Browsing by Author "Pries, Rachel, committee member"
Now showing 1 - 10 of 10
Results Per Page
Sort Options
Item Open Access Abelian surfaces with real multiplication over finite fields(Colorado State University. Libraries, 2014) Freese, Hilary, author; Achter, Jeffrey, advisor; Pries, Rachel, committee member; Peterson, Chris, committee member; Tavani, Daniele, committee memberGiven a simple abelian surface A/Fq, the endomorphism algebra, End(A) ⊗ Q, contains a unique real quadratic subfield. We explore two different but related questions about when a particular real quadratic subfield K+ is the maximal real subfield of the endomorphism algebra. First, we compute the number of principally polarized abelian surfaces A/Fq such that K+ ⊂ End(A) ⊗ Q. Second, we consider an abelian surface A/Q, and its reduction Ap = A mod p, then ask for which primes p is K+ ⊂ End(A) ⊗ Q. The result from the first question leads to a heuristic for the second question, namely that the number of p < χ for which K+ ⊂ End(A) ⊗ Q grows like √χ/log(c).Item Open Access Asymptotic enumeration of matrix groups(Colorado State University. Libraries, 2018) Tyburski, Brady A., author; Wilson, James B., advisor; Adams, Henry, committee member; Pries, Rachel, committee member; Wilson, Jesse W., committee memberWe prove that the general linear group GLd(pe) has between pd4e/64-O(d2) and pd4e2·log2p distinct isomorphism types of subgroups. The upper bound is obtained by elementary counting methods, where as the lower bound is found by counting the number of isomorphism types of subgroups of the generalized Heisenberg group. To count these subgroups, we use nuclei of a bilinear map alongside versor products - a division analog of the tensor product.Item Open Access Bridgeland stability of line bundles on smooth projective surfaces(Colorado State University. Libraries, 2014) Miles, Eric W., author; Cavalieri, Renzo, advisor; Achter, Jeff, committee member; Peterson, Chris, committee member; Prasad, Ashok, committee member; Pries, Rachel, committee memberBridgeland Stability Conditions can be thought of as tools for creating and varying moduli spaces parameterizing objects in the derived category of a variety X. Line bundles on the variety are fundamental objects in its derived category, and we characterize the Bridgeland stability of line bundles on certain surfaces. Evidence is provided for an analogous characterization in the general case. We find stability conditions for P1 × P1 which can be seen as giving the stability of representations of quivers, and we deduce projective structure on the Bridgeland moduli spaces in this situation. Finally, we prove a number of results on objects and a construction related to the quivers mentioned above.Item Open Access Continued exploration of nearly continuous Kakutani equivalence(Colorado State University. Libraries, 2013) Springer, Bethany Diane, author; Shipman, Patrick, advisor; del Junco, Andres, advisor; Eykholt, Richard, committee member; Dangelmayr, Gerhard, committee member; Pries, Rachel, committee memberNearly continuous dynamical systems, a relatively new field of study, blends together topological dynamics and measurable dynamics/ergodic theory by asking that properties hold modulo sets both meager and of measure zero. In the measure theoretic category, two dynamical systems (X, T) and (Y, S) are called Kakutani equivalent if there exists measurable subsets A subset of X and B subset of Y such that the induced transformations TA and SB are measurably conjugate. We say that a set A subset of X is nearly clopen if it is clopen in the relative topologyof a dense Gδ subset of full measure. Nearly continuous Kakutani equivalence refines the measure-theoretic notion by requiring the sets A and B to be nearly clopen and TA and SB to be nearly continuously conjugate. If A and B have the same measure, then we say that the systems are nearly continuously evenly Kakutani equivalent. All irrational rotations of the circle and all odometers belong to the same equivalence class for nearly continuous even Kakutani equivalence. For our first main result, we prove that if A and B are nearly clopen subsets of the same measure of X and Y respectively, and if ϕ is a nearly continuous conjugacy between TA and SB, then ϕ extends to a nearly continuous orbit equivalence between T and S. We also prove that if A subset of X and B subset of Y are nearly clopen sets such that the measure of A is larger than the measure of B, and if T is a nearly uniquely ergodic transformation and TA is nearly continuously conjugate to SB, then there exists B' subset of Y such that X is nearly continuously conjugate to SB'. We then introduce the natural topological analog of rank one transformations, called strongly rank one transformations, and show that all strongly rank one transformations are nearly continuously evenly Kakutani equivalent to the class containing all adding machines. Our main result proves that all minimal isometries of compact metric spaces are nearly continuously evenly Kakutani equivalent to the binary odometer.Item Open Access Counting isogeny classes of Drinfeld modules over finite fields via Frobenius distributions(Colorado State University. Libraries, 2024) Bray, Amie M., author; Achter, Jeffrey, advisor; Gillespie, Maria, committee member; Hulpke, Alexander, committee member; Pallickara, Shrideep, committee member; Pries, Rachel, committee memberClassically, the size of an isogeny class of an elliptic curve -- or more generally, a principally polarized abelian variety -- over a finite field is given by a suitable class number. Gekeler expressed the size of an isogeny class of an elliptic curve over a prime field in terms of a product over all primes of local density functions. These local density functions are what one might expect given a random matrix heuristic. In his proof, Gekeler shows that the product of these factors gives the size of an isogeny class by appealing to class numbers of imaginary quadratic orders. Achter, Altug, Garcia, and Gordon generalized Gekeler's product formula to higher dimensional abelian varieties over prime power fields without the calculation of class numbers. Their proof uses the formula of Langlands and Kottwitz that expresses the size of an isogeny class in terms of adelic orbital integrals. This dissertation focuses on the function field analog of the same problem. Due to Laumon, one can express the size of an isogeny class of Drinfeld modules over finite fields via adelic orbital integrals. Meanwhile, Gekeler proved a product formula for rank two Drinfeld modules using a similar argument to that for elliptic curves. We generalize Gekeler's formula to higher rank Drinfeld modules by the direct comparison of Gekeler-style density functions with orbital integralsItem Open Access Explicit and quantitative results for abelian varieties over finite fields(Colorado State University. Libraries, 2022) Krause, Elliot, author; Achter, Jeffrey, advisor; Pries, Rachel, committee member; Juul, Jamie, committee member; Ray, Indrajit, committee memberLet E be an ordinary elliptic curve over a prime field Fp. Attached to E is the characteristic polynomial of the Frobenius endomorphism, T2 − a1T + p, which controls several of the invariants of E, such as the point count and the size of the isogeny class. As we base change E over extensions Fpn, we may study the distribution of point counts for both of these invariants. Additionally, we look to quantify the rate at which these distributions converge to the expected distribution. More generally, one may consider these same questions for collections of ordinary elliptic curves and abelian varieties.Item Open Access Number-theoretic properties of the binomial distribution with applications in arithmetic geometry(Colorado State University. Libraries, 2014) Schmidt, Eric, author; Achter, Jeffrey, advisor; Pries, Rachel, committee member; Cavalieri, Renzo, committee member; Bohm, Wim, committee memberAlina Bucur et al. showed that the distribution of the number of points on a smooth projective plane curve of degree d over a finite field of order q is approximated by a particular binomial distribution. We generalize their arguments to obtain a similar theorem concerning hypersurfaces in projective m-space. We briefly describe Bucur and Kedlaya's generalization to complete intersections. We then prove theorems concerning the probability that a binomial distribution yields an integer of various certain properties, such as being prime or being squarefree. Finally, we show how to apply such a theorem, concerning a property P, to yield results concerning the probability that the numbers of points on random complete intersections possess property P.Item Open Access Properties of tautological classes and their intersections(Colorado State University. Libraries, 2019) Blankers, Vance T., author; Cavalieri, Renzo, advisor; Achter, Jeff, committee member; Pries, Rachel, committee member; Shoemaker, Mark, committee member; Heineman, Kristin, committee memberThe tautological ring of the moduli space of curves is an object of interest to algebraic geometers in Gromov-Witten theory and enumerative geometry more broadly. The intersection theory of this ring has a highly combinatorial structure, and we develop and exploit this structure for several ends. First, in Chapter 2 we show that hyperelliptic loci are rigid and extremal in the cone of effective classes on the moduli space of curves in genus two, while establishing the skeleton for similar results in higher genus. In Chapter 3 we connect the intersection theory of three families of important tautological classes (Ψ-, ω-, and κ-classes) at both the cycle and numerical level. We also show Witten's conjecture holds for κ-classes and reformulate the Virasoro operators in terms of κ-classes, allowing us to effectively compute relations in the κ-class subring. Finally, in Chapter 4 we generalize the results of the previous chapter to weighted Ψ-classes on Hassett spaces.Item Open Access Techniques in interpolation problems(Colorado State University. Libraries, 2010) Dumitrescu, Olivia Mirela, author; Miranda, Rick, advisor; Pries, Rachel, committee member; Iyer, Hariharan K., committee member; Peterson, Christopher Scott, 1963-, committee memberThis dissertation studies degeneration techniques in interpolation problems, that can be phrased as computing the dimension of the space of plane curves of degree d having general multiple points. The general interpolation problem goes back to the origin of algebraic geometry and is still far from being solved. We approach it using algebraic geometry techniques, by systematically exploiting degenerations of the projective plane. Degenerating the plane into a union of planes we prove the planar case of the interpolation problem for double points, and we present results obtained for higher multiplicities. We will generalize this technique and using toric geometry methods, we prove the interpolation problems for triple points. Using non-toric degenerations we prove the emptiness of a linear system with ten multiple points for different ratios, a result that approximates from below Nagata's bound by rational numbers. In the introduction we also state other results obtained and we mention different directions for further research.Item Open Access The conjugacy extension problem(Colorado State University. Libraries, 2021) Afandi, Rebecca, author; Hulpke, Alexander, advisor; Achter, Jeff, committee member; Pries, Rachel, committee member; Rajopadhye, Sanjay, committee memberIn this dissertation, we consider R-conjugacy of integral matrices for various commutative rings R. An existence theorem of Guralnick states that integral matrices which are Zp-conjugate for every prime p are conjugate over some algebraic extension of Z. We refer to the problem of determining this algebraic extension as the conjugacy extension problem. We will describe our contributions to solving this problem. We discuss how a correspondence between Z-conjugacy classes of matrices and certain fractional ideal classes can be extended to the context of R-conjugacy for R an integral domain. In the case of integral matrices with a fixed irreducible characteristic polynomial, this theory allows us to implement an algorithm which tests for conjugacy of these matrices over the ring of integers of a specified number field. We also describe how class fields can be used to solve the conjugacy extension problem in some examples.