Intersecting Complex Tori

Let $T=V/\Lambda$ be a complex torus, and suppose that $A=V_A/\Lambda_A$ and $B=V_B/\Lambda_B$ are subtori, so $V_A$ and $V_B$ are subspace of $V$, $\Lambda_A = V_A\cap \Lambda$, and $\Lambda_B = V_B\cap \Lambda$.

Proposition 3.2   If $A\cap B$ is finite, then there is an isomorphism

$$A\cap B \cong \left(\frac{\Lambda}{\Lambda_A + \Lambda_B}\right)_{\tor.}$$

Proof. Extend the exact sequence

$$0\rightarrow A\cap B \rightarrow A \oplus B \xrightarrow{(x,y)\mapsto x-y} J$$

to the following diagram:

$$\xymatrix{ & {\Lambda_A \oplus\Lambda_B}\ar[d] \ar[r] & {\Lambda}... ...V/(V_A+V_B)}\ar[d]\\ {A\cap B}\ar[r] & A\oplus B\ar[r] & J \ar[r] & J/(A+ B).}$$

Using the snake lemma, which connects the kernel $A\cap B$ of $A\oplus B \rightarrow J$ to the cokernel of $\Lambda_A \oplus \Lambda_B \rightarrow \Lambda$, we obtain an exact sequence

$$0 \rightarrow A\cap B \rightarrow \Lambda/(\Lambda_A + \Lambda_B) \rightarrow V/(V_A+V_B).$$

Since $V/(V_A+V_B)$ is a $\mathbf{C}$-vector space, the torsion part of $\Lambda/(\Lambda_A + \Lambda_B)$ must map to 0. No non-torsion in $\Lambda/(\Lambda_A + \Lambda_B)$ could map to 0, because if it did then $A\cap B$ would not be finite. The proposition follows. $\qedsymbol$

For abelian subvarieties of $J_0(N)$ attached to newforms, the proposition above is used as follows. The complex vector space $V=\Hom(S_2(\Gamma_0(N)),\mathbf{C})$ is the tangent space of  $J_0(N)(\mathbf{C})$ at the identity. Setting $\Lambda = H_1(X_0(N),\mathbf{Z})$ and considering $\Lambda$ as a lattice in $V$ via the integration pairing, we have $J_0(N)(\mathbf{C})\cong V/\Lambda$. Suppose $f$ and $g$ are non-conjugate newforms, and let $I_f$ and $I_g$ be their annihilators in the Hecke algebra $\mathbf{T}$. Then $V_A=V[I_f]$ and $V_B=V[I_g]$ are the tangent spaces to $A$ and $B$ at the identity, respectively. The above proposition shows that the group $\char93 (A\cap B)$ is canonically isomorphic $({\Lambda}/(\Lambda_A + \Lambda_B))_{\tor}$.

The following formula for the intersection of $n$ subtori is obtained in a similar way to that of Proposition 3.2.

Proposition 3.3   For $i=1,\ldots,n$ let $A_i = V_i/\Lambda_i$ be a subtorus of $J=V/\Lambda$, and assume that each pairwise intersection $A_i \cap A_j$ is finite. Then

$$A_1\cap \cdots \cap A_n \cong \left(\frac{\Lambda\oplus \cdots \oplus \Lambda} {f(\Lambda_1\oplus\cdots\oplus \Lambda_n)}\right),$$

where $f(x_1,\ldots,x_n)=(x_1-x_2,x_2-x_3,x_3-x_4,\ldots,x_{n-1}-x_n)$.