When Tate returned in the summer of 1966, I told him that I knew flat cohomology and he suggested that for my thesis I try to prove that the Brauer group of a product $E_{1}\times E_{2}$ of elliptic curves over a finite field $k$ is finite, or (a related problem) that the Tate-Shafarevich group of $E_{2}$ regarded as a constant elliptic curve over $k(E_{1})$ is finite. At the time, nothing at all was known about the $p$ components of these groups ($p=$ characteristic). In particular, the Tate-Shafarevich group was not known to be finite for any nonzero abelian variety over a global field.
By November, I had decided that the best approach to the question was through the flat cohomology groups $H^{1}(E_{1},E_{2,p^{n}})$, where $E_{2,p^{n}}$ is the finite flat group scheme $\mathrm{Ker}(p^{n}\colon E_{2}\rightarrow E_{2})$, but who knew (in 1966) what the finite group schemes $E_{p^{n}}$ were? Well, of course, Tate did. When he told me the structure of $E_{p}$ I was able to show that, for some examples, the $p$-components of the Tate-Shafarevich group and the Brauer group are zero and deduce that the entire groups are finite. Thus, the first such examples1 were found in November 1966.
During the rest of the academic year 1966/67 I was able to extend my results and show that the Tate-Shafarevich group of a constant abelian variety over a global field $k(C)$, $C$ a curve, is finite and has the order predicted by the conjecture of Birch and Swinnerton-Dyer.
At the time, it was not even clear (to some experts at least) that one should expect the $p$-component to be finite. As I was completing my thesis in spring 1967, my recollection is that Tate received a letter from André Weil in which he said that he thought he could show that the $p$-components of the Tate-Shafarevich groups of some geometrically-constant elliptic curves over global fields of characteristic $p$ are infinite. However, it follows from my thesis that if an abelian variety $A$ over $K$ becomes constant over a finite extension $L$ of $K$, then the Tate-Shafarevich group of $A/K$ is finite (because the exact sequence \[ 0\rightarrow H^{1}(L/K,A(L))\rightarrow H^{1}(K,A)\rightarrow H^{1}(L,A) \] and the Mordell-Weil theorem show that the kernel of the map on Tate-Shafarevich groups is finite).
The calculations in my thesis involved flat cohomology, Dieudonné modules, … all of which were purely characteric $p$ objects. At the time I was very struck by the fact that the results of my calculations had been predicted by a few computer calculations involving elliptic curves over $\mathbb{Q}$. Clearly, there was something very deep.
My thesis defence was attended by much of the Harvard department. Hartshorne asked why I needed flat cohomology. When he looked puzzled at my explanation, Tate suggested I begin by explaining what flat cohomology is. Afterwards Brauer warmly congratulated me, but claimed not to understand my thesis (the finiteness of the Brauer group of a product of two curves).
One note. Neither Artin nor Tate proved anything about the $p$-parts of groups in question. In the example on p.61 of my thesis, Tate showed only that the Artin-Tate conjecture predicts that the Brauer groups of $E_{1}\times E_{2}$ and $E\times E$ have the orders I state. That the prediction is correct, of course, requires my thesis. (This example is worked out in the notes for 1968a.)