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Browsing Department of Mathematics by Author "Achter, Jeffrey, advisor"
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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 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 memberThe 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.Item 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.