Elements in $K_4$ and regulator maps of Fermat curves

Technical Summary: Elements in $K_4$ and Regulator Maps of Fermat Curves

Problem Statement
The paper addresses the construction of explicit, non-trivial elements in the third Adams eigenspace $K_4^{(3)}(X_N)$ of Fermat curves $X_N$ defined by the equation $x^N + y^N = 1$ over $\mathbb{Q}$, for all integers $N \geq 3$. While the structure of $K_2(X_N)$ and the non-triviality of the Ross element $\{1-x, 1-y\}$ are well-established, the $K_4$ group of Fermat curves remains largely unexplored. The primary objective is to construct explicit elements in $K_4^{(3)}(X_N)$ and verify their non-triviality by computing their images under Beilinson's regulator map. Furthermore, the authors aim to numerically verify Beilinson's conjecture regarding the special values of $L$-functions at $s=3$ for specific Fermat motives.

Methodology
The authors employ a uniform construction strategy based on polylogarithmic complexes and the work of Goncharov and de Jeu.

1. Polylogarithmic Complexes: The construction utilizes Goncharov's weight 3 polylogarithmic complex $\Gamma(F, 3)$ for the function field $F = \mathbb{Q}(X_N)$. This complex involves the Bloch group $B_3(F)$ and $B_2(F) \otimes F^\times_{\mathbb{Q}}$.
2. Explicit Cocycles: The authors construct explicit 2-cocycles $\xi_{r,s}^N(\zeta)$ in $\Gamma(F, 3)$ of the form:
   $$\xi_{r,s}^N(\zeta) := \{x^N\}_2 \otimes (1 - \zeta x^r y^s) - \frac{N}{s} \{\zeta x^r y^s\}_2 \otimes x^N$$
   where $\zeta$ is a root of unity.
3. De Jeu's Map: They apply de Jeu's map $\phi_{dJ}: H_2(\Gamma(F, 3)) \to K_4^{(3)}(F)$ to these cocycles. To ensure the resulting elements lie in $K_4^{(3)}(X_N)$ rather than just the open curve $U$, they verify that the residues of the cocycles vanish at the cusps and zeros of the defining functions, utilizing Theorem 3.6 which relies on the base field being totally real.
4. Specific Choice: The authors focus on the specific case $(r, s) = (-2, 1)$ and $\zeta = 1$ (and its twists by roots of unity) to define the element $\Xi_N = \Xi_{-2,1}^N$.
5. Regulator Computation: To prove non-triviality, the authors compute the image of these elements under Beilinson's regulator map $r_B$. They relate $r_B$ to Goncharov's regulator $r_3(2)$ via de Jeu's compatibility theorem. The regulator integrals are computed over a specific homology cycle $\gamma_N$ that generates the homology of $X_N(\mathbb{C})$ as a module over the group ring $\mathbb{Q}[(\mathbb{Z}/N\mathbb{Z})^{\oplus 2}]$.
6. Analytic Tools: The evaluation of these integrals involves:
   • Zagier's trilogarithm function $L_3(z)$.
   • Hypergeometric functions $F(\alpha; x)$, generalizing previous results by Otsubo for $K_2$.
   • Asymptotic analysis as $N \to \infty$.
7. Numerical Verification: For $N \in \{3, 4, 6\}$, the authors use PARI/GP to numerically evaluate the regulator integrals and compare them with the derivatives of Hecke $L$-functions associated with specific elliptic curves with Complex Multiplication (CM), which are isogenous to the relevant Fermat motives.

Key Contributions and Results

• Construction of Non-Trivial Elements: The paper proves that the constructed element $\Xi_N \in K_4^{(3)}(X_N)$ is non-trivial for all $N \geq 3$ (Theorem 1.1). This is achieved by showing that the regulator integral of a twisted version of the element, $(\tau_\varepsilon)^*(\Xi_N)$, is non-zero.
• Explicit Regulator Formulas:
  • Trilogarithm Formula: The regulator integral is expressed explicitly in terms of Zagier's trilogarithm $L_3$ and an integral involving a function $f_{N,z}(t)$ (Theorem 1.3).
  • Hypergeometric Formula: The regulator form is expressed as a linear combination of differential forms weighted by hypergeometric functions $F(\alpha; -1)$, generalizing Otsubo's results for $K_2$ (Theorem 1.6).
• Asymptotic Behavior: The authors derive an asymptotic formula for the regulator integrals as $N \to \infty$:
  $$\int_{\gamma_N} r_B((\tau_\varepsilon)^*(\Xi_N)) \sim \frac{3}{2}\zeta(3)N^2$$
  This confirms the non-triviality of the elements for large $N$ and provides insight into their growth rate (Theorem 1.4).
• Linear Independence: For $N$ not divisible by 3, the elements $\Xi_N$, $\iota^*(\Xi_N)$, and $\tau_\varepsilon^*(\Xi_N)$ are shown to be non-zero and linearly independent in $K_4^{(3)}(X_N, \mathbb{Q}(\varepsilon))$ (Theorem 1.2).
• Numerical Verification of Beilinson's Conjecture: The paper numerically verifies the weak form of Beilinson's conjecture for the motives $X_{[-2,1]}^N$ where $N \in \{3, 4, 6\}$. In these cases, the motives correspond to elliptic curves $E_3, E_4, E_6$ with CM. The computed regulator integrals match the predicted values involving $L'(E, -1)$ up to rational factors and powers of $\pi$ (Theorem 1.8).

Significance and Claims
The authors position this work as the first to tackle Beilinson's conjecture for general Fermat curves beyond the $K_2$ level.

• Generalization: The work generalizes the construction of the Ross element for $K_2$ to $K_4$ and extends Otsubo's hypergeometric formulas from $K_2$ to $K_4$.
• Uniformity: The construction is uniform in $N$, providing a single family of elements for all degrees $N \geq 3$, unlike previous constructions for modular curves which often required case-by-case analysis.
• Verification: The paper provides the first explicit numerical verification of Beilinson's conjecture at $s=3$ for Fermat curves, bridging the gap between algebraic K-theory constructions and analytic $L$-value predictions.
• Modesty: The authors note that while they verify the conjecture for rank 2 motives (corresponding to elliptic curves), the conjecture remains open for higher rank Fermat motives (e.g., $N=5$), where constructing a full basis of $K_4$ elements is still an open problem. They also acknowledge that proving the linear independence of their elements unconditionally (without relying on the conjectural injectivity of the regulator) remains difficult.