Related papers: A counterexample to the local-global principle of …

For an abelian variety $A$ over a finitely generated field $K$ of characteristic $p > 0$, we prove that the algebraic rank of $A$ is at most a suitably defined analytic rank. Moreover, we prove that equality, i.e., the BSD rank conjecture,…

Let $O_F$ be the ring of integers of a totally real field $F$ of degree $g$. We study the reduction of the moduli space of separably polarized abelian $O_F$-varieties of dimension $g$ modulo $p$ for a fixed prime $p$. The invariants and…

Let K be any field and G be a finite group. We will prove that, if K is any field, p an odd prime number, and G is a non-abelian group of exponent p with |G|=p^3 or p^4 satisfying [K(\zeta_p):K] <= 2, then K(G) is rational over K. We will…

We show that local-global compatibility (at split primes) away from $p$ holds at all points of the $p$-adic eigenvariety of a definite $n$-variable unitary group. The novelty is we allow non-classical points, possibly non-\'{e}tale over…

Let K be a number field. We consider a local-global principle for elliptic curves E/K that admit (or do not admit) a rational isogeny of prime degree n. For suitable K (including K=Q), we prove that this principle holds when n = 1 mod 4,…

We use knowledge of local fields to adapt Jonathan Lubin and Michael Rosen's proof of Mazur's Proposition 4.39. This changes the result about abelian varieties from only working over local fields with a finite residue field to working with…

We show that the problem of whether the fixed point of a morphism avoids Abelian $k$-powers is decidable under rather general conditions

Let $f:X\rightarrow X$ be a quasi-finite endomorphism of an algebraic variety $X$ defined over a number field $K$ and fix an initial point $a\in X$. We consider a special case of the dynamical Mordell-Lang Conjecture, where the subvariety…

We consider a local to global principle for detecting linear dependence of nontorsion points, by reduction maps, in the Mordell-Weil group of an abelian variety over a number field.

Given an abelian variety $A$ defined over a finite field $k$, we say that $A$ is "cyclic" if its group $A(k)$ of rational points is cyclic. In this paper we give a bijection between cyclic abelian varieties of an ordinary isogeny class…

Let k be a field and f be a Siegel modular form of weight h \geq 0 and genus g>1 over k. Using f, we define an invariant of the k-isomorphism class of a principally polarized abelian variety (A,a)/k of dimension g. Moreover when (A,a) is…

A simple abelian variety $A$ defined over a number field $k$ is called of $\mathrm{GL}_n$-type if there exists a number field of degree $2\dim(A)/n$ which is a subalgebra of $\mathrm{End}^0(A)$. We say that $A$ is genuinely of…

Let ${\mathbb Z}_p$ denote the ring of all $p$-adic integers and call $${\mathcal U}=\{(x_1,\ldots,x_n):\,a_1x_1+\ldots+a_nx_n+b=0\}$$ a hyperplane over ${\mathbb Z}_p^n$, where at least one of $a_1,\ldots,a_n$ is not divisible by $p$. We…

Let $X$ be a four-dimensional projective variety defined over the field of complex numbers with only terminal singularities. We prove that if the intersection number of the canonical divisor $K$ with every very general curve is positive…

We prove that if $f:X \rightarrow A$ is a morphism from a smooth projective variety $X$ to an abelian variety $A$ over a number field $K$, and $G$ is a subgroup of automorphisms of $X$ satisfying certain properties, and if a prime $p$…

Let $\lambda\colon A\rightarrow A^{\vee}$ be a polarization on an abelian variety over a field $k$. If $k$ is not algebraically closed, there might not exist an ample line bundle on $A$ defined over $k$ that represents $\lambda$. To remedy…

Let K be a finitely generated field over Q, and A an abelian variety over K. Let <, > : A(K^a) x A(K^a) --> R be an arithmetic height pairing on A, where K^a is the algebric closure of K. For x_1,..., x_l \in A(K^a), we denote det(<x_i,…

We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For…

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…

Fix a poset $P$ and a natural number $n$. For various commutative local rings $\Lambda$, each of Loewy length $n$, consider the category $\textrm{sub}_\Lambda P$ of $\Lambda$-linear submodule representations of $P$. We give a criterion for…