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Let L be a number field and let E ⊂ OL∗ be any subgroup of the units of L. If rankZ(E) = 1, Lehmer’s conjecture predicts that the height of any non-torsion element of E is bounded below by an absolute positive constant. If rankZ(E) = rankZ(OL∗), Zimmert proved a lower bound on the regulator of E which grows exponentially with [L : Q]. By sharpening a 1997 conjecture of Daniel Bertrand’s, Fernando Rodriguez Villegas “interpolated” between these two extremes of rank with a new higher-dimensional version of Lehmer’s conjecture. Here we prove a high-rank case of the Bertrand-Rodriguez Villegas conjecture. Namely, it holds if L contains a subfield K for which [L : K] ≫ [K : Q] and E contains the kernel of the norm map from OL∗ to OK∗ .
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We study the vanishing of Massey products of order at least 3 for absolutely irreducible smooth projective curves over a perfect field with coefficients in Z/ℓ. We mainly focus on elliptic curves, for which we obtain a complete characterization of when triple Massey products do not vanish.more » « less
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The Iwasawa theory of CM fields has traditionally concerned Iwasawa modules that are abelian pro-p Galois groups with ramification allowed at a maximal set of primes over p such that the module is torsion. A main conjecture for such an Iwasawa module describes its codimension one support in terms of a p-adic L-function attached to the primes of ramification. In this paper, we study more general and potentially much smaller modules that are quotients of exterior powers of Iwasawa modules with ramification at a set of primes over p by sums of exterior powers of inertia subgroups. We show that the higher codimension support of such quotients can be measured by finite collections of characteristic ideals of classical Iwasawa modules, hence by p-adic L-functions under the relevant CM main conjectures.more » « less
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Let M be a compact 3-manifold and = π1(M). The work by Thurston and Culler– Shalen established the SL2(C) character variety X() as fundamental tool in the study of the geometry and topology of M. This is particularly the case when M is the exterior of a hyperbolic knot K in S3. The main goals of this paper are to bring to bear tools from algebraic and arithmetic geometry to understand algebraic and number theoretic properties of the so-called canonical component of X(), as well as distinguished points on the canonical component, when is a knot group. In particular, we study how the theory of quaternion Azumaya algebras can be used to obtain algebraic and arithmetic information about Dehn surgeries, and perhaps of most interest, to construct new knot invariants that lie in the Brauer groups of curves over number fields.more » « less
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By work of Belyi, the absolute Galois group G_Q of the rational numbers embeds into a subgroup \hat{GT} called the Grothendeick-Teichmuller group of the group A of continuous automorphisms of a profinite group on two generators. We show that a rich class of representations of G_Q lifts to \hat{GT} by showing they lift all the way to a finite index subgroup of A.more » « less
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In this paper we compute cup products of elements of the first étale cohomology group of μℓ over a smooth projective geometrically irreducible curve C over a finite field k when ℓ is a prime and #k≡1 mod ℓ. Over the algebraic closure of k, such products are values of the Weil pairing on the ℓ-torsion of the Jacobian of C. Over k, such cup products are more subtle due to the fact that they naturally take values in Pic(C)⊗Zμ~ℓ rather than in the group μ~ℓ of ℓth roots of unity in k∗.more » « less
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Coppersmith’s method for finding small solutions of multivariable congruences uses lattice techniques to find sufficiently many algebraically independent polynomials that must vanish on such solutions. We apply adelic capacity theory in the case of two variable linear congruences to determine when there is a second such auxiliary polynomial given one such polynomial. We show that in a positive proportion of cases, no such second polynomial exists, while in a different positive proportion one does exist. This has applications to learning with errors and to bounding the number of small solutions.more » « less
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