Picard lattice, over finite fields

$P_2(T) = \det(T - \operatorname{Frob} \mid H^2)$ ; roots $\alpha_i$ , $|\alpha_i| = q$ .
$q^{-22}P_2(qT)$ monic; roots $\zeta_i := \alpha_i/q$ , $|\zeta_i| = 1$ .

Tate classes correspond to roots of unity (Tate, a theorem for K3 surfaces over finite fields).

$\begin{gathered} q^{-22}P_2(qT) = h(T)\prod_i\Phi_{k_i}(T)^{\gamma_i} \\ \Phi_k\text{ the }k\text{-th cyclotomic polynomial};\quad h\text{ has no cyclotomic factor} \\ \rho(X_{\Bbb{F}_{q^r}}) = \sum_{k_i\mid r}\gamma_i\deg\Phi_{k_i} \end{gathered}$

Example: $X := Z(y^4 - x^3z + yz^3 + zw^3 + w^4) \subset \mathbf{P}^3$ , $p = 89$ .

$p^{-22}P_2(pT) = (T-1)(T+1)(T-1)^4(T^4+1)h(T), \qquad \deg h = 12$

$(T-1) = \Phi_1$ , degree $1$ ;
$(T+1) = \Phi_2$ , degree $1$ ;
$(T-1)^4 = \Phi_1^4$ , degree $4$ ;
$(T^4+1) = \Phi_8$ , degree $4$ .

Over $\Bbb{F}_{89}$ : only $k=1$ divides $r=1$ , so $\rho(X_{\Bbb{F}_{89}}) = 1+4 = 5$ .
Over $\Bbb{F}_{89^r}$ : $\Phi_2$ joins when $2\mid r$ , $\Phi_8$ when $8\mid r$ ; $\rho(\overline{X}_{89}) = 1+1+4+4 = 10$ , reached at $r=8$ .
$\operatorname{Pic}(\overline{X}_{89})$ decomposes as $P_{\zeta_1}\oplus P_{\zeta_2}\oplus P_{\zeta_8}$ .

For $p > 7$ , naive point counting is impractical; crystalline methods [Abbott--Kedlaya--Roe, C, C--Harvey--Kedlaya, Tuitman--Pancratz].

How to distinguish an elliptic curve with CM from one without?

non-CM	CM
$\operatorname{End}_{\Bbb{Q}} E^{\mathrm{al}} = \Bbb{Q}$	$\operatorname{End}_{\Bbb{Q}} E^{\mathrm{al}} = \Bbb{Q}(\sqrt{-d})$

$\operatorname{End}_{\Bbb{Q}} E^{\mathrm{al}} \hookrightarrow \operatorname{End}_{\Bbb{Q}} E_p^{\mathrm{al}} \hookleftarrow \Bbb{Q}(\operatorname{Frob}_p)$ .
$p\nmid a_p \Longleftrightarrow \operatorname{End}_{\Bbb{Q}} E_p^{\mathrm{al}} \simeq \Bbb{Q}[T]/(c_p(T))$ is a quadratic field; otherwise a quaternion algebra.
If $E$ has CM by $\Bbb{Q}(\sqrt{-d})$ , then
$\begin{aligned} a_p \equiv 0 \bmod p &\Longleftrightarrow p\text{ inert or ramified in }\Bbb{Q}(\sqrt{-d}) \\ &\Longleftrightarrow \operatorname{End}_{\Bbb{Q}} E^{\mathrm{al}} \not\simeq \operatorname{End}_{\Bbb{Q}} E_p^{\mathrm{al}} \end{aligned}$
If $E$ is non-CM, $\operatorname{End}_{\Bbb{Q}} E_p^{\mathrm{al}} \cap \operatorname{End}_{\Bbb{Q}} E_q^{\mathrm{al}} \simeq \Bbb{Q}$ with probability $1$ ;
heuristically, $\operatorname{Prob}(a_p \equiv 0 \bmod p) \sim 1/\sqrt{p}$ .

$p$	11.a2: $\operatorname{End}_{\Bbb{Q}} E_p^{\mathrm{al}}$	27.a2: $\operatorname{End}_{\Bbb{Q}} E_p^{\mathrm{al}}$
$2$	$B_{2,\infty}$ , $a_2=-2$ , supersingular	$B_{2,\infty}$ , $2\equiv 2\bmod 3$
$3$	$\Bbb{Q}(\sqrt{-11})$	bad reduction
$13$	$\Bbb{Q}(\sqrt{-1})$	$\Bbb{Q}(\sqrt{-3})$ , $13\equiv 1\bmod 3$
$\Rightarrow$	$\operatorname{End}_{\Bbb{Q}} E^{\mathrm{al}} = \Bbb{Q}$	$\operatorname{End}_{\Bbb{Q}} E^{\mathrm{al}} = \Bbb{Q}(\sqrt{-3})$

Left column wanders: two non-isomorphic quadratic fields force $\Bbb{Q}$ .
Right column is constant where it is a field: CM by $\Bbb{Q}(\sqrt{-3})$ .

Picard lattice, over finite fields P 2 ( T ) = det ⁡ ( T − Frob ⁡ ∣ H 2 ) ; roots α i , ∣ α i ∣ = q . q − 22 P 2 ( q T ) monic; roots ζ i : = α i / q , ∣ ζ i ∣ = 1 . Tate classes correspond to roots of unity (Tate, a theorem for K3 surfaces over finite fields). q − 22 P 2 ( q T ) = h ( T ) ∏ i Φ k i ( T ) γ i Φ k the k -th cyclotomic polynomial ; h has no cyclotomic factor ρ ( X F q r ) = ∑ k i ∣ r γ i deg ⁡ Φ k i Example: X : = Z ( y 4 − x 3 z + y z 3 + z w 3 + w 4 ) ⊂ P 3 , p = 89 . p − 22 P 2 ( p T ) = ( T − 1 ) ( T + 1 ) ( T − 1 ) 4 ( T 4 + 1 ) h ( T ) , deg ⁡ h = 12 ( T − 1 ) = Φ 1 , degree 1 ; ( T + 1 ) = Φ 2 , degree 1 ; ( T − 1 ) 4 = Φ 1 4 , degree 4 ; ( T 4 + 1 ) = Φ 8 , degree 4 . Over F 89 : only k = 1 divides r = 1 , so ρ ( X F 89 ) = 1 + 4 = 5 . Over F 8 9 r : Φ 2 joins when 2 ∣ r , Φ 8 when 8 ∣ r ; ρ ( X ‾ 89 ) = 1 + 1 + 4 + 4 = 10 , reached at r = 8 . Pic ⁡ ( X ‾ 89 ) decomposes as P ζ 1 ⊕ P ζ 2 ⊕ P ζ 8 . For p > 7 , naive point counting is impractical; crystalline methods [Abbott--Kedlaya--Roe, C, C--Harvey--Kedlaya, Tuitman--Pancratz].

The Picard lattice

The lattice structure

Pic inside $H^2$

Geometric versus ground-field Picard group

Picard lattice, over finite fields

What the characteristic polynomial gives you

Reduction to finite characteristic

Pic plays the role of $\operatorname{End}(A)$

How to distinguish an elliptic curve with CM from one without?