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Department of Mathematics

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These digital collections include faculty/student publications, theses, and dissertations from the Department of Mathematics.

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Browsing Department of Mathematics by Author "Achter, Jeffrey, committee member"

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    Arithmetic properties of curves and Jacobians
    (Colorado State University. Libraries, 2020) Bisogno, Dean M., author; Pries, Rachel, advisor; Achter, Jeffrey, committee member; Cavalieri, Renzo, committee member; Tavani, Daniele, committee member
    This thesis is about algebraic curves and their Jacobians. The first chapter concerns Abhyankar's Inertia Conjecture which is about the existence of unramified covers of the affine line in positive characteristic with prescribed ramification behavior. The second chapter demonstrates the existence of a curve C for which a particular algebraic cycle, called the Ceresa cycle, is torsion in the Jacobian variety of C. The final chapter is a study of supersingular Hurwitz curves in positive characteristic.
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    Maximal curves, zeta functions, and digital signatures
    (Colorado State University. Libraries, 2011) Malmskog, Beth, author; Pries, Rachel, advisor; Achter, Jeffrey, committee member; Penttila, Tim, committee member; Roberts, Jacob, committee member
    Curves with as many points as possible over a finite field Fq under the Hasse-Weil bound are called maximal curves. Besides being interesting as extremal objects, maximal curves have applications in coding theory. A maximal curves may also have a great deal of symmetry, i.e. have an automorphism group which is large compared to the curve's genus. In Part 1, we study certain families of maximal curves and find a large subgroup of each curve's automorphism group. We also give an upper bound for the size of the automorphism group. In Part 2, we study the zeta functions of graphs. The Ihara zeta function of a graph was defined by Ihara in the 1960s. It was modeled on other zeta functions in its form, an infinite product over primes, and has some analogous properties, for example convergence to a rational function. The knowledge of the zeta function of a regular graph is equivalent to knowledge of the eigenvalues of its adjacency matrix. We calculate the Ihara zeta function for an infinite family of irregular graphs and consider how the same technique could be applied to other irregular families. We also discuss ramified coverings of graphs and a joint result with Michelle Manes on the divisibility properties of zeta functions for graphs in ramified covers. Part 3 is joint work with Jeremy Muskat. Gauss's curve, with equation x2t2 + y2t2 + x2y2 - t4 = 0 defined over Fp was the subject of the last entry in Gauss' mathematical diary. For p equivalent to 3 ≡ 4, we give a proof that the zeta function of C is ZC(u) = (1 + pu2)(1 + u)2/(1 - pu)(1 - u). Using this, we find the global zeta function for C. The best algorithms for solving some lattice problems, like finding the shortest vector in an arbitrary lattice, are exponential in run-time. This makes lattice problems a potentially good basis for cryptographic protocols. Right now, lattices are especially important in information security because there are no known quantum computer algorithms that solve lattice problems any faster than traditional computing. The learning with errors problem (LWE) is provably as hard as certain lattice problems. Part 4 of the dissertation is a description of a digital signature scheme based on the learning with errors problem over polynomial rings. The search version of LWE is to find a hidden vector s, given access to many pairs of noisy inner products with random vectors (ai, bi = ai • s + ei). The context can be shifted to a polynomial ring over Z/q, giving rise to the problem of learning with errors over a ring (R-LWE). In this joint work with Kristin Lauter, Michael Naehrig, and Vinod Vaikuntanathan, we devise a digital signature scheme based on R-LWE and outline a proof of security for certain parameter choices.
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    Relative oriented class groups of quadratic extensions
    (Colorado State University. Libraries, 2024) O'Connor, Kelly A., author; Pries, Rachel, advisor; Achter, Jeffrey, committee member; Shoemaker, Mark, committee member; Rugenstein, Maria, committee member
    In 2018 Zemková defined relative oriented class groups associated to quadratic extensions of number fields L/K, extending work of Bhargava concerning composition laws for binary quadratic forms over number fields of higher degree. This work generalized the classical correspondence between ideal classes of quadratic orders and classes of integral binary quadratic forms to any base number field of narrow class number 1. Zemková explicitly computed these relative oriented class groups for quadratic extensions of the rationals. We consider extended versions of this work and develop general strategies to compute relative oriented class groups for quadratic extensions of higher degree number fields by way of the action of Gal(K/Q) on the set of real embeddings of K. We also investigate the binary quadratic forms side of Zemková's bijection and determine conditions for representability of elements of K. Another project comprising work done jointly with Lian Duan, Ning Ma, and Xiyuan Wang is included in this thesis. Our project investigates a principal version of the Chebotarev density theorem, a famous theorem in algebraic number theory which describes the splitting of primes in number field extensions. We provide an overview of the formulation of the principal density and describe its connection to the splitting behavior of the Hilbert exact sequence.
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    Three projects in arithmetic geometry: torsion points and curves of low genus
    (Colorado State University. Libraries, 2019) Camacho-Navarro, Catalina, author; Pries, Rachel, advisor; Achter, Jeffrey, committee member; Cavalieri, Renzo, committee member; Peterson, Chris, committee member; Velasco, Marcela, committee member
    This paper is an exposition of three different projects in arithmetic geometry. All of them consider problems related to smooth curves with low genus and the torsion points of their Jacobians. The first project studies curves over finite fields and two invariants of the p-torsion part of their Jacobians: the a-number (a) and p-rank (f). There are many open questions in the literature about the existence of curves with a certain genus g and given values of a and f. In particular, not much is known when g = 4 and the curve is non-hyperelliptic. This is the case that we focus on here; we collect and analyze statistical data of curves over Fp for p = 3, 5, 7, 11 and their invariants. Then, we study the existence of Cartier points, which are also related to the structure of J[p]. For curves with 0 ≤ a < g, the number of Cartier points is bounded, and it depends on a and f. The second project addresses the problem of computing the endomorphism ring of a supersingular elliptic curve. This question has gained recent interest as the basis of alternative cryptosystems that hope to be resistant to quantum attacks. Our strategy is to generate these endomorphism rings by finding cycles in the l-isogeny graph which correspond to generators of the ring. We were able to find a condition for cycles to be linearly independent and an obstruction for two of them to be generators. Finally, the last chapter considers the Galois representations associated to the n-torsion points of elliptic curves over Q. In concrete, we construct models for the modular curves associated to applicable subgroups of GL₂(Z/nZ) and find the rational points on all of those which result in genus 0 or 1 curves, or prove that they have infinitely many. We also analyze the curves with a hyperelliptic genus 2 model and provably find the rational points on all but seven of them.