For families of smooth complex projective varieties we show that normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. As a consequence, we prove a conjecture of Charles and Kerr--Pearlstein, that zero loci of normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. In particular, this gives a short proof of a special, algebraically motivated case of a result of Saito, Brosnan--Pearlstein, and Schnell, conjectured by Green--Griffiths, on zero loci of admissible normal functions.
Let X be a curve in positive characteristic. The Hasse--Witt matrix represents the action of the Frobenius operator on the cohomology group H^{1}(X,O_{X}). The Cartier--Manin matrix represents the action of the Cartier operator on the space of holomorphic differentials of X. The operators that these matrices represent are dual to one another, so the Hasse--Witt matrix and the Cartier--Manin matrix are related to one another, but they should not be viewed as being identical. There seems to be a fair amount of confusion in the literature about terminology, about whether matrices act on the left or the right, and about the proper formulae for iterating semi-linear operators. Unfortunately, this confusion has led to the publication of incorrect results. In this paper we present the issues involved as clearly as we can, and we look through the literature to see where there may be problems. We encourage future authors to clearly distinguish between the Hasse--Witt and Cartier--Manin matrices, in the hope that further errors can be avoided.
A conjecture of Orlov predicts that derived equivalent smooth projective varieties have isomorphic Chow motives. We show a result in that direction: two derived equivalent threefolds over a field of characteristic zero have isogenous intermediate Jacobians. For threefolds over an arbitrary perfect field, we show that the isogeny class of the algebraic representative (for algebraically trivial cycles of codimension two) is a derived invariant.
We show that the image of the Abel--Jacobi map admits a model over the field of definition, with the property that the Abel--Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level one.
A cycle is algebraically trivial if it can be exhibited as the difference of two fibers in a family of cycles parameterized by a smooth scheme. Over an algebraically closed field, it is a result of Weil that it suffices to consider families of cycles parameterized by curves, or by abelian varieties. In this paper, we extend these results to arbitrary base fields. The strengthening of these results turns out to be a key step in our work elsewhere extending Murre's results on algebraic representatives for varieties over algebraically closed fields to arbitrary perfect fields.
Popa and Schnell have proved that derived equivalent complex varieties have isogenous Picard varieties. We give a proof of the analogous statement (for reduced Picard schemes) over an arbitrary field.
In this paper, motivated by a problem posed by Barry Mazur, we show that for smooth projective varieties over the rationals, the odd cohomology groups of degree less than or equal to the dimension can be modeled by the cohomology of an abelian variety, provided the geometric coniveau is maximal. This provides an affirmative answer to Mazur's question for all uni-ruled threefolds, for instance. Concerning cohomology in degree three, we show that the image of the Abel--Jacobi map admits a distinguished model over the rationals.
For prime powers q, let s(q) denote the probability that a randomly-chosen principally-polarized abelian surface over the finite field F_q is not simple. We show that there are positive constants B and C such that for all q, B (log q)^{-3}(log log q)^{-4} < s(q)sqrt(q) < C (log q)^4(log log q)^2, and we obtain better estimates under the assumption of the generalized Riemann hypothesis. If A is a principally-polarized abelian surface over a number field K, let pi_split(A/K, z) denote the number of prime ideals p of K of norm at most z such that A has good reduction at p and A_p is not simple. We conjecture that for sufficiently general A, the counting function pi_split(A/K, z) grows like sqrt(z)/log z. We indicate why our theorem on the rate of growth of s(q) gives us reason to hope that our conjecture is true.
An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. Gekeler has given a product formula, in terms of congruence considerations involving that polynomial, for the size of such an isogeny class. In this paper, we give a new, transparent proof of this formula; it turns out that this product actually computes an adelic orbital integral which visibly counts the desired cardinality.
We consider the distribution of p-power group schemes among the torsion of abelian varieties over finite fields of characteristic p, as follows. Fix natural numbers g and n, and let ξ be a non-supersingular principally quasipolarized Barsotti-Tate group of level n. We classify the F_{q}-rational forms ξ^{α} of ξ Among all principally polarized abelian varieties X/F_{q} of dimension g with p^{n}-torsion geometrically isomorphic to ξ, we compute the frequency with which X[p^{n}] is isomorphic to ξ^{α}. The error in our estimate is bounded by D/q^{1/2} where D depends on g, n and p, but not on ξ.
An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. In this paper, the superspecial condition is generalized by defining the superspecial rank of an abelian variety, which is an invariant of its p-torsion. The main results in this paper are about the superspecial rank of supersingular abelian varieties and Jacobians of curves. For example, it turns out that the superspecial rank determines information about the decomposition of a supersingular abelian variety up to isomorphism; namely it is a bound for the maximal number of supersingular elliptic curves appearing in such a decomposition.
Consider a quartic q-Weil polynomial f. Motivated by equidistribution considerations we define, for each prime l, a local factor which measures the relative frequency with which f mod l occurs as the characteristic polynomial of a symplectic similitude over F_{l}. For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over F_{q} with Weil polynomial f.
How many rational points are there on a random algebraic curve of large genus g over a given finite field F_{q}? We propose a heuristic for this question motivated by a (now-proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean q+1+1/(q-1). We prove a weaker version of this statement in which g and q tend to infinity, with q much larger than g.
A polarized abelian variety (X,λ) of dimension g and good reduction over a local field K determines an admissible representation of GSpin_{2g+1}(K). We show that the restriction of this representation to Spin_{2g+1}(K) is reducible if and only if X is isogenous to its twist by the quadratic unramified extension of K. When g=1 and K = Q_{p}, we recover the well-known fact that the admissible GL_{2}(K) representation attached to an elliptic curve E with good reduction is reducible upon restriction to SL_{2}(K) if and only if E has supersingular reduction.
We survey results and open questions about the p-ranks and Newton polygons of Jacobians of curves in positive characteristic p. We prove some geometric results about the p-rank stratification of the moduli space of (hyperelliptic) curves. For example, if 0 ≤ f ≤ g-1, we prove that every component of the p-rank f+1 stratum of M_{g} contains a component of the p-rank f stratum in its closure. We prove that the p-rank f stratum of ̅M_{g} is connected. For all primes p and all g ≥ 4, we demonstrate the existence of a Jacobian of a smooth curve, defined over ̅F_{p}, whose Newton polygon has slopes {0, 1/4, 3/4, 1}. We include partial results about the generic Newton polygons of curves of given genus g and p-rank f.
It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base. This answers, away from the prime 2, an old question of Deligne and a recent question of Kudla and Rapoport.
Let L be a quadratic imaginary field, inert at the rational prime p. Fix an integer n ≥ 3 and let M be the moduli space (in characteristic p) of principally polarized abelian varieties of dimension n equipped with an action by O_{L} of signature of (1,n-1). We show that each Newton stratum of M, other than the supersingular stratum, is irreducible.
To a family of smooth projective cubic surfaces one can canonically associate a family of abelian fivefolds. In characteristic zero, we calculate the Hodge groups of the abelian varieties which arise in this way. In arbitrary characteristic we calculate the monodromy group of the universal family of abelian varieties, and thus show that the Galois group of the 27 lines on a general cubic surface in positive characteristic is as large as possible.
Fix a prime l, and let F_{q} be a finite field with q≡ 1 mod l. If l> 2 and q>>_{l} 1, we show that asymptotically (l-1)^{2}/2l^{2} of the elliptic curves E/F_{q} with complete rational l-torsion are such that E/P does not have complete rational l-torsion for any point P ∈ E(F_{q}) of order l For l=2 the asymptotic density is 0 or 1/4, depending whether q≡ 1 mod 4 or 3 mod 4. We also show that for any l, if E/F_{q} has an F_{q}-rational point R of order l^{2}, then E/lR always has complete rational l-torsion.
Let X/K be an absolutely simple abelian variety over a number field. We show that if End(X) is a definite quaternion algebra, then the reduction X_{p} is geometrically isogenous to the self-product of an absolutely simple abelian variety for p in a set of positive density, while if X is of Mumford type, then X_{p} is simple for almost all p. For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound for the growth of the set of primes of non-simple reduction.
We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p ≥ 3. Using this, we prove that the Z/l-monodromy of every irreducible component of the stratum H^{f}_{g} of hyperelliptic curves of genus g and p-rank f is the symplectic group Sp_{2g}(Z/l) if g ≥ 3, f > 0, and l is an odd prime distinct from p. These results yield applications about the generic behavior of hyperelliptic curves of given genus and p-rank. The first application is that a generic hyperelliptic curve of genus g ≥ 3 and p-rank 0 is not supersingular. Other applications are about absolutely simple Jacobians and the generic behavior of class groups and zeta functions of hyperelliptic curves of given genus and p-rank over finite fields.
Consider a finite morphism f:X → Y of
smooth, projective varieties
over a finite field F. Suppose X is the vanishing locus in
P^{N} 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) →
Jones and Rouse define a quantity F(l,g), in terms of an integral on GSp_{2g}(Z_{l}), which measures how often the reduction of a suitably general point on a suitably general $g$-dimensional abelian variety has order prime to l. They conjecture that lim_{g→ ∞}F(l,g) exists; we prove this for l odd.
Let F_{q} be a finite field of odd characteristic, and let N be an odd natural number. An explicit fiber product construction shows that if N divides the class number of some quadratic function field over F_{q}, then it does so for infinitely many such function fields.
A p-divisible group over a field K admits a slope decomposition; associated to each slope λ is an integer m and a representation Gal(K) → GL_{m}(D_{λ}), where D__{λ} is the Q_{p}-division algebra with Brauer invariant [λ]. We call m the multiplicity of λ in the p-divisible group. Let G_{0} be a p-divisible group over a field k. Suppose that λ is not a slope of G_{0}, but that there exists a deformation of G in which λ appears with multiplicity one. Assume that λ ≠ (s-1)/s for any natural number s > 1. We show that there exists a deformation G/R of G_{0}/k such that the representation Gal(Frac(R) → GL_{1}(D_{λ}) has large image.
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 X_{p} is absolutely simple for almost all primes p. Conversely, if the absolute endomorphism ring of X is noncommutative, then X_{p} is reducible for p in a set of positive density.
Let k be an algebraically closed field of characteristic p > 0. Suppose g ≥ 3 and 0 ≤ f ≤ 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 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) ≅ L.
We compute the Z/l and Z_{l} monodromy of every irreducible component of the moduli space M^{f}_{g} 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 M^{f}_{g} is the symplectic group Sp_{2g}(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.
We calculate the chance that an elliptic curve over a finite field has a specified number of l-isogenies which emanate from it. We give a partial answer for abelian varieties of arbitrary dimension.
We compute the Z/l and Z_{l} 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 Sp_{2g}(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 F_{q} 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](F_{q}) = H. We also give estimates for the l-part of the class group of the affine coordinate ring of a curve over F_{q}. 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 dissertation 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 O_{K} 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 O_{L} be the ring of integers in a totally real field dimension g over Q, let N_{0} and N be relatively prime square-free integers, and let k be a finite field of characteristic relatively prime to both N_{0}N and disc(L,Q). Finally, let (X/k,i,a) be a g-dimensional abelian variety over k equipped with an action by O_{L} and a &Gamma_{0}(N_{0},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!