1,720,989 research outputs found
Some Topics in the Arithmetic of Hodge Structures and an Ax-Scanuel Theorem for
Get PDFIn the first part of this thesis, we consider smooth projective morphisms of -varieties with an open curve and a number field. We bound the Weil height polynomially from above by at points that are ``exceptional'' with respect to the variation of Hodge structures , where . For this variation, we assume that its generic special Mumford-Tate group is , that it degenerates strongly over some point of a curve containing , the Hodge conjecture holds, and that what we define as a ``good arithmetic model'' exists for the morphism over .\\
In the second part of this thesis, we prove an Ax-Schanuel type result for the exponential functions of general linear groups over . We prove the result first for the group of upper triangular matrices and then for the group of all invertible matrices over . As a corollary we obtain Ax-Lindemann type results for these maps, characterizing the bi-algebraic subsets of these maps.Ph.D
Algebraic Cycle Loci at the Integral Level
Get PDFLet be a smooth projective family defined over , where is a number field and is a finite set of primes. For each prime with residue field , we consider the algebraic loci in above which cohomological cycle conjectures predict the existence of non-trivial families of algebraic cycles, generalizing the Hodge loci of the generic fibre . We develop a technique for studying all such loci, together, at the integral level. As a consequence we give a non-Zariski density criterion for the union of non-trivial ordinary algebraic cycle loci in . The criterion is quite general, depending only on the level of the Hodge flag in a fixed cohomological degree and the Zariski density of the associated geometric monodromy representation.Ph.D
Functional Transcendence in Mixed Hodge Theory
Get PDFThe Ax-Schanuel theorem is a function field analogue of the Schanuel’s conjecture in transcendental number theory. Building on the works of Bakker, Gao, Klingler, Mok, Pila, Tsimerman, Ullmo and Yafaev, we extend the Ax-Schanuel theorem to mixed period mappings. Using this together with the Ax-Schanuel theorem for foliated principal bundles by Blázquez-Sanz, Casale, Freitag, and Nagloo, we furthur extend the Ax-Schanuel theorem to the derivatives of mixed period mappings. Linear subspaces in the Ax-Schanuel theorem are replaced by weak Mumford-Tate domains, which are certain group orbits of mixed Hodge structures. Likewise, algebraic subtori are replaced by a notion called weakly special subvarieties. We prove that if an irreducible analytic subset of the jet space of germs of local liftings of a mixed period mapping satisfies a certain property about the dimensions of this subset and its Zariski closure, then the projection of it lies in a proper weakly special subvariety. The proof uses o-minimal geometry, namely the definable Chow theorem and the Pila-Wilkie counting theorem. Preliminarily, we prove that the weak Mumford-Tate domains have complex structures, and that their real-split retractions can be decomposed into semisimple and unipotent parts. We also prove that the image of a mixed period mapping is contained in the weak Mumford-Tate domain that arises from the monodromy group of the variation.Ph.D
Distribution of Cokernels of (n+u) × n Matrices Over Zp and Expected Number of Fields Using Local Mass Formulas
No full textIn the first part of the thesis, let n and u be nonnegative, Mn be (n+u) x n matrices over Zp, and G be a finite abelian p-group. We find that the probability that the p-torsion of the cokernel of Mn is isomorphic to G, as n goes to infinity, is equal to the Cohen and Lenstra measure mu u of G. In the second part of the thesis, using Bhargava's heuristics and local mass formulas, we compute the expected number of cubic and sextic fields using discriminant square classes, as well as the expected number of number fields ordered by conductor.Ph.D
A Non-archimedean Definable Chow Theorem
Get PDFO-minimality has had some striking applications to number theory. The utility of o-minimal structures originates from the remarkably tame topological properties satisfied by sets definable in such structures. Despite the rigidity that it imposes, the theory is sufficiently flexible to allow for a range of analytic constructions. An illustration of this `tame' property is the following surprising generalization of Chow's theorem proved by Peterzil and Starchenko - A closed analytic subset of a complex algebraic variety that is also definable in an o-minimal structure, is in fact algebraic. While the o-minimal machinery aims to capture the archimedean order topology of the real line, it is natural to wonder if such a machinery can be set up over non-archimedean fields. In this thesis, we explore a non-archimedean analogue of an o-minimal structure and prove a version of the definable Chow theorem in this context.Ph.D
Statistics of class groups and related topics
No full textWe prove several results concering class groups of number fields and function fields.
Firstly we compute all the moments of the -torsion in the first step of a filtration of the class group defined by Gerth for cyclic number fields of degree , unconditionally for and underGRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen-Lenstra-Martinet conjectures. In the case this gives the distribution of the -torsion of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Kluners in their proof of the distribution of the -torsion in quadratic fields.
Secondly, we compute all the moments of a normalization of the function which counts unramified -extensions of quadratic number fields, where is the quaternion group of order , and show that the values of this function determine a point mass distribution. Furthermore we propose a similar modification to the non-abelian Cohen-Lenstra heuristics for unramified -extensions of quadratic fields for several other 2-groups , which we conjecture will give finite moments which determine a distribution. These are all cases in which the unnormalized average is known or conjectured to be infinite. Our method additionally can be used to determine the asymptotics of the unnormalized counting function, which we also do for unramified -extensions. This part of the thesis is joint work with Brandon Alberts.
Thirdly we present several new examples of reflection principles which apply to both class groups of number fields and picard groups of of curves over . This proves a conjecture of Lemmermeyer about equality of 2-rank in subfields of , up to a constant not depending on the discriminant in the number field case, and exactly in the function field case. More generally we prove similar relations for subfields of a Galois extension with group for the cases when is , , , and . The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.Ph.D
Hyperbolicity and Rational Points on Complex Ball Quotients
Get PDFLet be an -dimensional complex ball quotient by a torsion-free non-uniform lattice whose parabolic subgroups are unipotent. Let be the unique toroidal complication of
In the first part of this thesis we prove positivity properties of and \Omega^{1}_{\Xb}\big(\log(D)\big) depending intrinsically on . We prove that \Omega^{1}_{\Xb}\big(\log(D)\big) \langle} -r D \rangle} is ample for all sufficiently small rational numbers , and \Omega^{1}_{\Xb}\big(\log(D)\big) is ample modulo Further, we conclude that if the cusps of have uniform depth greater than , then \Omega^{1}_{\Xb} is semi-ample and is ample modulo , and all subvarieties of are of general type.
In the second part of this thesis we prove that the volumes of subvarieties of are controlled by
the systole of which is the length of the shortest closed geodesic of . There are a number of arithmetic and geometric consequences: the systole of controls the growth rate of rational points on uniformly in the field of definition of Also, we obtain
effective global generation and very ampleness results for multiples of the canonical bundle These results follow from the bound we find for the Seshadri constant of in terms of the systole.Ph.D
Hirzebruch Riemann Roch Theorem
Get PDFThis paper is an exposition on the Hirzebruch-Riemann-Roch Theorem, a generalization of the classicalRiemann Roch theorem for curves and Riemann surfaces. The Classical Riemann Roch Theorem asserts that if D
is a (Weil) divisor of a curve/Riemann surface C, the dimension of the vector space of functions on C
satisfying D is dependent on deg(D) and the genus g. The goal of this paper is to
show that the result generalizes to a projective variety M and show that the appropriate generalization of the
divisor D is to consider vector bundles E on M. Afterwards, the paper will present the progress t further generalize the result in various different directions.M.Sc
Field Extensions Generated by Kernels of Isogenies
No full textGiven an odd prime , A technique due to Jean-Fran\c{c}ois Mestre allows one to construct infinitely many quadratic fields for which the ideal class group has --rank at least , using a degree isogeny between elliptic curves such that the kernel has a rational point. This technique only works for primes ; we attempt to generalize the construction for larger primes. One line of approach uses higher degree isogenies (which have no rational point in the kernel), from which we obtain higher-degree number fields with --rank at least two. In the process, we collect data on the number fields generated by the points in the kernel of an isogeny, and make a series of conjectures based on the data. We also discuss the possibilities and limitations of replacing the elliptic curves in Mestre's technique with more general abelian varieties.M.Sc
Statistics of class groups and related topics
Get PDFWe prove several results concerning class groups of number fields and function fields.
Firstly we compute all the moments of the p-torsion in the first step of a filtration of the class group defined by Gerth for cyclic number fields of degree p, unconditionally for p = 3 and under GRH in general. We show that it satisfies a distribution which Gerth conjectured as an extension of the Cohen-Lenstra-Martinet conjectures. In the p = 3 case this gives the distribution of the 3-torsion of the class group modulo the Galois invariant part. We follow the strategy used by Fouvry and Kluners in their proof of the distribution of the 4-torsion in quadratic fields.
Secondly, we compute all the moments of a normalization of the function which counts unramified H8-extensions of quadratic number fields, where H8 is the quaternion group of order 8, and show that the values of this function determine a point mass distribution. Furthermore we propose a similar modification to the non-abelian Cohen-Lenstra heuristics for unramified G-extensions of quadratic fields for several other 2-groups G, which we conjecture will give finite moments which determine a distribution. These are all cases in which the unnormalized average is known or conjectured to be infinite. Our method additionally can be used to determine the asymptotics of the unnormalized counting function, which we also do for unramified H8-extensions. This part of the thesis is joint work with Brandon Alberts.
Thirdly we present several new examples of reflection principles which apply to both class groups of number fields and picard groups of curves over P1/Fp. This proves a conjecture of Lemmermeyer about equality of 2-rank in subfields of A4, up to a constant not depending on the discriminant in the number field case, and exactly in the function field case. More generally we prove similar relations for subfields of a Galois extension with group G for the cases when G is S3, S4, A4, D2l and Z/ lZ × Z/rZ. The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.Ph.D
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