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. ------------ 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. ------------ 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. ------------ BLACKBOARD 4: Definition of Visibility.