Let σ(k) denote the maximum of the number of squares in a+b, . . . , a+kb as we vary over positive integers a and b. Erdős conjectured that σ(k) = o(k) which Szemerédi [30] elegantly proved as follows: If there are more than δk squares amongst the integers a+b, . . . , a+kb (where k is sufficiently large) then there exists four indices 1 ≤ i1 < i2 < i3 < i4 ≤ k in arithmetic progression such th...

In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R in a finite étale extension of R[p−1] is “almost” finite étale over R. Here, we use almost purity to lift the finite étale extension of R[p−1] to a finite étale extension of rin...

In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of `-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper...

For a fixed odd prime p and a representation ̺ of the absolute Galois group of Q into the projective group PGL2(Fp), we provide the twisted modular curves whose rational points supply the quadratic Q-curves of degree N prime to p that realize ̺ through the Galois action on their p-torsion modules. The modular curve to twist is either the fiber product of the modular curves X0(N) and X(p) or a cer...

For a (connected) smooth projective curve C over the rational numbers Q, it is known that the rational points CpQq depends on the genus g “ gpCq of C: (1) If g “ 0, then the local-global principle holds for C, i.e.: CpQq ‰ H if and only if CpQpq ‰ H for all primes p ď 8 (we understand Qp “ R when p “ 8). In other words, C is globally solvable if and only if it is locally solvable everywhere. An...

The starting point of this method is Falting’s article in which he proves the Mordell-Weil theorem. He remarked and Serre turned it into a working method, the fact that the equivalence of two λ-adic representations is something that can be basically determined on some finite extension of the base field (even though the representations might not factor through a finite quotient). Let Oλ be the r...

We introduce a new variant of tight closure associated to any fixed ideal a, which we call a-tight closure, and study various properties thereof. In our theory, the annihilator ideal τ(a) of all a-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal τ(a) and the multi...

Cayley-Dickson algebras are non-associative R-algebras that generalize the well-known algebras R, C, H, and O. We study zero-divisors in these algebras. In particular, we show that the annihilator of any element of the 2n-dimensional Cayley-Dickson algebra has dimension at most 2n−4n+4. Moreover, every multiple of 4 between 0 and this upper bound occurs as the dimension of some annihilator. Alt...