Research Articles from WIN Conferences

Here we collect links to research papers that have come about as a result of WIN conferences, but were not published in the Proceedings.  If you have a paper to submit, please follow these instructions.

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1   Link   Bucur, Alina; David, Chantal; Feigon, Brooke; Lalín, Matilde. Statistics for traces of cyclic trigonal curves over finite fields. Int. Math. Res. Not. IMRN 2010, no. 5, 932–967.
We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over Graphic as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of Frobenius equals the sum of q + 1 independent random variables taking the value 0 with probability 2/(q + 2) and 1, e2π i/3, e4π i/3 each with probability q/(3(q + 2)). This extends the work of Kurlberg and Rudnick who considered the same limit for hyperelliptic curves. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to p-fold covers of the projective line.
2   Link   Bucur, Alina; David, Chantal; Feigon, Brooke; Lalín, Matilde. Fluctuations in the number of points on smooth plane curves over finite fields. J. Number Theory 130 (2010), no. 11, 2528–2541.
In this note, we study the fluctuations in the number of points on smooth projective plane curves over a finite field Fq as q is fixed and the genus varies. More precisely, we show that these fluctuations are predicted by a natural probabilistic model, in which the points of the projective plane impose independent conditions on the curve. The main tool we use is a geometric sieving process introduced by Poonen (2004).
3   Link   Malmskog, Beth; Manes, Michelle. Almost divisibility in the Ihara zeta functions of certain ramified covers of $q+1$-regular graphs. Linear Algebra Appl. 432 (2010), no. 10, 2486–2506.
Given a (q+1)-regular graph X and a second graph Y formed by taking k copies of X and identifying them at a common vertex, we form a ramified cover of the original graph. We prove that the reciprocal of the zeta function for X “almost divides” the reciprocal of the zeta function for Y, in the following sense. The reciprocal of the zeta function of X divides the product of the reciprocal of the zeta function of Y and some polynomial of bounded degree (which depends only on the graph X, not on the number of copies). Two specific examples show that in fact “almost divisibility” is the best that can be hoped for.
4   Link   Garthwaite, Sharon; Long, Ling; Swisher, Holly; Treneer, Stephanie. Zeros of some level 2 Eisenstein series. Proc. Amer. Math. Soc. 138 (2010), 467-480
The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property. In this paper we extend these distribution properties to a particular family of Eisenstein series on Γ(2) because of its elegant connection to a classical Jacobi elliptic function cn(u) which satisfies a differential equation. As part of this study we recursively define a sequence of polynomials from the differential equation mentioned above that allows us to calculate zeros of these Eisenstein series. We end with a result linking the zeros of these Eisenstein series to an L-series.
5   Link   An arithmetic intersection formula for denominators of Igusa class polynomials - Kristin Lauter, Bianca Viray
In this paper we prove an explicit formula for the arithmetic intersection number on the Siegel moduli space of abelian surfaces, generalizing the work of Bruinier-Yang and Yang. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography.
Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for the intersection number under strong assumptions on the ramification of the primitive quartic CM field K. Yang later proved this conjecture assuming that the ring of integers is freely generated by one element over the ring of integers of the real quadratic subfield. In this paper, we prove a formula for the intersection number for more general primitive quartic CM fields, and we use a different method of proof than Yang. We prove a tight bound on this intersection number which holds for all primitive quartic CM fields. As a consequence, we obtain a formula for a multiple of the denominators of the Igusa class polynomials for an arbitrary primitive quartic CM field. Our proof entails studying the Embedding Problem posed by Goren and Lauter and counting solutions using our previous article that generalized work of Gross-Zagier and Dorman to arbitrary discriminants.
6   Link   On singular moduli for arbitrary discriminants - Kristin Lauter, Bianca Viray
Let d1 and d2 be discriminants of quadratic imaginary orders and let J(d1,d2)^2 denote the product of differences of CM j-invariants with discriminants d1 and d2. In 1985, Gross and Zagier gave an elegant formula for the factorization of the integer J(d1,d2) in the case that d1 and d2 are relatively prime and discriminants of maximal orders. We generalize their methods and give a complete factorization in the case that d1 is squarefree and d2 is any discriminant. We also give a partial factorization in all other cases, and give a conjectural formula when the conductors of d1 and d2 are relatively prime.