The Existence of an Abelian Variety over the Algebraic Numbers isogenous to no Jacobian

We prove the existence of an Abelian variety $A$ of dimension $g$ over \Qa which is not isogenous to any Jacobian, subject to the necessary condition $g>3$. Recently, C.Chai and F.Oort gave such a proof assuming the Andr\'e-Oort conjecture. We modify their proof by constructing a special sequence of CM points for which we can avoid any unproven hypotheses. We make use of various techniques from the recent work of Klingler-Yafaev et al

The Kodaira dimension of complex hyperbolic manifolds with cusps

We prove a bound relating the volume of a curve near a cusp in a hyperbolic manifold to its multiplicity at the cusp. The proof uses a hybrid technique employing both the geometry of the uniformizing group and the algebraic geometry of the toroidal compactification. There are a number of consequences: we show that for an $n$-dimensional toroidal compactification $\bar X$ with boundary $D$, $K_{\bar X}+(1-\frac{n+1}{2\pi}) D$ is nef, and in particular that $K_{\bar X}$ is ample for $n\geq 6$. By an independent algebraic argument, we prove that every hyperbolic manifold of dimension $n\geq 3$ is of general type, and conclude that the phenomena famously exhibited by Hirzebruch in dimension 2 do not occur in higher dimensions. Finally, we investigate the applications to the problem of bounding the number of cusps and to the Green--Griffiths conjecture.Comment: Minor typos corrected. Comments welcom

Non-split Sums of Coefficients of GL(2)-Automorphic Forms

Given a cuspidal automorphic form $\pi$ on \GL_2, we study smoothed sums of the form $\sum_{n\in\mathbb{N}} a_{\pi}(n^2+d)W(\frac{n}{Y})$. The error term we get is sharp in that it is uniform in both $d$ and $Y$ and depends directly on bounds towards Ramanujan for forms of half-integral weight and Selberg eigenvalue conjecture. Moreover, we identify (at least in the case where the level is square-free) the main term as a simple factor times the residue as $s=1$ of the symmetric square L-function L(s,\Msym^2\pi). In particular there is no main term unless $d>0$ and $\pi$ is a dihedral form

An analysis of a war-like card game

In his book "Mathematical Mind-Benders", Peter Winkler poses the following open problem, originally due to the first author: "[In the game Peer Pressure,] two players are dealt some number of cards, initially face up, each card carrying a different integer. In each round, the players simultaneously play a card; the higher card is discarded and the lower card passed to the other player. The player who runs out of cards loses. As the number of cards dealt becomes larger, what is the limiting probability that one of the players will have a winning strategy?" We show that the answer to this question is zero, as Winkler suspected. Moreover, assume the cards are dealt so that one player receives r >= 1 cards for every one card of the other. Then if r < phi = (1+sqrt 5)/2, the limiting probability that either player has a winning strategy is still zero, while if r > phi, it is one.Comment: 5 pages, 1 figur

Sum-product estimates for rational functions

We establish several sum-product estimates over finite fields that involve polynomials and rational functions. First, |f(A)+f(A)|+|AA| is substantially larger than |A| for an arbitrary polynomial f over F_p. Second, a characterization is given for the rational functions f and g for which |f(A)+f(A)|+|g(A,A)| can be as small as |A|, for large |A|. Third, we show that under mild conditions on f, |f(A,A)| is substantially larger than |A|, provided |A| is large. We also present a conjecture on what the general sum-product result should be.Comment: 32 pages, small additions, several typos fixe

How Large is Ag(Fq)A_g(\mathbb{F}_q)?

Let $B(g,p)$ denote the number of isomorphism classes of $g$-dimensional abelian varieties over the finite field of size $p.$ Let $A(g,p)$ denote the number of isomorphism classes of principally polarized $g$ dimensional abelian varieties over the finite field of size $p.$ We derive upper bounds for $B(g,p)$ and lower bounds for $A(g,p)$ for $p$ fixed and $g$ increasing. The extremely large gap between the lower bound for $A(g,p)$ and the upper bound $B(g,p)$ implies some statistically counterintuitive behavior for abelian varieties of large dimension over a fixed finite field.Comment: 38 page

Counting S5S_5-fields with a power saving error term

We show how the Selberg $\Lambda^2$-sieve can be used to obtain power saving error terms in a wide class of counting problems which are tackled using geometry of numbers. Specifically, we give such an error term for the counting function of $S_5$-quintic fields.Comment: 7 page

Multiplicative relations among singular moduli

We consider some Diophantine problems of mixed modular-multiplicative type associated with the Zilber-Pink conjecture. In particular, we prove a finiteness statement for the number of multiplicative relations between singular moduli (j-invariants of elliptic curves with complex multiplication.)Comment: Comments Welcome

The Ax-Schanuel conjecture for variations of Hodge structures

We extend the Ax-Schanuel theorem recently proven for Shimura varieties by Mok-Pila-Tsimerman to all varieties supporting a pure polarized integral variation of Hodge structures. The essential new ingredient is a volume bound on Griffiths transverse subvarieties of period domains

Ax-Lindemann for \mathcal{A}_g

We prove the Ax-Lindemann theorem for the coarse moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g\ge 1$, and affirm the Andr\'e-Oort conjecture unconditionally for $\mathcal{A}_{g}$ for $g\le 6$.Comment: 21 pages. Accepted for publication in Annals of Mathematic