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Theses and Dissertations

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Browsing Theses and Dissertations by Author "Achter, Jeffrey, advisor"

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    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 member
    Given 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).
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    Conjugacy classes of matrix groups over local rings and an application to the enumeration of abelian varieties
    (Colorado State University. Libraries, 2012) Williams, Cassandra L., author; Achter, Jeffrey, advisor; Eykholt, Richard, committee member; Hulpke, Alexander, committee member; Penttila, Tim, committee member
    The Frobenius endomorphism of an abelian variety over a finite field Fq of dimension g can be considered as an element of the finite matrix group GSp2g(Z/lr). The characteristic polynomial of such a matrix defines a union of conjugacy classes in the group, as well as a totally imaginary number field K of degree 2g over Q. Suppose g = 1 or 2. We compute the proportion of matrices with a fixed characteristic polynomial by first computing the sizes of conjugacy classes in GL2(Z/lr) and GSp4(Z/lr. Then we use an equidistribution assumption to show that this proportion is related to the number of abelian varieties over a finite field with complex multiplication by the maximal order of K via a theorem of Everett Howe.
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    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 member
    Classically, 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 integrals
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    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 member
    Let 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.
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    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 member
    Alina 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.