We prove that the stable homotopy of any Γ-module F is the homology of a bicomplex Ξ(F), in which the (q − 1)st row is the twosided bar construction B(Lieq,Σq, F[q]). This gives a natural homotopical cotangent bicomplex for graded commutative algebras, in a form suitable for use in a new obstruction theory for classifying E∞ ring structures on spectra. The E∞ structure on certain Lubin-Tate spe...

The Weil and Tate pairings have found several new applications in cryptography. To eÆciently implement these cryptosystems it is necessary to optimise the computation time for the Tate pairing. This paper provides methods to achieve fast computation of the Tate pairing. We also give division-free formulae for point tripling on a family of elliptic curves in characteristic three. Examples of the...

2 Preliminaries 4 2.1 Syntomic cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Syntomic cohomology with log poles . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Symbol maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Tate curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5...

The Tate-Shafarevich groups of certain elliptic curves over Fq(t) are related, via étale cohomology, to the group of points of an elliptic curve with complex multiplication. The Cassels-Tate pairing is computed under this identification.

OBJECTIVES The increasing global burden of oral cancer has driven much of the focus of research to the determination of reliable prognostic markers which may have significant effects on survival and the control of post-treatment morbidity. This study was undertaken to evaluate tumour associated tissue eosinophilia (TATE) quantitatively in oral cancer specimens and observe for its possible assoc...

A Drinfeld module has a $\mathfrak{p}$-adic Tate not only for every finite place $\mathfrak{p}$ of the coefficient ring but also $\mathfrak{p} = \infty$. This was discovered by J.-K. Yu in form representation Weil group. Following an insight Taelman we construct $\infty$-adic means theory isocrystals. applies more generally to pure $A$-motives and $F$-isocrystals $p$-adic cohomology theory. We ...