Joint Linnik problems

This paper proves a conjecture by Michel and Venkatesh regarding the joinings of distinct Linnik problems through simultaneous quaternionic embeddings of imaginary quadratic fields, while also addressing a related non-equivariant conjecture by Aka, Einsiedler, and Shapira.

The Gist

Imagine you are a detective trying to solve a cosmic mystery involving two different types of "dots" floating in space.

The Mystery: The "Joining" of Two Worlds

For a long time, mathematicians have studied two separate groups of dots:

1. The Sphere Dots: Imagine integer points (like coordinates on a 3D grid) that sit exactly on the surface of a giant sphere. As the sphere gets bigger, these dots spread out. A famous mathematician named Linnik proved that if you look at enough of them, they eventually spread out perfectly evenly across the sphere, like sprinkles on a donut.
2. The Modular Dots: Imagine a different set of dots floating on a strange, curved surface called a "modular surface." These are related to complex numbers and quadratic equations. They also spread out evenly as they get larger.

The Big Question:
What happens if you look at these two sets of dots at the same time? Specifically, is there a hidden rule that links a specific Sphere Dot to a specific Modular Dot? If you plot these pairs on a giant 4D map (Sphere × Modular Surface), do they eventually spread out evenly across the whole map, or do they clump together in weird patterns?

This is the "Joint Linnik Problem." It's like asking: "If I pick a random person from New York and a random person from Tokyo, is there a hidden rule connecting them, or are they just randomly distributed across the globe?"

The Old Detective Work (The Problem)
Previous attempts to solve this (by Michel and Venkatesh, and later by the authors' own team in a paper called [BB]) were stuck. They could only prove the dots spread out evenly if they made some very strict, unrealistic assumptions:

• They had to assume a massive, unproven theory called the Generalized Riemann Hypothesis (GRH) was true. This is like saying, "We can only solve this mystery if we assume the universe follows a specific, unproven law of physics."
• They had to assume the dots were "cuspidal," which is a fancy way of saying they had to ignore the "background noise" (the continuous spectrum) of the universe.

The New Breakthrough (The Solution)
In this paper, Valentin Blomer, Farrell Brumley, and Maksym Radziwiłł act like a new team of detectives who found a better way to solve the case without needing those impossible assumptions.

Here is how they did it, using simple analogies:

1. The "Mollifier" (The Noise-Canceling Headphones)

The authors realized that the "noise" in the system was making it hard to hear the signal. They used a technique called mollification.

• Analogy: Imagine trying to hear a whisper in a loud stadium. Instead of shouting louder, you put on noise-canceling headphones that filter out the specific frequencies of the crowd noise.
• In the paper: They created mathematical "filters" (mollifiers) that smooth out the erratic behavior of the dots, allowing them to see the underlying pattern without needing to assume the Riemann Hypothesis is true.

2. The "Splitting Primes" (The Key to the Door)

The old methods required the dots to be in a very specific, rare location. The new method only requires that the "universe" has enough small split primes.

• Analogy: Imagine trying to open a locked door. The old key required a very specific, rare gemstone. The new team found that if you just have enough common, small keys (primes that "split" in a certain way), you can pick the lock.
• The Result: They proved that for almost all numbers (all but a tiny, negligible fraction), these "small keys" exist. This means their solution works for the vast majority of cases, not just the rare, perfect ones.

3. The "Gauss Orthogonal Complement" (The Shadow Connection)

One of the most beautiful parts of their work involves a classic idea from the mathematician Carl Friedrich Gauss.

• The Setup: Gauss discovered a way to take a point on a sphere (like a 3D coordinate) and turn it into a "shadow" on the modular surface.
• The New Proof: The authors proved that if you take a point on the sphere and its corresponding "shadow" on the modular surface, and you plot all of them together, they eventually fill the entire 4D space evenly.
• Why it matters: This confirms a conjecture that had been hanging over the field for years. It shows that the "sphere world" and the "modular world" are deeply, randomly connected, not just in isolated cases.

4. The "No Siegel Zero" Condition (The Safety Net)

The only condition they do need is that there are no "Siegel zeros."

• Analogy: Think of a "Siegel zero" as a glitch in the matrix—a weird, rogue number that breaks the rules of how numbers behave.
• The Result: The authors say, "As long as there are no glitches in the matrix (no Siegel zeros), our proof works." This is a much weaker condition than assuming the entire Riemann Hypothesis is true. It's like saying, "As long as the car doesn't have a flat tire, we can drive to the destination," rather than "As long as the car is a perfect, unbreakable machine."

The Big Picture

This paper is a massive step forward in Number Theory and Ergodic Theory (the study of how things move and spread out).

• Before: We could only prove the dots spread out evenly if we assumed a "miracle" (GRH) or if we ignored half the data.
• Now: We can prove it for almost all cases using only the fact that the universe has enough "small keys" (split primes) and no "glitches" (Siegel zeros).

In a Nutshell:
The authors took a problem that seemed to require a magic wand (unproven hypotheses) and replaced it with a sturdy, reliable tool (spectral theory and mollification). They showed that the universe of numbers is far more orderly and interconnected than we thought, and that two seemingly different worlds of mathematics are actually dancing together in perfect, random harmony.

Technical Summary

Here is a detailed technical summary of the paper "Joint Linnik Problems" by Valentin Blomer, Farrell Brumley, and Maksym Radziwiłł.

1. Problem Statement and Context

The paper addresses the simultaneous equidistribution of Linnik points, a problem concerning the joint distribution of arithmetic objects associated with imaginary quadratic fields. Specifically, it investigates the distribution of pairs of points on product spaces arising from two distinct "Linnik problems."

• Classical Linnik Problems:
  1. Sphere: The equidistribution of integer points on the sphere $S^2$ with large norm $d$ (solutions to $x_1^2 + x_2^2 + x_3^2 = d$).
  2. Modular Curve: The equidistribution of CM points (complex multiplication points) on the modular curve $Y_0(1)$ associated with a discriminant $-D$.
• The "Joining" Problem: The central question is whether these two sets of points, when mapped to a product space (e.g., $S^2 \times Y_0(1)$), become equidistributed with respect to the product of their respective invariant measures.
• Previous Limitations: Previous work by the authors (Blomer–Brumley, [BB]) and others (Michel–Venkatesh, Aka–Einsiedler–Shapira) established partial results under strong assumptions:
  • Generalized Riemann Hypothesis (GRH): Required for automorphic $L$-functions up to degree 8.
  • Cuspidality: Required testing functions to be cuspidal, excluding the continuous spectrum (Eisenstein series).
  • Splitting Conditions: Required specific primes to split in the quadratic field, often leading to large exceptional sets.

2. Methodology and Novel Techniques

The authors introduce a new analytic framework that replaces the reliance on GRH and cuspidality with mollification, period formulae, and spectral theory.

A. Spectral Expansion and Periods

Instead of converting Heegner periods directly into central $L$-values (via Waldspurger's formula) and applying GRH, the authors work directly with the spectral expansion of periods.

• They analyze twisted moments of Heegner periods: $\sum_{\chi} W_{D}(f; \chi) W_{D}(g; \chi)$.
• They utilize incomplete Eisenstein series ($E_\Psi$) to handle the continuous spectrum, a significant departure from previous works that were restricted to cusp forms. This allows the treatment of non-compact factors (like the modular curve) without the cuspidality restriction.

B. Mollification Schemes

To control the correlation between the two factors in the product space, the authors employ sophisticated mollification techniques to make the factors "statistically independent" or "almost parallel."

1. Equivariant Case ($\nu=1$): When the class group acts with the same speed on both factors, they use a symmetric mollifier based on truncated exponentials of prime sums. This exploits the statistical independence of $\chi$-twisted periods.
2. Off-Speed Case ($\nu=2$): When the class group acts with different speeds (e.g., $\chi$ vs. $\chi^2$), they introduce a novel asymmetric mollification and a probabilistic conditioning argument.
   • They define a "typical" set of characters $X$ where the first factor has large mass.
   • They show that the second factor has negligible mass on this set $X$ due to the mismatch in character frequencies ($\chi$ vs $\chi^2$).
   • This avoids the need for GRH to handle the non-bijectivity of the squaring map on the class group.

C. Zero-Free Regions and Siegel Zeros

The paper replaces the GRH assumption with a much weaker condition related to Siegel zeros.

• Hypothesis: The quadratic Dirichlet $L$-function $L(s, \chi_{-D})$ has no zeros in the region $|s-1| \leq \psi(D)/\log D$ for some $\psi(D) \to \infty$.
• Implication: This is equivalent to the abundance of small split primes in the quadratic field. The authors prove that this condition holds for all but a very small set of discriminants (specifically, $O((\log \log X)^{1+o(1)})$ discriminants up to $X$).
• Technical Tool: They prove a new result (Theorem 8.1) establishing cancellation in short character sums over primes under these weak zero-free conditions, bypassing the need for strong zero-free regions like $\sigma > 1 - c/\log D$.

D. Automorphic Input (Symmetric Powers)

To ensure that Hecke eigenvalues do not conspire to vanish (which would break equidistribution), the authors rely on the automorphy and cuspidality of symmetric power lifts ($sym^k \pi$ for $k=2,3,4$) proved by Gelbart–Jacquet, Kim–Shahidi, and Kim.

• They use a linear programming argument (Lemma 7.2) involving polynomials in Hecke eigenvalues to show that the set of primes where eigenvalues differ has a logarithmic density strictly greater than $1/2$. This is crucial for the "pigeonhole" step in the proof.

3. Key Results

Main Theorem (Theorem 2.5)

The $\nu$-diagonal Heegner packets equidistribute on the product space $Y_1 \times Y_2$ as the discriminant $D \to \infty$, provided:

1. The quaternion algebras involved satisfy specific non-isomorphism or non-split conditions.
2. The discriminant $-D$ satisfies the no-Siegel-zero condition (no zeros in the specified region).
3. The error term is effective: $O(\psi(D)^{-\rho})$ for the equivariant case and $O((\log \psi(D))^{-1/2})$ for the off-speed case.

Specific Applications

1. Gauss Orthogonal Complement (Theorem 2.1):
   • Proves that the set of pairs $(\text{Arg}(x), \tau(\phi_x))$—linking integer points on the sphere to CM points on the modular surface via Gauss's orthogonal complement construction—equidistributes.
   • This resolves a conjecture by Aka–Einsiedler–Shapira [AES] under the weak splitting condition, removing the need for fixed auxiliary primes.
2. Simultaneous Quaternionic Embeddings (Theorem 2.3):
   • Proves equidistribution for pairs of points arising from embeddings into two different definite quaternion algebras (e.g., $B(2,8)$ and $B(5,8)$).
   • Handles cases where the class group acts with "double speed" on one component.

4. Significance and Impact

• Removal of GRH: This is the most significant breakthrough. The paper demonstrates that deep equidistribution results in number theory can be achieved without assuming the Generalized Riemann Hypothesis, relying instead on the abundance of small split primes (a condition known to hold for almost all discriminants).
• Continuous Spectrum: By successfully treating incomplete Eisenstein series, the authors extend the scope of Linnik-type problems to include non-compact settings (like the modular curve) without restricting to cusp forms.
• Quantitative Splitting: The work provides a precise quantitative link between the analytic condition of "no Siegel zeros" and the arithmetic condition of "many small split primes," unifying ergodic and analytic approaches.
• André-Oort Conjecture: The results imply a strong form of the André-Oort conjecture for products of Shimura varieties, replacing Zariski density with equidistribution.
• New Analytic Tools: The development of asymmetric mollification and the probabilistic conditioning on character sets for the "off-speed" case offers new techniques for handling non-equivariant joinings in ergodic theory and analytic number theory.

In summary, this paper represents a major advancement in the theory of simultaneous equidistribution, resolving long-standing conjectures by Michel–Venkatesh and Aka–Einsiedler–Shapira by replacing heavy automorphic assumptions with refined analytic number theory and spectral methods.