Let l be a prime. We show that there do not exist any non-zero semi-stable abelian varieties over Q with good reduction outside l if and only if l=2, 3, 5, 7 or 13. We show that any semi-stable abelian variety over Q with good reduction outside l=11 is isogenous to a power of the Jacobian variety of the modular curve X_0(11). In addition we show that there do not exist any non-zero abelian varieties over Q with good reduction outside l acquiring semi-stable reduction at l over a tamely ramified extension if and only if l \le 5.
We show that certain abelian varieties over $\Q$ with bad reduction at one prime only are modular by using methods based on the tables of Odlyzko and class field theory.
We show that up to potential isogeny, there are only finitely many abelian varieties of dimension $d$ defined over a number field $K$, such that for any finite place $v$ outside a fixed finite set $S$ of places of $K$ containing the archimedean places, it has either good reduction at $v$, or totally bad reduction at $v$ and good reduction over a quadratic extension of the completion of $K$ at $v$.
We determine the Galois representations inside the $l$-adic cohomology of some quaternionic and related unitary Shimura varieties at ramified places. The main results generalize the previous works of Reimann and Kottwitz in this setting to arbitrary levels at $p$, and confirm the expected description of the cohomology due to Langlands and Kottwitz.
We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a formula by Colmez and Obus specific to the CM case and valid when the CM field is an abelian extension of the rationals. This formula links the height and the logarithmic derivatives of an $L$-function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use results of Igusa, Liu, and Saito t…
An abelian surface $A_{/{\mathbb Q}}$ of prime conductor $N$ is favorable if its 2-division field $F$ is an ${\mathcal S}_5$-extension with ramification index 5 over ${\mathbb Q}_2$. Let $A$ be favorable and let $B$ be any semistable abelian variety of dimension $2d$ and conductor $N^d$ such that $B[2]$ is filtered by copies of $A[2]$. We give a sufficient class field theoretic criterion on $F$ to guarantee that $B$ is isogenous to $A^d$. As expected from our paramodular conjecture, we conclude that there is one isogeny class of abelian surfaces for each conductor in $\{277, 349,461,797,971\}$…
This survey article explains the construction of Rapoport-Zink local models and their use in understanding various questions relating to the singularities in the reduction modulo p of certain Shimura varieties with parahoric level structure at p.
In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.
We study monodromy action on abelian varieties satisfying certain bad reduction conditions. These conditions allow us to get some control over the Galois image. As a consequence we verify the Mumford--Tate conjecture for such abelian varieties.
We give a formula for the class number of an arbitrary CM algebraic torus over $\mathbb{Q}$. This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.
In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasi-projective varieties. Then, in terms of Prym varieties associated to the cyclic covers, we prove a structure theorem on their Mordell-Weil group. Our results give an explicit method for construction of elliptic curves, hyper- and super-elliptic Jacobians that have large ranks over function fields of certain varieties.
In this paper, we study a conjecture of Rasmussen and Tamagawa, on the finiteness of the set of isomorphism classes of abelian varieties with constrained prime power torsion. Our result is related with abelian varieties which have complex multiplication over their fields of definition.
We prove that if a curve of a non-isotrivial family of abelian varieties over a curve contains infinitely many isogeny orbits of a finitely generated subgroup of a simple abelian variety then it is special.
We provide a characterization of almost ordinary abelian varieties over finite fields, and use this characterization to provide lower bounds for the sizes of some almost ordinary isogeny classes.
In this paper we determine the number of isomorphism classes of superspecial abelian varieties $A$ over the prime field $\Fp$ such that the relative Frobenius morphism $π_A$ satisfying $π_A^2=-p$.
We study a certain class of simple abelian varieties of type $\mathrm{IV}$ (in Albert's classification) over number fields with Mumford-Tate groups of type $A$. In particular, we show that such abelian varieties have ordinary reduction away from a set of places of Dirichlet density zero, thus confirming a special case of a broader conjecture of Serre's. We also study the splitting types and Newton polygons of the reductions of the abelian varieties of this type with small dimension (nine).
We characterize the abelian varieties arising as absolutely simple factors of GL2-type varieties over a number field k. In order to obtain this result, we study a wider class of abelian varieties: the k-varieties A/k satisfying that $\End_k^0(A)$ is a maximal subfield of $\End_k^0(A)$. We call them Ribet-Pyle varieties over k. We see that every Ribet-Pyle variety over k is isogenous over $\bar k$ to a power of an abelian k-variety and, conversely, that every abelian k-variety occurs as the absolutely simple factor of some Ribet-Pyle variety over k. We deduce from this correspondence a precise …
We prove that Shimura varieties of abelian type with infinite level at $p$ are perfectoid. As a corollary, the moduli spaces of polarized K3 surfaces with infinite level at $p$ are also perfectoid.
If A and B are abelian varieties over a number field K such that there are non-trivial geometric homomorphisms of abelian varieties between reductions of A and B at most primes of K, then there exists a non-trivial (geometric) homomorphism from A to B defined over an algebraic closure of K.
We discuss heuristic asymptotic formulae for the number of pairing-friendly abelian varieties over prime fields, generalizing previous work of one of the authors arXiv:math1107.0307
In this paper we give conditions under which two abelian varieties that are defined over a finite field $F$, and are isogenous over some larger field, are $F$-isogenous. Further, we give conditions under which a given isogeny is defined over $F$.
In an earlier paper the author proved one divisibility of Perrin- Riou's Iwasawa main conjecture for Heegner points on elliptic curves. In the present paper, that result is generalized to abelian varieties of GL2-type (i.e. abelian varieties with real multiplication defined over totally real fields) under the hypothesis that the abelian variety is associated to a Hilbert modular form via a construction of Zhang.
The Mumford-Tate conjecture is first proved for CM abelian varieties by H. Pohlmann [Ann. Math., 1968]. In this note we give another proof of this result and extend it to CM motives.
We prove the Iwasawa main conjecture over the arithmetic $\mathbb{Z}_p$-extension for semistable abelian varieties over function fields of characteristic $p>0$.
We study the special fibers of a certain class of absolutely simple abelian varieties over number fields with endomorphism rings $\bz$ and possessing $l$-adic monodromy groups of the least possible rank. We also study the Dirichlet density of the places at which the possible reductions occur and confirm a special case of a broader conjecture for the splitting of reductions of abelian varieties over number fields.
This survey describe Hodge, Tate and Mumford-Tate conjectures for abelian varieties. After some preliminaries on endomorphism ring, polarization and algebraic cycles, we state the three conjectures and provide a list of know results. Finally, we explain some links between these conjectures.
The algebraic degeneracy of holomorphic curves in a semi-Abelian variety omitting a divisor is proved (Lang's conjecture generalized to semi-Abelian varieties) by making use of the {\it jet-projection method} and the logarithmic Wronskian jet differential after Siu-Yeung. We also prove a structure theorem for the locus which contains all possible image of non-constant entire holomorphic curves in a semi-Abelian variety omitting a divisor.
In this paper we consider certain local-global principles for Mordell-Weil type groups over number fields like S-units, abelian varieties and algebraic K-theory groups
This paper surveys the methods that have been used to attack the conjecture, still open, that an abelian variety over a characteristic $0$ field with finitely generated Galois group is always of infinite rank.
We formulate a question regarding uniform versions of "large Galois image properties" for modular abelian varieties of higher dimension, generalizing the well-known case of elliptic curves. We then answer our question affirmatively in the exceptional image case, and provide lower estimates for uniform bounds in the remaining cases.
We prove a diophantine result on generalized Fermat equations of the form $x^p + y^p = z^r$ which for the first time requires the use of Frey abelian varieties of dimension $\geq 2$ in Darmon's program. For that, we provide an irreducibility criterion for the mod $\mathfrak{p}$ representations attached to certain abelian varieties of $\text{GL}_2$-type over totally real fields.
A theorem of Chow concerns homomorphisms of two abelian varieties under a primary field extension base change. In this paper we generalize Chow's theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori.
We study a geometric analogue of the Iwasawa Main Conjecture for constant ordinary abelian varieties over $\ZZ_p^d$-extensions of function fields ramifying at a finite set of places.
If A is an abelian variety over a number field K, and L is a (possibly infinite) extension of K generated by torsion points of A, then the quotient of A(L) by its torsion subgroup is a free abelian group.
Suppose F=W(k)[1/p] where W(k) is the ring of Witt vectors with coefficients in algebraically closed field k of characteristic p>2. We construct integral theory of p-adic semi-stable representations of the absolute Galois group of F with Hodge-Tate weights from [0,p). This modification of Breuil's theory results in the following application in the spirit of Shafarevich's Conjecture. If Y is a projective algebraic variety over the field of rational numbers with good reduction modulo all primes different from 3 and semi-stable reduction modulo 3 then for the Hodge numbers of the complexification…
We assign functorially a $\mathbb{Z}$-lattice with semisimple Frobenius action to each abelian variety over $\mathbb{F}_p$. This establishes an equivalence of categories that describes abelian varieties over $\mathbb{F}_p$ avoiding $\sqrt{p}$ as an eigenvalue of Frobenius in terms of simple commutative algebra. The result extends the isomorphism classification of Waterhouse and Deligne's equivalence for ordinary abelian varieties.
The category of abelian varieties over $\mathbb{F}_q$ is shown to be anti-equivalent to a category of $\mathbb{Z}$-lattices that are modules for a non-commutative pro-ring of endomorphisms of a suitably chosen direct system of abelian varieties over $\mathbb{F}_q$. On full subcategories cut out by a finite set $w$ of conjugacy classes of Weil $q$-numbers, the anti-equivalence is represented by what we call $w$-locally projective abelian varieties.
We study (weak) log abelian varieties with constant degeneration in the log flat topology. If the base is a log point, we further study the endomorphism algebras of log abelian varieties. In particular, we prove the dual short exact sequence for isogenies, Poincaré complete reducibility theorem for log abelian varieties, and the semi-simplicity of the endomorphism algebras of log abelian varieties.
In this paper we study super-isolated abelian varieties, that is, abelian varieties over finite fields whose isogeny class contains a single isomorphism class. The goal of this paper is to (1) characterize whether a product of super-isolated varieties is super-isolated, and (2) characterize which super-isolated abelian varieties admit principal polarizations, and how many up to polarized isomorphisms.
Let $E$ be a quadratic imaginary field and let $p$ be a prime which is inert in $E.$ We study three types of Picard modular surfaces in positive characteristic $p$ and the morphisms between them. The first Picard surface, denoted $S$, parametrizes triples $(A,ϕ,ι)$ comprised of an abelian threefold $A$ with an action $ι$ of the ring of integers $\mathcal{O}_{E}$, and a principal polarization $ϕ$. The second surface, $S_{0}(p)$, parametrizes, in addition, a suitably restricted choice of a subgroup $H\subset A[p]$ of rank $p^{2}$. The third Picard surface, $\widetilde{S}$, parametrizes triples $…
In this paper we discuss multiplicative relations between eigenvalues of Frobenius endomorphism of abelian varieties of small dimension over finite fields.
We prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero.
We study analogues of Tate's conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of abelian varieties without nontrivial endomorphisms.
Analogy between Anderson T-motives and abelian varieties with multiplication by an imaginary quadratic field (MIQF) is a source of 2 results: 1. A description of abelian varieties with MIQF of dimension $r$ and signature $(n, r-n)$ in terms of "lattices" of dimension $r$ in $\Bbb C^n$; 2. A construction of exterior powers of abelian varieties with MIQF having $n=1$.
In two earlier articles, we proved that, if the Hodge conjecture is true for ALL CM abelian varieties over the complex numbers, then both the Tate conjecture and the standard conjectures are true for abelian varieties over finite fields. Here we rework the proofs so that they apply to a single abelian variety. As a consequence, we prove (unconditionally) that the Tate and standard conjectures are true for many abelian varieties over finite fields, including abelian varieties for which the algebra of Tate classes is not generated by divisor classes.
We study in this paper some criterions to get polarized morphisms between abelian varieties. We deduce explicit dynamical systems with particular intersection properties.
We prove that torsion codimension 2 algebraic cycles modulo rational equivalence on supersingular abelian varieties are algebraically equivalent to zero. As a consequence, we prove that homological equivalence coincides with algebraic equivalence for algebraic cycles of codimension 2 on supersingular abelian varieties over the algebraic closure of finite fields.
We prove a conjecture by Kawaguchi-Silverman on arithmetic and dynamical degrees, for self-morphisms of semi-abelian varieties. Moreover, we determine the set of the arithmetic degrees of orbits and the (first) dynamical degrees of self-morphisms of semi-abelian varieties.
The results in this paper imply that for every number field F and positive integer r, there exists an F-isogeny class of abelian varieties such that r divides the degree of every F-polarization on every abelian variety in the isogeny class.
We prove formulas for the p-adic logarithm of quaternionic Darmon points on p-adic tori and modular abelian varieties over Q having purely multiplicative reduction at p. These formulas are amenable to explicit computations and are the first to treat Stark-Heegner type points on higher-dimensional abelian varieties.
