*If we take a
-basis of
and
take the inverse image via the top chain of arrows in
the commutative diagram above, we get
a
-basis of
; let
denote the volume of
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
instead of working with
.
If one works with the easier-to-compute volume
instead
of
, it is necessary to obtain information about
in
order to make conclusions regarding the conjecture of Birch and
Swinnerton-Dyer, since
.
For example, see [AS05, §4.2]
when
and [GZ86, p. 310-311] when
; 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
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
-curves). This emphasizes the need for general theorems
regarding the Manin constant of quotients of dimension bigger than one.
*

William Stein 2006-06-25