The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be changed to ensure that the rank of the altered matrix is at most r. Since its introduction by Valiant [Val77], rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all n × n matrices over an infinite field have a rigidity of (n − r) 2 . It is a long-standing open question to construct infinite families of explicit matrices even with superlinear rigidity when r = Ω(n).In this paper, we construct an infinite family of complex matrices with the largest possible, i.e., (n − r) 2 , rigidity. The entries of an n × n matrix in this family are distinct primitive roots of unity of orders roughly exp(n 2 log n). To the best of our knowledge, this is the first family of concrete (but not entirely explicit) matrices having maximal rigidity and a succinct algebraic description.Our construction is based on elimination theory of polynomial ideals. In particular, we use results on the existence of polynomials in elimination ideals with effective degree upper bounds (effective Nullstellensatz). Using elementary algebraic geometry, we prove that the dimension of the affine variety of matrices of rigidity at most k is exactly n 2 − (n − r) 2 + k. Finally, we use elimination theory to examine whether the rigidity function is semicontinuous.

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Abstract. We show that a two dimensional ℓ-adic representation of the absolute Galois group of a number field which is locally potentially equivalent to a GLp2q-ℓ-adic representation ρ at a set of places of K of positive upper density is potentially equivalent to ρ.For an elliptic curver E defined over a number field K and for a place v of K of good reduction for E, let F pE; vq denote the Frobenius field of E at v, given by the splitting field of the characteristic polynomial of the Frobenius automorphism at v acting on the Tate module of E.As an application, suppose E 1 and E 2 defined over a number field K, with at least one of them without complex multiplication. We prove that the set of places v of K of good reduction such that the corresponding Frobenius fields are equal has positive upper density if and only if E 1 and E 2 are isogenous over some extension of K.For an elliptic curve E defined over a number field K, we show that the set of finite places of K such that the Frobenius field F pE, vq at v equals a fixed imaginary quadratic field F has positive upper density if and only if E has complex multiplication by F .

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Abstract. We consider Tate cycles on an Abelian variety A defined over a sufficiently large number field K and having complex multiplication. We show that there is an effective bound C = C(A, K) so that to check whether a given cohomology class is a Tate class on A, it suffices to check the action of Frobenius elements at primes v of norm ≤ C. We also show that for a set of primes v of K of density 1, the space of Tate cycles on the special fibre A v of the Néron model of A is isomorphic to the space of Tate cycles on A itself.