We survey the progress that has been made on the arithmetic of elliptic curves in the past twenty-five years, with particular attention to the questions highlighted in Tateâ€™s 1974 Inventiones paper.

We give a conjectural description of the restriction of an irreducible representation of a unitary group U(n) to a subgroup U(n âˆ’ 1) over a local or global field. We formulate analogous conjecturesâ€¦ (More)

The local Langlands correspondence can be used as a tool for making verifiable predictions about irreducible complex representations of p-adic groups and their Langlands parameters, which areâ€¦ (More)

In this paper we study the characteristic polynomials S(x) = det(xI âˆ’ F | IIp,q) of automorphisms of even, unimodular lattices with signature (p, q). In particular we show any Salem polynomial ofâ€¦ (More)

Let l â‰¥ 3 be a prime, and let p = 2 âˆ’ 1 be the corresponding Mersenne number. The Lucas-Lehmer test for the primality of p goes as follows. Define the sequence of integers xk by the recursion x0 = 4,â€¦ (More)

We prove that when all hyperelliptic curves of genus n â‰¥ 1 having a rational Weierstrass point are ordered by height, the average size of the 2-Selmer group of their Jacobians is equal to 3. Itâ€¦ (More)

Using theta correspondence, we classify the irreducible representations of Mp2n in terms of the irreducible representations of SO2n+1 and determine many properties of this classification. This is aâ€¦ (More)

In this paper, we review the theory of minuscule coweights Î» for a simple adjoint group G over C, as presented by Deligne [D]. We then decompose the associated irreducible representation VÎ» of theâ€¦ (More)