The Visibility Cross

William A. Stein

January 22, 2000

1 The visibility cross

Let A and B be abelian subvarieties of an abelian variety J over a field K; let C and D be the corresponding quotients:

\xymatrix{ & B \ar[d] \\ A \ar[r] & J \ar[r] \ar[d] & D \\  & C.}  (1)

Denote by A, B, C, D, and J the corresponding Néron models and Ã, B̃, etc., the corresponding sheaves on the fpqf site. Let p be a prime and make the following assumptions:

1. The sequences

0 → Ã → J̃ → D̃ → 0

and

0 → B̃ → J̃ → C̃ → 0

are exact. (This is probably only necessary up to groups of order prime-to-p; if so, then this might just be an assumption about component groups having order prime-to-p.)

2. (a) B[p] ⊂ A, as they sit inside of J,

(b) p ∤ #A(K)_{tor},

(c) p ∤ #� _{B}^{∘},

(d) p ∤ #Φ_{B}.

We have the following possibly false proposition, which is close to being true.

Proposition 1.1 (???). With the above hypothesis,

#(B(K)/pB(K)) | #� _{A}^{∘} · #Φ_{A}.

Proof. Expand the cross (1) by passing to fpqf-cohomology, observe that B → D factors through [p]_{B}, and use Mazur’s computation of H_{fpqf}^{1} to obtain the following diagram:

[figure: 0; B(K); [p]; i; B(K); j; 0; A(K); J(K); π; D(K); H_{fpqf}^{1}(O, A)^{∘}; 0; C(K); � _{A}^{∘} ⊕ Φ_{A}; H_{fpqf}^{1}(O, B)^{∘}; � _{B}^{∘} ⊕ Φ_{B}; 0.]