We give a purely local proof, in the depth 0 case, of the result by HarrisTaylor which asserts that the local Langlands correspondence for GLn is realized in the vanishing cycle cohomology of the deformation spaces of one-dimensional formal modules of height n. Our proof is given by establishing the direct geometric link with the Deligne-Lusztig theory for GLn(Fq).

We analyze the spectrum of tensor-triangulated category Artin-Tate motives over base field R real numbers, with integral coefficients. Away from 2, we obtain same as for complex Tate motives, previously studied by second-named author. So novelty is concentrated at prime where modular representation theory enters picture via work Positselski, based on Voevodsky's resolution Milnor Conjecture. Wi...

This article explains the Hasse principle and gives a self-contained development of certain counterexamples to this principle. The counterexamples considered are similar to the earliest counterexample discovered by Lind and Reichardt. This type of counterexample is important in the theory of elliptic curves: today they are interpreted as nontrivial elements in Tate– Shafarevich groups.

We survey recent developments in Hodge theory which are closely tied to families of CY varieties, including Mumford-Tate groups and boundary components, as well as limits of normal functions and generalized Abel-Jacobi maps. While many of the techniques are representation-theoretic rather than motivic, emphasis is placed throughout on the (known and conjectural) arithmetic properties accruing t...

We study the Madsen-Tillmann spectrum CP∞ −1 as a quotient of the Mahowald pro-object CP∞ −∞ , which is closely related to the Tate cohomology of circle actions. That theory has an associated symplectic structure, whose symmetries define the Virasoro operations on the cohomology of moduli space constructed by Kontsevich and Witten.

Let us assume every Chebyshev matrix is bounded and multiply Poisson. Recent developments in absolute graph theory [14] have raised the question of whether V −1 (C −∞) ⊂ ∑ ξ (∅+ π, . . . , ∅ ∨ i) . We show that |τ̄ | 6= −1. A central problem in advanced algebra is the derivation of Tate fields. This could shed important light on a conjecture of Desargues.

Let E be an elliptic curve over Q. Using Iwasawa theory, we give what seems to be the first general upper bound for the order of vanishing of the p-adic L-function at s = 0, and the Zp-corank of the Tate-Shafarevich group for all sufficiently large good ordinary primes p.

We explain how to combine deep results from Iwasawa theory with explicit computation to obtain information about p-parts of Tate-Shafarevich groups of elliptic curves over Q. This method provides a practical way to compute #X(E/Q)(p) in many cases when traditional p-descent methods are completely impractical and also in situations where results of Kolyvagin and Kato do not apply.