R-triviality for adjoint classical groups of type C

1. Problem Statement

The paper investigates the R-triviality of the algebraic group of projective similitudes, denoted as $PSim(A, \sigma)$, associated with a central simple algebra $A$ of degree $2n$ over a field $K$ (characteristic $\neq 2$) equipped with a symplectic involution $\sigma$.

• Context: By Weil's classification, absolutely simple adjoint algebraic groups of type $C_n$ are isomorphic to $PSim(A, \sigma)$.
• Known Results:
  • If $n$ is odd, $PSim(A, \sigma)$ is known to be stably rational and thus R-trivial.
  • If $n=2$ (degree 4), the group is rational.
  • If $n=4$ (degree 8) and the index of $A$ is 4, counterexamples exist where the group is not R-trivial.
• The Gap: The paper addresses the case where $n$ is even (specifically $n=4$ and $n=6$, corresponding to degrees 8 and 12) to determine under what conditions $PSim(A, \sigma)$ remains R-trivial.

2. Methodology

The author employs a combination of algebraic group theory, quadratic form theory, and the theory of central simple algebras with involution. The core methodology relies on Merkurjev's Theorem, which provides a criterion for R-triviality.

• Merkurjev's Criterion (Theorem 2.1):
  The group $PSim(A, \sigma)$ is R-trivial if and only if for every field extension $K'/K$:
  $$G(A_{K'}, \sigma_{K'}) = K'^{\times 2} \cdot Hyp(A_{K'}, \sigma_{K'})$$
  Where:

  • $G(A, \sigma)$ is the group of multipliers of similitudes.
  • $Hyp(A, \sigma)$ is the subgroup of $K^\times$ generated by norms from extensions where the involution becomes hyperbolic.
  • Essentially, the problem reduces to showing that every multiplier of similitude is a product of a square and a norm from a "hyperbolicizing" extension.

• Reduction to Quadratic Forms:
  The author utilizes the fact that when the index of the algebra is small ($\leq 2$), the algebra with involution $(A, \sigma)$ can be decomposed as a tensor product of an adjoint algebra of a quadratic form and a quaternion algebra with its canonical involution:
  $$(A, \sigma) \simeq Ad(\phi) \otimes (Q, can_Q)$$
  This allows the translation of the problem from algebras with involution to properties of the associated quadratic form $\psi = N_Q \otimes \phi$ (where $N_Q$ is the norm form of the quaternion algebra).

• Tools Used:

  • Pfister Forms: Analysis of $n$-fold Pfister forms and their properties.
  • Witt Ring Theory: Utilization of the fundamental ideal $I_n K$ and the Arason-Pfister Hauptsatz.
  • Cohomological Invariants: Use of the discriminant $\Delta(\sigma) \in H^3(K, \mu_2)$ and cup products in Galois cohomology.
  • Field Extensions: Constructing specific quadratic or biquadratic extensions to force hyperbolicity or increase the Witt index.

3. Key Contributions and Results

The paper establishes R-triviality in two specific new cases and provides a counterexample context.

A. Case: Degree 8 ($n=4$) with Index $\leq 2$

• Theorem 3.2: If $(A, \sigma)$ is a central simple algebra of degree 8 with a symplectic involution and $\text{ind}(A) \leq 2$, then $PSim(A, \sigma)$ is R-trivial.
• Mechanism:
  1. The algebra decomposes into $Ad(\phi) \otimes (Q, can_Q)$.
  2. The associated quadratic form $\psi = \langle\langle a, b \rangle\rangle \otimes \phi$ (where $\langle\langle a, b \rangle\rangle$ is the norm form of $Q$) is shown to be Witt equivalent to a sum of a scaled 4-fold and a scaled 3-fold Pfister form.
  3. Citing prior results on such forms, the author concludes $PSim(\psi)$ is R-trivial, which implies $PSim(A, \sigma)$ is R-trivial via Proposition 3.1.
• Counterexample Context (Example 3.3): The paper reiterates that if $\text{ind}(A) = 4$ (maximal index), R-triviality fails. This highlights the sharpness of the condition $\text{ind}(A) \leq 2$.

B. Case: Degree 12 ($n=6$) with Trivial Discriminant

• Theorem 3.7: If $(A, \sigma)$ is a central simple algebra of degree 12 with a symplectic involution, $\text{ind}(A) \leq 2$, and the discriminant $\Delta(\sigma)$ is trivial, then $PSim(A, \sigma)$ is R-trivial.
• Mechanism:
  1. Similar decomposition: $(A, \sigma) \simeq Ad(\phi) \otimes (Q, can_Q)$ where $\phi$ is a 6-dimensional form.
  2. The condition of trivial discriminant implies a specific cohomological relation: $\Delta(\sigma) = [Q] \cup \text{disc}(\phi) = 0$.
  3. This vanishing condition forces the associated form $\psi = N_Q \otimes \phi$ to lie in the fourth power of the fundamental ideal ($I^4 K$) and be hyperbolic under specific conditions.
  4. Lemma 3.6 is crucial here: It proves that for a form $\phi \in I^4 K$ constructed as a tensor product of a 2-fold Pfister form and a 6-dimensional form, the group of multipliers $G(\phi)$ equals $K^{\times 2} \cdot Hyp(\phi)$.
  5. This equality satisfies Merkurjev's criterion, proving R-triviality.

C. Open Problem

• Problem 3.8: The paper leaves open the case of degree 12 where $\text{ind}(A) = 4$. It challenges the reader to either prove R-triviality or construct a counterexample in this specific high-index scenario.

4. Significance

• Classification of Type $C_n$: The paper significantly advances the classification of R-triviality for adjoint groups of type $C_n$ (symplectic groups). It clarifies the boundary between rationality and non-rationality based on the index of the underlying algebra and the discriminant of the involution.
• Interplay of Invariants: It demonstrates a deep connection between the R-triviality of algebraic groups and the arithmetic properties of quadratic forms (specifically Pfister forms and the Witt ring) and Galois cohomology (discriminants in $H^3$).
• Refinement of Known Bounds: By proving R-triviality for degree 8 with index $\leq 2$ and degree 12 with trivial discriminant, the author narrows the scope of where "exotic" non-R-trivial behavior can occur, suggesting that high index and non-trivial cohomological invariants are the primary obstructions to R-triviality in these classical groups.

In summary, Archita provides a rigorous proof that for symplectic involutions on central simple algebras, R-triviality holds provided the algebra is "close" to being split (low index) or the involution satisfies specific cohomological vanishing conditions (trivial discriminant), effectively mapping the landscape of R-triviality for Type $C$ groups in even dimensions.