Quadratic Points On Modular Curves: In this paper we determine the quadratic points on the modular curves X_0(N), where the curve is non-hyperelliptic, the genus is 3, 4 or 5, and the Mordell--Weil group of J_0(N) is finite. The values of N are 34, 38, 42, 44, 45, 51, 52, 54, 55, 56, 63, 64, 72, 75, 81. As well as determining the non-cuspidal quadratic points, we give the j-invariants of the elliptic curves parametrized by those points, and determine if they have complex multiplication or are quadratic \Q-curves. Joint work with S. Siksek

A bound on the primes of bad reduction for CM curves of genus 3: We give a bound on the primes of stable bad reduction for curves of genus three of primitive CM type in terms of the CM order. The genus one case follows from the fact that CM elliptic curves are CM abelian varieties which have potential good reduction everywhere. However, for genus at least two, the curve can have bad reduction at a prime although the Jacobian has potential good reduction. Goren and Lauter gave the first bound in the genus two case.

In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are essential for algorithmic construction of curves with given characteristic polynomials over finite fields. Joint work with P.Kilicer, K.Lauter,E.Garcia,R.Newton and M.Streng


(Note that the actual published versions may be slightly different than the arXiv versions given below)

On the existence of ordinary and almost ordinary Prym varieties: We study the relationship between the p-rank of a curve and the p-ranks of the Prym varieties of its cyclic covers in characteristic p>0. For arbitrary p,g3 and 0≤fg, we generalize a result of Nakajima by proving that the Prym varieties of all unramified cyclic degree ℓ≠p covers of a generic curve X of genus g and p-rank f are ordinary. Furthermore, when p5, we prove that there exists a curve of genus g and p-rank f having an unramified degree =2 cover whose Prym is almost ordinary. Using work of Raynaud, we use these two theorems to prove results about the (non)-intersection of the -torsion group scheme with the theta divisor of the Jacobian of a generic curve X of genus g and p-rank f. The proofs involve geometric results about the p-rank stratification of the moduli space of Prym varieties. Joint work with R. Pries, to appear in Asian Journal of Mathematics

Non-ordinary curves with a Prym variety of low p -rank: If π:Y→X is an unramified double cover of a smooth curve of genus g, then the Prym variety Pπ is a principally polarized abelian variety of dimension g−1. When X is defined over an algebraically closed field k of characteristic p, it is not known in general which p-ranks can occur for Pπ under restrictions on the p-rank of X. In this paper, when X is a non-hyperelliptic curve of genus g=3, we analyze the relationship between the Hasse-Witt matrices of X and Pπ. As an application, when p≡5mod6, we prove that there exists a curve X of genus 3 and p-rank f=3 having an unramified double cover π:Y→X for which Pπ has p-rank 0 (and is thus supersingular); for 3≤p≤19, we verify the same for each 0≤f≤3. Using theoretical results about p-rank stratifications of moduli spaces, we prove, for small p and arbitrary g≥3, that there exists an unramified double cover π:Y→X such that both X and Pπ have small p-rank. Joint work with TO. Celik, Y. Elias, B. Gunes, R. Newton, R. Pries and L. Thomas, to appear in: Women in Numbers Europe 2, Research Directions in Number Theory, Association for Women in Mathematics Series Vol. 11, Springer, 2018

RLWE Cryptography for the Number Theorist: In this paper, we survey the status of attacks on the ringand polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems gives rise to interesting questions about number fields. We extend these attacks and survey related open problems in number theory, including spectral distortion of an algebraic number and its relationship to Mahler measure, the monogenic property for the ring of integers of a number field, and the size of elements of small order modulo q. Joint work with Y. Elias, K. Lauter and K. Stange, Directions in Number Theory,Association for Women in Mathematics Series, Volume 3, Springer, 2016, DOI:10.1007/978-3-319-30976-7

The distribution of F_q-points on cyclic \ell-covers of genus g: We study fluctuations in the number of points of ℓ-cyclic covers of the projective line over the finite field F_q when q1modℓ is fixed and the genus tends to infinity. The distribution is given as a sum of q+1 i.i.d. random variables. This was settled for hyperelliptic curves by Kurlberg and Rudnick, while statistics were obtained for certain components of the moduli space of ℓ-cyclic covers by Bucur, David, Feigon and Lal\'{i}n. In this paper, we obtain statistics for the distribution of the number of points as the covers vary over the full moduli space of ℓ-cyclic covers of genus g. This is achieved by relating ℓ-covers to cyclic function field extensions, and counting such extensions with prescribed ramification and splitting conditions at a finite number of primes. Joint work with A. Bucur, C. David, B. Feigon, N. Kaplan, M. Lal\'{i}n and M. M. Wood, International Math Research Notices, Vol. 2016, No. 14, pp. 4297–4340, DOI:10.1093/imrn/rnv279.

Provably weak instances of Ring-LWE: The ring and polynomial learning with errors problems (Ring-LWE and Poly-LWE) have been proposed as hard problems to form the basis for cryptosystems, and various security reductions to hard lattice problems have been presented. So far these problems have been stated for general (number) rings but have only been closely examined for cyclotomic number rings. In this paper, we state and examine the Ring-LWE problem for general number rings and demonstrate provably weak instances of Ring-LWE. We construct an explicit family of number fields for which we have an efficient attack. We demonstrate the attack in both theory and practice, providing code and running times for the attack. The attack runs in time linear in q, where q is the modulus. Our attack is based on the attack on Poly-LWE which was presented in [Eisentr\"ager-Hallgren-Lauter]. We extend the EHL-attack to apply to a larger class of number fields, and show how it applies to attack Ring-LWE for a heuristically large class of fields. Certain Ring-LWE instances can be transformed into Poly-LWE instances without distorting the error too much, and thus provide the first weak instances of the Ring-LWE problem. We also provide additional examples of fields which are vulnerable to our attacks on Poly-LWE, including power-of-2 cyclotomic fields. Joint work with Y. Elias, K. Lauter and K. Stange, Advances in Cryptology -- CRYPTO 2015, 63-92, Springer (2015), DOI: 10.1007/978-3-662-47989-6

Bad reduction of genus-3 curves with complex multiplication: Let C be a smooth, absolutely irreducible genus-3 curve over a number field M. Suppose that the Jacobian of C has complex multiplication by a sextic CM-field K. Suppose further that K contains no imaginary quadratic subfield. We give a bound on the primes v of M such that the stable reduction of C at v contains three irreducible components of genus 1. Joint with I.Bouw, J.Cooley, K. Lauter, E. Garcia,M. Manes and R. Newton. Research Directions in Number Theory, Association for Women in Mathematics Series, Volume 2, Springer, DOI: 10.1007/978-3-319-17987-2

Local to Global Trace Questions and Twists of Genus One Curves: Let E be an elliptic curve defined over a number field F and K/F be a quadratic extension. For a point P in E(F) that is a local trace for every completion of K/F, we find necessary and sufficient conditions for P to lie in the image of the global trace map. These conditions can then be used to determine whether a quadratic twist of E, as a genus one curve, has rational points. In the case of quadratic twists of genus one modular curves X_0(N) with squarefree N, the existence of rational points corresponds to the existence of Q-curves of degree N defined over K. Joint work with M. Çiperiani, Proc. Amer. Math. Soc. 143 (2015), no. 9, 3815–3826.

Unramified Brauer classes on cyclic covers of the projective plane: Let X --> P^2 be a p-cyclic cover branched over a smooth, connected curve C of degree divisible by p, defined over a separably closed field of prime-to-p characteristic. We show that all (unramified) p-torsion Brauer classes on X that are fixed by Aut(X/P^2) arise as pullbacks of certain Brauer classes on k(P^2) that are unramified away from C and a fixed line L. We completely characterize these Brauer classes on k(P^2) and relate the kernel of the pullback map to the Picard group of X. If p = 2, we give a second construction, which works over any base field of characteristic not 2, that uses Clifford algebras arising from symmetric resolutions of line bundles on C to yield Azumaya representatives for the 2-torsion Brauer classes on X. We show that, when p=2 and sqrt{-1} is in our base field, both constructions give the same result.Joint work with with C. Ingalls, A. Obus and B. Viray, Progr. Math., 320, Brauer groups and obstruction problems, 115–153, Birkhäuser/Springer, Cham, 2017.

Newton and Hodge polygons for a variant of the Kloosterman family: We study the p-adic valuations of roots of L-functions associated with certain families of exponential sums of Laurent polynomials in n variables over a finite field. The families we consider are reflection and Kloosterman variants of diagonal polynomials. Using decomposition theorems of Wan, we determine the Newton and Hodge polygons of a non-degenerate Laurent polynomial in one of these families. Joint work with R. Bellovin, S. Garthwaite, R. Pries, C. Williams and H. J.Zhu, to appear in in Women in Numbers 2: Research Directions in Number Theory Contemporary Mathematics, 606. AMS, 2013, 206 pp.

On Polyquadratic Twists of X_0(N): In this paper, we study the Q_p-rational points of the twist of X_0(N) by a polyquadratic field and give an algorithm to produce such twists which has Q_p-rational points for all primes p. Then we investigate violations of the Hasse Principle for these curves and give an asymptotic for the number of such violations. Finally, we study reasons of such violations, Journal of Number Theory, Volume 133, pp. 3325-3338, 2013

Points on Quadratic Twists of X_0(N): In this paper we give necessary and sufficient conditions for the existence of a Q_p-rational point on quadratic twists of X_0(N), whenever p is not simultaneously ramified in Q(sqrt{d}) and Q(sqrt{-N}). The main theorem yields a population of curves which have local points everywhere but no points over Q; in several cases we show that this obstruction to the Hasse Principle is explained by the Brauer-Manin obstruction. Acta Arithmetica, Volume 152, Number 4, pages 323-348, 2012

Semi-direct product Galois covers of curves in characteristic p: Let k be an algebraically closed field with characteristic p>0. Raynaud showed in that the finite quotients of the algebraic fundamental group of the affine line over k is equal to the set of quasi p-groups. In this paper, we consider (Z/lZ)^b semi-direct Z/pZ Galois covers of the affine line ramified only at infinity, where l is a prime different from p. We show that the minimal genus of such a cover depends only on l, p, and and the order a of l modulo p. Moreover, we show that the number of such minimal genus covers equals (p-1)/a. The results of this paper, which are based on a project which took place in BANFF as a part of the conference WIN, are joint work with L. Gruendken, L. Hall-Seelig, B.H. Im, R. Pries and K. Stevenson. WIN—women in numbers, 201–210, Fields Inst. Commun., 60, Amer. Math. Soc., 2011.

Points on quadratic twists of the classical modular curve. Ph.D. ThesisThe University of Wisconsin - Madison. 2010. 73 pp. ISBN: 978-1124-37082


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