staple (Eisenstein series) if and only if two specific conditions are met regarding the 'flavor numbers' of the ingredients."

1. Condition 1 (The Global Flavor): The "flavor numbers" (related to special values of L-functions, which are like the nutritional labels of these mathematical dishes) must be divisible by a specific prime number. This is the main reason the match happens.
2. Condition 2 (The Local Flavor): For every "extra" spice in the big kitchen (the level $M$), the flavor numbers must also align in a specific way. This ensures the match isn't just a fluke of the smaller kitchen, but a true match for the bigger one.

What They Proved

• The Easy Cases: They fully proved their rule works when the kitchen is small ($M=1$) or when the extra complexity is just one single spice ($M=p$, a prime number).
• The Hard Case: They couldn't fully prove it for any combination of extra spices (composite $M$). They can prove that if the match exists, the conditions must be true. But proving that "if the conditions are true, the match must exist" is still a work in progress for the most complex kitchens.

Why Does This Matter? (The Bloch-Kato Connection)
Why do we care if two mathematical dishes taste the same?

Because these "taste matches" are the key to unlocking a massive, unsolved mystery in mathematics called the Bloch-Kato Conjecture. This conjecture is like a "Theory of Everything" for number theory. It tries to connect the flavor of these mathematical dishes to the hidden structure of numbers (specifically, how many solutions certain equations have).

The authors show that their "taste matches" are actually evidence of deep, hidden structures in the universe of numbers. If you find a match, you've found a clue that helps solve the bigger puzzle.

The "Local" Bottleneck
The paper highlights a tricky problem. Sometimes, the conditions are met for each individual extra spice, but the authors can't yet prove that there is one single "Master Dish" that matches all of them at once. It's like knowing that a cake matches a pie in flavor if you add strawberries, and it matches if you add blueberries, but you can't yet prove there is a cake that matches a pie if you add both strawberries and blueberries simultaneously.

In Summary
This paper is a map for finding hidden connections between complex and simple mathematical objects.

• The Metaphor: It's about finding when a gourmet meal tastes exactly like a basic meal through a special lens.
• The Goal: To create a rulebook that predicts exactly when this happens.
• The Impact: These predictions help mathematicians solve the biggest open problems in number theory, connecting the taste of numbers to the very fabric of the universe.

The authors dedicate this work to Lynne Walling, a mentor who encouraged them and many others, reminding us that behind every complex equation is a community of people supporting each other.

Technical Summary

1. Problem Statement

The paper addresses the existence of Eisenstein congruences between newforms (cusp forms) and Eisenstein series in the context of modular forms with arbitrary Dirichlet characters and square-free levels.

• Context: Historically, Eisenstein congruences (e.g., Ramanujan's $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$) relate the Fourier coefficients of a cusp form to those of an Eisenstein series modulo a prime $l$. These congruences are linked to the divisibility of special values of $L$-functions (Bernoulli numbers, Dirichlet $L$-values).
• The Gap: While congruences for trivial characters or specific levels are well-understood, a general conjecture for newforms of square-free level $NM$ (where $N$ is the conductor of the character and $M$ is an additional square-free integer) with arbitrary character $\chi$ of conductor $N$ was missing.
• Specific Challenge: The authors investigate when a newform $f \in S_k^{new}(\Gamma_0(NM), \tilde{\chi})$ satisfies a congruence $a_q(f) \equiv \psi(q) + \phi(q)q^{k-1} \pmod{\lambda}$ against an Eisenstein series $E_{\psi, \phi}^k$ lifted from level $N$ to level $NM$. The term "local origin" refers to congruences where the modulus divides local Euler factors rather than just the global $L$-value.

2. Methodology

The authors employ a combination of modular form theory, Galois representations, and cohomological methods.

• Construction of Eisenstein Series: They construct specific Eisenstein series $E_\delta$ of level $NM$ by applying operators $(T_p - \delta_p)$ to a base Eisenstein series $E_{\psi, \phi}^k$ of level $N$. This allows them to control the constant terms at various cusps.
• Galois Representations: They utilize the $\lambda$-adic Galois representations $\rho_{f, \lambda}$ associated with newforms. The congruence condition implies that the residual representation $\bar{\rho}_{f, \lambda}$ is reducible, splitting into characters $\psi$ and $\phi \chi_l^{k-1}$.
• Local Analysis: By analyzing the local behavior of these representations at primes $p | M$, they derive necessary conditions on the eigenvalues.
• Bloch-Kato Conjecture: They relate the existence of these congruences to the Bloch-Kato conjecture, specifically examining the $p$-torsion in Selmer groups and the divisibility of partial $L$-values.
• Lifting Lemmas: They use the Deligne-Serre lifting lemma and Carayol's lemma to lift modular forms from characteristic $p$ (residual) to characteristic 0, ensuring the existence of eigenforms.

3. Key Contributions and Conjectures

The General Conjecture (Conjecture 1.1 / 3.1)

The paper proposes a necessary and sufficient condition for the existence of a newform $f$ satisfying the Eisenstein congruence. Let $N, M$ be coprime square-free integers, $k > 2$, and $l > k+1$ a prime. Let $\psi, \phi$ be characters with conductors $u, v$ such that $uv=N$ and $\psi\phi = \chi$.

A newform $f \in S_k^{new}(\Gamma_0(NM), \tilde{\chi})$ satisfying $a_q(f) \equiv \psi(q) + \phi(q)q^{k-1} \pmod \lambda$ exists if and only if for some prime $\lambda' | l$:

1. Global/Local $L$-value Condition:
   $$\text{ord}_{\lambda'} \left( L(1-k, \psi^{-1}\phi) \prod_{p|M} (\psi(p) - \phi(p)p^k) \right) > 0$$
   This ensures the existence of some eigenform (cusp or Eisenstein) satisfying the congruence.
2. Local "Newness" Condition:
   For each prime $p | M$, either:
   • $\text{ord}_{\lambda'}(\psi(p) - \phi(p)p^k) > 0$ (Case A), OR
   • $\text{ord}_{\lambda'}(\psi(p) - \phi(p)p^{k-2}) > 0$ (Case B).
     This condition distinguishes between forms that are merely oldforms lifted from lower levels and genuine newforms at level $NM$.

Main Theoretical Results

• Theorem 3.5 (Necessity): Proves that if such a newform exists, then Conditions (1) and (2) must hold. This relies on analyzing the local composition factors of the Galois representation at primes dividing $M$.
• Theorem 3.6 (Sufficiency - Partial): Proves that Condition (1) alone guarantees the existence of a normalized Hecke eigenform (not necessarily new) satisfying the congruence. This extends previous results by Dummigan and Spencer.
• Theorem 3.7 (Full Proof for $M=p$): Proves the full conjecture when $M$ is a single prime. The authors show that under Conditions (1) and (2), one can construct a genuine $p$-newform satisfying the congruence.
• The Bottleneck: The paper identifies that while Conditions (1) and (2) guarantee the existence of a $p$-newform for each $p|M$ individually, it remains unproven whether a single global newform (new at all $p|M$ simultaneously) exists when $M$ is composite.

4. Relation to the Bloch-Kato Conjecture

The authors establish a deep link between their conjecture and the Bloch-Kato conjecture:

• The congruence implies the existence of a non-trivial extension class $c'$ in a specific Galois cohomology group $H^1_{PNM}(Q, C_{\psi, \phi}^{k, l})$.
• Proposition 4.3: Condition (1) of the conjecture is equivalent to the divisibility of a partial Dirichlet $L$-value, which by the Bloch-Kato conjecture predicts the size of the Selmer group.
• Conjecture 4.5 & 4.7: The authors conjecture that the specific "local" conditions (Case A vs. Case B in Condition 2) correspond to the existence of "p-new" elements in the Selmer group. Specifically, Case A corresponds to new elements arising from weight $k$, while Case B corresponds to new elements arising from weight $k-2$.

5. Computational Examples

The paper provides computational evidence using the LMFDB (L-functions and Modular Forms Database) and MAGMA:

• Examples 5.1 & 5.2: Demonstrate Theorem 3.7 where $M$ is prime ($M=2$). In these cases, the conjecture holds, and explicit newforms are found satisfying the congruences.
• Example 5.3: Demonstrates the conjecture for a composite $M$ ($M=6$). A newform of level 42 is found satisfying the congruence, providing evidence for the general case, though a theoretical proof for composite $M$ remains open.

6. Significance

• Generalization: This work significantly generalizes the theory of Eisenstein congruences to arbitrary square-free levels and arbitrary characters, moving beyond the classical trivial character case.
• Bridge to Arithmetic Geometry: By linking these congruences to the Bloch-Kato conjecture and Selmer groups, the paper provides a concrete framework for understanding how local Euler factors influence global arithmetic invariants.
• Newform Existence: It clarifies the subtle distinction between the existence of any eigenform satisfying a congruence versus a newform. The "local origin" terminology highlights that the obstruction to finding a newform is often local (related to specific primes dividing the level).
• Open Problems: The paper explicitly highlights the difficulty of proving the existence of a global newform when $M$ is composite, suggesting that current techniques (which work prime-by-prime) are insufficient for the general case. This sets a clear direction for future research in the field.