This is exactly what I will say during my Princeton talk on
Visibility of Shafarevich-Tate Groups.
BLACKBOARD 1: Modular abelian varieties
Hello.
* Thanks for your generous hospitality in inviting me to talk about
Visibility of Shafarevich-Tate Groups
* I will primarily talk about visibility in the context of modular
abelian varieties, which are constructed as follows.
* Let f equal sum a-n q-to-the-n be a weight two newform on Gamma-0-N;
in particular, f is an element of S-2 gamma-0-of-N; this space is
canonically isomorphic to the global differentials on the modular
curve X-0-of-N.
* Here X-0-of-N is the compactified quotient of the upper
half plane by the action of Gamma-0-of-N (the 2-by-2 integer determinant-1
matrices that are upper triangular modulo N), endowed with a canonical
structure of algebraic curve over the rational numbers.
* X-0-of-N sits inside its Jacobian J_0(N) (draw arrow)
* The Hecke algebra T, which equals Z adjoin the T_n, for each positive n,
acts as a ring of endomorphisms on J_0(N) and hence on cusp forms.
* Furthermore, because we assumed that f is a newform, it is an eigenform
for the action of each element of T.
* Let I_f denote the annihilator of f in T.
* The quotient T modulo I_f is the ring Z adjoin the a-sub-n.
* We now attach an arithmetic and an analytic object to f.
* The quotient of J_0(N) by I_f*J_0(N) is an optimal abelian variety
quotient A_f of dimension equal to the number of Galois conjugates of f.
* This is an important class of abelian varieties--all elliptic curves
over Q appear, up to isogeny, as such quotients.
* The L-series of A_f is the product of the Dirichlet series
sum a_n^{(i)}/n^s attached to the Galois conjugates of f,
analytically continued to all of C.
* The conjecture of Birch and Swinnerton-Dyer connects these two objects.
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BLACKBOARD 2: BSD Conjecture
* A special case of their more general conjecture is that
the special value at 1 of the L-series attached to A_f equals
the following product of invariants of A_f:
* It equals Omega (the real volume) times the product of the
Tamagawa numbers c_p (the orders of component groups) times
the order of the Shafarevich-Tate group divided by the product
of the orders of A_f(Q) and A_f-dual-(Q).
* There are two caveats: One. Part of the conjecture is that Sha(A_f) is
finite, which is far from being known in general. Two. The right
hand side should be interpreted as zero if A_f(Q) is infinite.
* Two selected partial results:
* Using the formalism of modular symbols one can show that
L(A_f,1)/Omega is a (computable) rational number. For example,
Manin made this observation this when A_f has dimension 1.
* Kolyvagin and others: L(A_f,1)=/=0 then A_f(Q) finite.
The converse remains an enticing open problem: If L(A_f,1)=0,
we do not in general know how to construct points in A_f(Q),
except when L'(A_f,1)=/=0. See Wiles' Clay Math Inst. paper.
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BLACKBOARD 3: Mazur's imperative: Visualize Sha!
* A few years ago I saw Barry Mazur give a few lectures on a clever
way of thinking about Shafarevich-Tate groups of elliptic curves.
He said: "Visualize Sha!"
* Let A be an abelian variety over Q. (Everything works over general
number fields, but let's choose Q to fix ideas.)
* Then Sha(A) is by definition the kernel of the natural restriction map
from the first Galois cohomology group H^1(Q,A) to the product over all
places v of Q of the local Galois cohomology groups H^1(Q_v,A).
* Here, H^1(Q,A) is the direct limit of the finite group cohomology
H^1(Gal(K/Q), A(K)).
* Alternatively, we can view H^1(Q,A) as the group of principal homogeneous
spaces X for A (modulo suitable equivalence). Such a space is equipped
with a map A x X --> X satisfying axioms similar to those of a simply
transitive group action.
* Then Sha(A) equals the subgroup of homogeneous spaces X that have a point
in every completion of Q.
* Given a cohomology class c, here is one possible recipe to construct X.
* There is a finite extension K of Q such that res_K(c)=0.
* By Shapiro's lemma, H^1(K,A) is canonically isomorphic to H^1(Q,Res(A)).
* Furthermore, there is a canonical injection A\->Res(A).
* Then c maps to 0 in H^1(Q,Res(A)).
* Consider the long exact sequence of Galois cohomology attached to
the short exact sequence
0 ---> A --> Res_{K/Q} A_K -pi--> B.
* We have B(Q) maps to H^1(Q,A) maps to H^1(Q,Res{K/Q}A_K), and c must
be hit by a point P in B(Q). Then X = pi^(-1)(P) is the fiber over P.
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BLACKBOARD 4: Definition of Visibility.