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On the Bloch-Kato conjecture for GSp(4)

This paper establishes an explicit reciprocity law for the Euler system associated with the spin motive of a genus 2 Siegel modular form, thereby proving one inclusion of the Iwasawa Main Conjecture and verifying the Bloch-Kato conjecture in analytic rank 0 for its critical twists.

Original authors: David Loeffler, Sarah Livia Zerbes

Published 2026-07-23

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: David Loeffler, Sarah Livia Zerbes

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 universe of numbers as a vast, silent library where every book is a mathematical object, and hidden inside each book are secret codes called "L-functions." These codes are like the DNA of numbers; they hold the answers to some of the most stubborn puzzles in mathematics, such as how many solutions exist for certain equations or how prime numbers are distributed. For decades, mathematicians have been trying to crack these codes, but the books are written in a language so complex that most people can't even read the cover. To make progress, they use powerful tools called "Euler systems," which act like a master key, allowing them to unlock specific doors in the library and peek inside. However, for a very important class of these mathematical objects—specifically those related to a shape called a "Siegel modular form"—the key was broken. The mathematicians knew the key existed, but they couldn't prove it actually worked. They were stuck in a loop: they needed the key to work to prove the key worked.

This paper is the story of how two mathematicians, David Loeffler and Sarah Livia Zerbes, finally fixed that broken key. They didn't just guess; they built a massive, intricate bridge made of advanced geometry and algebra to prove that the key fits perfectly. Their work confirms a deep connection between the shape of these number-theoretic objects and the values of their L-functions. By proving this connection, they have unlocked new ways to understand the "arithmetic" of these objects, specifically proving that in certain cases, the number of solutions to these equations is exactly what the L-functions predicted. It's a bit like finally proving that a specific map leads to a hidden treasure, rather than just hoping the map is real.

The Story of the Broken Key

In the world of number theory, there is a famous hypothesis called the Bloch–Kato conjecture. Think of this conjecture as a promise: it says that if you look at a specific mathematical object (like a complex shape made of numbers), the number of "holes" or "loops" it has (which mathematicians call the rank of its solution set) is directly tied to a specific number you get from its L-function. If the L-function is zero at a certain point, the object should have a certain number of solutions. If it's not zero, it should have none.

To prove this, mathematicians use a tool called an Euler system. Imagine an Euler system as a set of "clues" scattered across different mathematical landscapes. If you can find these clues and show they are not empty (not zero), you can use them to bound the number of solutions. The problem is, for a long time, the authors of this paper had built a set of clues for a specific type of shape called a genus 2 Siegel modular form (a fancy, high-dimensional generalization of a donut shape), but they couldn't prove the clues were actually useful. They had a "reciprocity law"—a formula that should link the clues to the L-function—but they couldn't prove the formula was true. Without that proof, the entire Euler system could have been zero, rendering it useless.

The Big Breakthrough

The main achievement of this paper is the proof of an explicit reciprocity law. The authors successfully demonstrated that the Euler system they built for these Siegel modular forms is indeed non-zero and that it connects perfectly to the values of the L-function.

Here is how they did it, using a few creative metaphors:

The Problem of the "Ordinary" Locus: The authors needed to calculate a specific value (a "regulator") that links their clues to the L-function. The math was too messy to do everywhere, so they decided to focus on a specific, cleaner area called the "multiplicative-ordinary locus." Imagine trying to hear a whisper in a noisy stadium; instead of shouting over the crowd, they found a quiet, soundproof room (the ordinary locus) where the whisper was clear.
The Bridge of "Partial Support": To get to this quiet room, they had to cross a bridge made of a new mathematical technique called "cohomology with partial compact support." Think of this as a special kind of net. Usually, nets catch everything, but this net is designed to catch only the "fish" (mathematical data) that are swimming in a specific direction, while letting the rest pass through. This allowed them to ignore the messy, chaotic parts of the problem and focus only on the clean, structured parts.
The "Poznań Spectral Sequence": Along the way, they discovered a new mathematical tool they jokingly named the "Poznań spectral sequence" (named after a conference in Poland where they had the idea). This tool acts like a translator. It takes a message written in one difficult language (rigid cohomology) and translates it into another language (coherent cohomology) that is much easier to read and calculate. This translation was crucial because it allowed them to use known formulas to solve the problem.
The Final Calculation: Once they had translated the problem into the easier language, they performed a series of calculations involving "Eisenstein series" (which are like special, repeating patterns in the number world). They found that one part of the calculation vanished (became zero), and the remaining part matched exactly with a specific value of the L-function.

What They Proved (and What They Didn't)

The paper proves two major things, but with different levels of strictness:

The Main Result (Theorem A): They proved the explicit reciprocity law for a wide range of these shapes, assuming the shapes are "Klingen-ordinary" at a prime number $p$ . This means the law holds true for a very broad set of conditions, provided the shape behaves nicely at that specific prime. This result is proven and holds for arbitrary levels (meaning the complexity of the shape doesn't have to be minimal).
The Stronger Result (Theorem B): They also proved a stronger version of the result, which leads to a proof of the Iwasawa Main Conjecture for these shapes. However, this stronger result requires stricter conditions: the shape must have "level 1" (the simplest possible complexity), be "Borel-ordinary" (a very specific type of nice behavior), and satisfy a "big image" condition (a technical requirement about the size of the solution set). They also assume the difference between two weight numbers, $r_1 - r_2$ , is at least 6. Under these specific, somewhat restrictive conditions, they proved that the Euler system works and that the Iwasawa Main Conjecture holds (specifically, one inclusion of the conjecture).

Why It Matters

By proving this reciprocity law, the authors have unlocked the Bloch–Kato conjecture for the "analytic rank 0" case. In plain English, this means they have proven that for these specific shapes, if the L-function is not zero at a critical point, then the set of solutions to the corresponding equation is empty (or trivial). This is a massive step forward in understanding the deep structure of numbers.

They also established one side of the Iwasawa Main Conjecture, which is a grand unification of different areas of number theory. This conjecture predicts a relationship between the algebraic structure of solutions and the analytic properties of L-functions. Proving even one side of this is a significant victory.

The authors are careful to note that while they have solved the problem for the "rank 0" case (where the L-function is non-zero), the more difficult cases (where the L-function is zero and solutions might exist) are still open. They also mention that their methods could be applied to other similar problems, such as those involving quadratic Hilbert modular forms or the Birch–Swinnerton-Dyer conjecture for abelian surfaces, suggesting that this "broken key" they fixed might open many other doors in the library of mathematics.

In summary, this paper is a triumph of modern number theory. It takes a complex, abstract problem that had been stuck for years, builds a new bridge of mathematical logic to cross the gap, and proves that the connection between the shape of numbers and their hidden codes is real and precise. It's a reminder that even in the most abstract corners of math, persistence and clever new tools can turn a "maybe" into a "definitely."

Technical Summary: On the Bloch–Kato conjecture for GSp(4)

1. Problem Statement and Context

This paper addresses the explicit reciprocity law for the Euler system attached to the spin motive of a genus 2 Siegel modular form. The authors, David Loeffler and Sarah Livia Zerbes, build upon their previous work in [LSZ22] and [LZ24], where they constructed an Euler system for the 4-dimensional $p$ -adic spin Galois representations ( $V_\Pi$ ) arising from automorphic representations $\Pi$ of $GSp_4/\mathbb{Q}$ .

In [LSZ22], the construction of the Euler system was conditional on the assumption that the system satisfied an explicit reciprocity law relating the localization of the Euler system classes at $p$ to critical values of $L$ -functions. Without this law, it was impossible to rule out the possibility that the entire Euler system was zero, thereby limiting the arithmetic applications.

The primary goal of this paper is to prove this missing explicit reciprocity law. Specifically, the authors aim to compute the image of the cohomology class $z[\Pi,q,r]_{can}(\chi)$ under the Bloch–Kato logarithm map at $p$ , expressing it as a non-critical value of a $p$ -adic spin $L$ -function.

2. Methodology and Strategy

The proof of the explicit reciprocity law (Theorem A) is achieved through a multi-step strategy that bridges étale cohomology, rigid cohomology, and coherent cohomology. The strategy involves five main stages:

Step 1: Reduction to a Pairing in Finite-Polynomial Cohomology

The authors first express the Bloch–Kato logarithm of the Euler system class as a pairing in Nekovář–Nizioł finite-polynomial (fp) cohomology. Using the equivariance properties of the Lemma–Flach classes, they reduce the problem to a specific type of test data ("Klingen-type"). The left-hand side of the reciprocity law is reformulated as a cup-product between the pushforward of a syntomic $GL_2 \times GL_2$ Eisenstein class and a lifting of a de Rham cohomology class $\eta_{dR}$ to Nekovář–Nizioł fp-cohomology.

Step 2: Reduction to the Multiplicative-Ordinary Locus

To make the computation explicit, the authors replace Nekovář–Nizioł cohomology with Besser's rigid syntomic cohomology. A crucial technical innovation is the introduction of cohomology with partial compact support. This allows the authors to restrict the computation to the "multiplicative-ordinary" locus ( $Y^m_{G,Kl}$ ) of the Shimura variety, where the special fiber is smooth. This step bypasses the lack of a global Frobenius lifting over the entire Shimura variety by working in the cohomology of the ordinary locus with appropriate support conditions.

Step 3: Reduction to Coherent Cohomology

The pairing in rigid syntomic cohomology is further reduced to a pairing in coherent cohomology. This relies on two novel ingredients:

Partial Compact Support Theory: A formalism allowing cohomology with compact support towards some boundary strata but not others, developed in Section 14.
The Poznań Spectral Sequence: A new spectral sequence (Proposition 18.4.3) relating syntomic (or finite-polynomial) cohomology to the mapping fiber of a polynomial in Frobenius over coherent cohomology. This serves as a syntomic analogue of the Frölicher spectral sequence.

This step connects the regulator to classes in the coherent cohomology groups studied in Pilloni's higher Hida theory.

Step 4: Computation of the Regulator

The authors compute the resulting pairing in coherent cohomology. They utilize an identity relating Hecke operators on $GSp_4$ and $GL_2 \times GL_1 GL_2$ to simplify the expression. The computation reduces the regulator to a linear combination of two terms, $L_1$ and $L_2$ , which are special values of $p$ -adic measures.

Through local zeta-integral computations, they show that $L_2$ vanishes identically.
They demonstrate that $L_1$ corresponds to a non-critical value of the $p$ -adic $L$ -function constructed in [LP+21].

Step 5: Deformation to Critical Values (Theorem B)

To deduce the stronger Theorem B (which establishes the Euler system for the cyclotomic twist), the authors employ $p$ -adic families. They consider "Siegel-type" Hida families where the weights vary.

They compare the "motivic" $p$ -adic $L$ -function (derived from the Euler system via the Perrin-Riou regulator) with the "analytic" $p$ -adic $L$ -function constructed via functorial transfer to $GL_4$ (using results from [BD+25]).
By showing that the ratio of these two functions is a scalar that cannot vanish (using a contradiction argument involving Mazur and Rubin's results on Euler systems and the "big image" condition), they prove that the Euler system class is non-zero and its regulator is exactly the $p$ -adic $L$ -function.

3. Key Contributions and Technical Innovations

Explicit Reciprocity Law: The paper provides the first proof of an explicit reciprocity law for the spin motive of genus 2 Siegel modular forms, linking Euler system classes to $p$ -adic $L$ -values.
Partial Compact Support: The development of a theory of rigid and coherent cohomology with partial compact support (Section 14) allows for computations on non-proper, non-smooth schemes by isolating smooth open subvarieties (the ordinary locus) while controlling boundary behavior.
The Poznań Spectral Sequence: The introduction of a spectral sequence relating syntomic cohomology to coherent cohomology (Proposition 18.4.3) provides a new tool for computing regulators in settings where degree-zero coherent cohomology is insufficient.
Comparison of $p$ -adic $L$ -functions: The paper establishes a bridge between the "coherent" $p$ -adic $L$ -function (constructed via higher Hida theory) and the "Betti" $p$ -adic $L$ -function (constructed via functorial transfer to $GL_4$ ), proving they are proportional.

4. Main Results

Theorem A (Explicit Reciprocity Law)

For a non-endoscopic, non-CAP, globally generic automorphic representation $\Pi$ of $GSp_4$ that is Klingen-ordinary at $p$ with $r_1 - r_2 \ge 3$ , the image of the cohomology class $z[\Pi,q,r]_{can}(\chi)$ under the Bloch–Kato logarithm is given by:
$\langle \nu_{dR}, \log_{BK} z[\Pi,q,r]_{can}(\chi) \rangle = (\star) \times L_{p,\nu}(\Pi, \chi; -1-r_2+q, r)$
where $(\star)$ is an explicit non-zero factor and $L_{p,\nu}$ is the 2-variable spin $p$ -adic $L$ -function from [LP+21].

Theorem B (Euler System and Perrin-Riou Regulator)

Under stronger hypotheses (Borel-ordinary, level 1, $r_1 - r_2 \ge 6$ , and the "big image" condition), there exists an Euler system for $V_\Pi^*(-1-r_2)$ whose image under the Perrin-Riou cyclotomic regulator map is the single-variable $p$ -adic $L$ -function $L_p(\Pi, j)$ .

Theorem C (Iwasawa Main Conjecture)

Assuming the conditions of Theorem B, the module $\tilde{H}^2_{Iw}(\mathbb{Q}(\mu_{p^\infty}), V)$ is torsion over the Iwasawa algebra, and its characteristic ideal divides the $p$ -adic $L$ -function $L_p(\Pi, j)$ . This establishes one inclusion of the Iwasawa Main Conjecture for these motives.

Theorem D (Bloch–Kato Conjecture)

Assuming the conditions of Theorem B, if the complex $L$ -value $L(\Pi \otimes \rho, \frac{1-r_1+r_2}{2} + j) \neq 0$ for a finite-order character $\rho$ , then the Selmer group $H^1_f(\mathbb{Q}, V(-j-\rho))$ vanishes. This proves the Bloch–Kato conjecture in analytic rank 0 for critical twists.

5. Significance and Scope

The paper claims to resolve a critical gap in the arithmetic of Siegel modular forms by proving the explicit reciprocity law necessary to activate the Euler system machinery. The results are significant for:

Iwasawa Theory: Providing the first divisibility result for the Iwasawa Main Conjecture for spin motives of $GSp_4$ .
Bloch–Kato Conjecture: Proving the conjecture for analytic rank 0 twists of these representations.
Methodology: The techniques developed, particularly the use of partial compact support and the Poznań spectral sequence, are presented as applicable to other cases where Euler systems exist but $L$ -values cannot be expressed purely in terms of degree-zero coherent cohomology (e.g., Asai representations of Hilbert modular forms).

The authors note that the simplifying hypotheses (level 1, high regularity) are imposed primarily to keep the arguments concise; the core regulator formula (Theorem A) is proved for arbitrary levels under a mild weight condition ( $r_1 - r_2 \ge 3$ ). Future work is anticipated to relax these conditions and apply the methods to other groups and motives.