2008 Western Number Theory Conference
Consider an absolutely simple abelian variety X over a number field. We show that if the absolute endomorphism ring of X is commutative and satisfies certain parity conditions, then Xp is absolutely simple for almost all primes p. Conversely, if the absolute endomorphism ring of X is noncommutative, then Xp is reducible for p in a set of positive density.
A p-divisible group over a field K admits a slope decomposition; associated to each slope &lambda is an integer m and a representation Gal(K) &rarr GLm(D&lambda), where D_&lambda is the Qp-division algebra with Brauer invariant [&lambda]. We call m the multiplicity of &lambda in the p-divisible group. Let G0 be a p-divisible group over a field k. Suppose that &lambda is not a slope of G0, but that there exists a deformation of G in which &\lambda appears with multiplicity one. Assume that &lambda &ne (s-1)/s for any natural number s > 1. We show that there exists a deformation G/R of G0/k such that the representation Gal(Frac(R) &rarr GL1(D&lambda) has large image.
We compute the Z/l and Zl monodromy of every irreducible component of the moduli space Mfg of curves of genus g and p-rank f in characteristic p. In particular, we prove that the Z/l-monodromy of every component of Mfg is the symplectic group Sp2g(Z/l) if g≥ 3 and l is a prime distinct from p. We give applications to the generic behavior of automorphism groups, Jacobians, class groups, and zeta functions of curves of given genus and p-rank.
Consider hyperelliptic curves C of fixed genus over a finite field F. Let L be a finite abelian group of exponent dividing N. We give an asymptotic formula in F, with explicit error term, for the proportion of C for which Jac(C)[N](F) &cong L.
Let k be an algebraically closed field of characteristic p > 0. Suppose g &ge 3 and 0 &le f &le g. We prove there is a smooth projective k-curve of genus g and p-rank f with no non-trivial automorphisms. In addition, we prove there is a smooth projective hyperelliptic k-curve of genus g and p-rank f whose only non-trivial automorphism is the hyperelliptic involution.
Consider a finite morphism f:X &rarr Y of
smooth, projective varieties
over a finite field F. Suppose X is the vanishing locus in
PN of r forms of degree at most
d. We show that there is a
constant C depending only on (N,r,d) and
deg(f) such that if |F| > C, then
f(F): X(F) &rarr
We calculate the chance that an elliptic curve over a finite field has a specified number of $\ell$-isogenies which emanate from it. We give a partial answer for abelian varieties of arbitrary dimension.
We compute the Z/l and Zl monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the Z/l monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group Sp2g(Z/l). We prove that the Z/l monodromy of the moduli space of trielliptic curves with signature (r,s) is a special unitary group.
By combining an effective Chebotarev density theorem and finiteness theorems of Faltings, we give an algorithm to determine if two abelian varieties over a number field are isogenous. As an application, we explain how to decide if an elliptic curve over a number field has complex multiplication.
For any sufficiently general family of curves over a finite field Fq and any elementary abelian l-group H with l relatively prime to q, we give an explicit formula for the proportion of curves C for which Jac(C)[l](Fq) = H. We also give estimates for the l-part of the class group of the affine coordinate ring of a curve over Fq. In doing so, we prove a conjecture of Friedman and Washington.
For a given natural number N, Corso constructs a graph with vertices {2,...,N}. He analyzes this family of graphs with computer calculations; I get exact answers using number theory.
We investigate the proportion of function fields F over a finite field k with the following property: Let F' be obtained from F by taking the maximal l extension of the base field k. If L is an unramified l-adic extension of F, and L does not contain F', then L is a finite extension of F. As a pleasant side effect, we calculate the proportion of abelian varieties over k with a k-rational point of order l.
This paper is based on my disseration work at the University of Pennsylvania, condensed in some respects and expanded in others. The first paragraph explains: Hilbert-Siegel varieties are moduli spaces for abelian varieties equipped with an action by an order OK in a fixed, totally real field K. As such, they include both the Siegel moduli spaces (use K = Q and the action is the standard one) and Hilbert-Blumenthal varieties (where the dimension of K is the same as that of the abelian varieties in question). In this paper we study certain phenomena associated to Hilbert-Siegel varieties in positive characteristic. Specifically, we show that ordinary points are dense in moduli spaces of mildly inseparably polarized abelian varieties with action by a given totally real field. Moreover, we introduce a combinatorial invariant of the first cohomology of an abelian variety which allows us to compute and explain the singularities of such a moduli space.
This paper gives an explicit formula for the size of the isogeny class of a Hilbert-Blumenthal abelian variety over a finite field. More precisely, let OL be the ring of integers in a totally real field dimension g over Q, let N0 and N be relatively prime square-free integers, and let k be a finite field of characteristic relatively prime to both N0N and disc(L,Q). Finally, let (X/k,i,a) be a g-dimensional abelian variety over k equipped with an action by OL and a Gamma0(N0,N)-level structure. Using work of Kottwitz, we express the number of (X'/k,i',a') which are isogenous to (X,i,a) as a product of local orbital integrals on GL(2); then, using work of Arthur-Clozel and the affine Bruhat decomposition we evaluate all the relevant orbital integrals, thereby finding the cardinality of the isogeny class.
My doctoral thesis at the University of Pennsylvania. Much of this turned into the RMJM paper described above. Abstract: Over a field of positive characteristic p, we consider moduli spaces of polarized abelian varieties equipped with an action by a ring unramified at p. Using deformation theory, we show that ordinary points are dense in each of the following situations: the polarization is separable; the polarization is mildly inseparable, and the ring of endomorphisms is a totally real number field; or the polarization is arbitrary, and the ring is a real quadratic field acting on abelian fourfolds. We introduce a new invariant which measures the extent to which a polarized Dieudonne module admits an isotropic splitting lifting the Hodge filtration, and use it to explain the singularities arising from mildly inseparable polarizations.
This was my senior thesis at Brown University. Mine was a combined degree in computer science and mathematics. In my thesis, I discuss the Mordell-Weil groups of abelian varieties and, in particular, elliptic curves. I wrote code to compute the rank of about a hundred thousand elliptic curves. From a computational perspective, the most interesting feature was the coarse parallelism -- at any given time, the program was running on a couple dozen SPARCstations. From a mathematical perspective, the observed rank distributions line up suprisingly well with those computed by Brumer and McGuinness. I say surprising, since we examined very different families of elliptic curves. If you can explain these observations please, let me know!