Motivation: connection with the conjecture of Birch and Swinnerton-Dyer

On a Néron model, the global differentials are the same as the invariant differentials, so $H^0(A_{{\bf {Z}}},\Omega^1_{A/{\bf {Z}}})$ is a free ${\bf {Z}}$ -module of rank $d=\dim(A)$ . The real measure $\Omega_A$ of $A$ is the measure of  $A({\bf {R}})$ with respect to the volume given by a generator of  $\bigwedge^d H^0(A_{{\bf {Z}}},\Omega^1_{A/{\bf {Z}}}) \simeq \H ^0(A_{{\bf {Z}}}, \Omega^d_{A_{{\bf {Z}}}/{\bf {Z}}})$ . This quantity is of interest because it appears in the conjecture of Birch and Swinnerton-Dyer, which expresses the ratio $L^{(r)}(A,1)/\Omega_A$ , in terms of arithmetic invariants of $A$ , where $r={\mathrm{ord}}_{s=1} L(A,s)$ (see, e.g., [Lan91, Chap. III, §5] and [AS05, §2.3]).

If we take a ${\bf {Z}}$ -basis of $S_2({\bf {Z}})[I]$ and take the inverse image via the top chain of arrows in the commutative diagram above, we get a ${\bf {Q}}$ -basis of  $H^0(A,\Omega^1_{A/{\bf {Q}}})$ ; let $\Omega_A'$ denote the volume of  $A({\bf {R}})$ with respect to the wedge product of the elements in the latter basis (this is independent of the choice of the former basis). In doing calculations or proving formulas regarding the ratio in the Birch and Swinnerton-Dyer conjecture mentioned above, it is easier to work with the volume $\Omega_A'$ instead of working with $\Omega_A$ . If one works with the easier-to-compute volume $\Omega_A'$ instead of $\Omega_A$ , it is necessary to obtain information about  ${c_{\scriptscriptstyle{A}}}$ in order to make conclusions regarding the conjecture of Birch and Swinnerton-Dyer, since $\Omega_A = {c_{\scriptscriptstyle{A}}}\cdot \Omega_{A}'$ . For example, see [AS05, §4.2] when $r=0$ and [GZ86, p. 310-311] when $r=1$ ; in each case, one gets a formula for computing the Birch and Swinnerton-Dyer conjectural order of the Shafarevich-Tate group, and the formula contains the Manin constant (see, e.g., [Mc91]).

The method of Section 5 for verifying that ${c_{\scriptscriptstyle{A}}}=1$ for specific elliptic curves is of little use when applied to general abelian varieties, since there is no simple analogue of the minimal Weierstrass model (but see [GL01] for ${\bf {Q}}$ -curves). This emphasizes the need for general theorems regarding the Manin constant of quotients of dimension bigger than one.