Paper XI: Odd residue characteristic phases

12 Odd residue characteristic phases

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Let \(p\) be odd, let \((V,b)\) be a finite symplectic \({\mathbb F}_p\) space of dimension \(2r\), and let

\[ g\in{\operatorname{Sp}}(V) \]

have no fixed vector. Fix one of the standard finite Weil normalizations used in the literature [15], [25], [16].

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Standard fact 12.1: Fixed point free Weil trace

Under the declared normalization,

\[ \boxed{ {\operatorname{Tr}}\omega(g) = \left(\frac{-1}{p}\right)^r \left(\frac{\det(g-I)}{p}\right). } \]

Here \((\frac{\cdot}{p})\) denotes the quadratic character of \({\mathbb F}_p^{\times}\).

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Definition 12.2: Odd finite Weil phase

For fixed point free \(g\), define

\[ \Phi_p(g,V) = \left(\frac{-1}{p}\right)^r \left(\frac{\det(g-I)}{p}\right). \]

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Proposition 12.3: Symplectic invariance

\[ \iota:(V,b)\longrightarrow(V',b') \]

is a symplectic isometry and \(g'=\iota g\iota^{-1}\), then

\[ \Phi_p(g,V)=\Phi_p(g',V'). \]

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Proof of Proposition 12.3

The dimensions agree, and conjugate matrices have the same determinant of \(g-I\).

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Proposition 12.4: Orthogonal multiplicativity

Suppose

\[ V=V_1{\mathop{\perp}\limits}\cdots{\mathop{\perp}\limits} V_s \]

and \(g\) preserves every summand. If no \(g|_{V_j}\) has a fixed vector, then

\[ \boxed{ \Phi_p(g,V) = \prod_{j=1}^{s} \Phi_p(g|_{V_j},V_j). } \]

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Proof of Proposition 12.4

The half dimensions add, and

\[ \det(g-I\mid V) = \prod_j\det(g-I\mid V_j). \]

The quadratic character is multiplicative.

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Corollary 12.5: Characteristic polynomial certificate

Let

\[ P_g(X)=\det(XI-g\mid V). \]

Then

\[ \boxed{ \Phi_p(g,V) = \left(\frac{-1}{p}\right)^r \left(\frac{P_g(1)}{p}\right). } \]

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Proof of Corollary 12.5

Since \(\dim V\) is even,

\[ \det(g-I)=\det(I-g)=P_g(1). \]

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Proposition 12.6: Fixed and moving decomposition

Assume that \(g\in{\operatorname{Sp}}(V)\) has order prime to \(p\). Then

\[ V=\ker(g-I){\mathop{\perp}\limits}{\operatorname{im}}(g-I). \]

The restriction of the symplectic form to \({\operatorname{im}}(g-I)\) is nondegenerate, and \(g-I\) is invertible on that moving summand.

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Proof of Proposition 12.6

The order hypothesis makes \(g\) semisimple, so

\[ V=\ker(g-I)\oplus{\operatorname{im}}(g-I). \]

If \(x\in\ker(g-I)\) and \(y=(g-I)z\), then

\[ b(x,y)=b(x,gz)-b(x,z)=b(g^{-1}x,z)-b(x,z)=0. \]

Thus the two summands are orthogonal. Nondegeneracy on each summand follows from nondegeneracy on \(V\). The last statement is immediate from the direct sum decomposition.

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Definition 12.7: Moving phase

Under the hypotheses of Proposition 12.6, define

\[ \Phi_p^{\mathrm{mov}}(g,V) = \Phi_p\left(g|_{{\operatorname{im}}(g-I)},{\operatorname{im}}(g-I)\right). \]

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Proposition 12.8: Hyperbolic phase formula

Let \(U\) be a finite dimensional \({\mathbb F}_p\) space, let \(a\in{\operatorname{GL}}(U)\) have no eigenvalue one, and let \(g\) act on

\[ H(U)=U\oplus U^* \]

\[ g=(a,(a^{-1})^*). \]

Then

\[ \boxed{ \Phi_p(g,H(U)) = \left(\frac{\det a}{p}\right). } \]

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Proof of Proposition 12.8

Put \(d=\dim U\). We have

\[ \det\bigl((a^{-1})^*-I\bigr) = (-1)^d(\det a)^{-1}\det(a-I). \]

Hence

\[ \det(g-I) = (-1)^d(\det a)^{-1}\det(a-I)^2. \]

Multiplying its quadratic character by the normalization factor \((\frac{-1}{p})^d\) cancels the sign and the square. Since a quadratic character equals its inverse, the result is \((\frac{\det a}{p})\).

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Corollary 12.9: Phase only bridge test

Two odd characteristic moving packets have the same normalized phase whenever their moving half dimensions agree and the determinants of \(g-I\) lie in the same square class.

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Proof of Corollary 12.9

Apply the definition of \(\Phi_p\).

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Odd finite Weil phases multiply across orthogonal atomic blocks.

Remark 12.10

When \(g\) has fixed vectors, one first separates the fixed part and applies the formula to the moving quotient. Additional normalization data may enter the full trace on the fixed part.

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