| April 5, 2002 |
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| April 12, 2002 |
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| April 19, 2002 |
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| April 26, 2002 |
Abstract: Let K/Q be the extension obtained by adjoining the n-th roots of unity to Q (where n is assumed to be squarefree), and let G denote the Galois group of this extension. Hilbert's Theorem 132 states that the ring of integers O_K of K is freely generated over the integral group ring Z[G] by a primitive n-th root of unity. In this talk, I shall describe recent work of W. Bley and D. Burns which gives a conjectural generalisation of Hibert's Theorem 132, in which K/Q is replaced by an arbitrary finite abelian extension M/L. The conjecture involves invariants which arise from the etale cohomology of G_m over Spec(L) for certain non-archimedean completions of L, and it is suggested by the study of equivariant Tamagawa numbers of motives. |
| May 3, 2002 |
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| May 10, 2002 |
Abstract: In my previous talk (in the Winter) I proved some lower bounds to the degrees of liftings of hyperelliptic curves in characteristic $p>2$ (to elliptic curves in characteristic 0) and gave a necessary condition to achieve those bounds. The proof that the condition is also sufficient in the special case of elliptic curves (modulo $p^3$) depended on a conjecture that appeared in my Ph.D. thesis. Such conjecture can be stated in very elementary terms and involves only cubics in characteristic $p>3$. John Tate later found a more general conjecture and proved it. I will present Tate's proof and show the implications of it to this theory of minimal degree lifts of elliptic curves. If time allows, I will briefly discuss the case of Mochizuki's canonical lifts and a possible connection to minimal degree lifts of hyperelliptic curves. |
| May 17, 2002 |
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| May 24, 2002 |
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| May 31, 2002 |
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| June 7, 2002 |
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