Abelian Varieties Of Dimension Research Articles

Efficient computation of $$(2^n,2^n)$$-isogenies

AbstractElliptic curves are abelian varieties of dimension one; the two-dimensional analogues are abelian surfaces. In this work we present an algorithm to compute $$(2^n,2^n)$$ ( 2 n , 2 n ) -isogenies between abelian surfaces defined over finite fields. These isogenies are the natural generalization of $$2^n$$ 2 n -isogenies of elliptic curves. The efficient computation of such isogeny chains gained a lot of attention as the runtime of the attacks on SIDH (Castryck–Decru, Maino–Martindale, Robert) depends on this computation. Different results deduced in the development of our algorithm are also interesting beyond these applications. For instance, we derive a formula for the evaluation of (2, 2)-isogenies. Given an element in Mumford coordinates, this formula outputs the (unreduced) Mumford coordinates of its image under the (2, 2)-isogeny. Furthermore, we study 4-torsion points on Jacobians of hyperelliptic curves and explain how to extract square roots of coefficients of 2-torsion points from these points.

On the $\lambda$-invariant of Selmer Groups Arising from Certain Quadratic Twists of Gross Curves

Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, and we write $\mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_\infty$ be the unique $\mathbb{Z}_2$-extension of $K$ unramified outside $\mathfrak{p}$. For certain quadratic and biquadratic extensions $\mathfrak{F}/K$, we prove a simple exact formula for the $\lambda$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $\mathfrak{p}$ of the field $\mathfrak{F}_\infty = \mathfrak{F} K_\infty$. Equivalently, our result determines the exact $\mathbb{Z}_2$-corank of certain Selmer groups over $\mathfrak{F}_\infty$ of a large family of quadratic twists of the higher dimensional abelian variety with complex multiplication, which is the restriction of scalars to $K$ of the Gross curve with complex multiplication defined over the Hilbert class field of $K$. We also exhibit computations of the associated Selmer groups over $K_n$ in the case when the $\lambda$-invariant is equal to $1$; here $K_n$ denotes the $n$-th layer of $K_\infty/K$.

Abelian varieties with prescribed embedding and full embedding degrees

We study the problem of the embedding degree of an abelian variety over a finite field and extend some of the results of [1]. In particular, we show that for a prescribed CM field L of degree ≥4, prescribed integers m, n and a prescribed prime ℓ≡1(modm⋅n) that splits completely in L, there exists an ordinary abelian variety over a prime finite field with endomorphism algebra L, embedding degree n with respect to ℓ and full embedding degree m⋅n with respect to ℓ. We also study a class of absolutely simple higher dimensional abelian varieties whose endomorphism algebras are central over imaginary quadratic fields.

Higher dimensional Shimura varieties in the Prym loci of ramified double covers

AbstractIn this paper, we construct Shimura subvarieties of dimension bigger than one of the moduli space of δ‐polarized abelian varieties of dimension p, which are generically contained in the Prym loci of (ramified) double covers. The idea is to adapt the techniques already used to construct Shimura curves in the Prym loci to the higher dimensional case, namely, to use families of Galois covers of . The case of abelian covers is treated in detail, since in this case, it is possible to make explicit computations that allow to verify a sufficient condition for such a family to yield a Shimura subvariety of .

Tamagawa numbers of elliptic curves with torsion points

Let K be a global field and let E/K be an elliptic curve with a K-rational point of prime order p. In this paper, we are interested in how often the (global) Tamagawa number c(E/K) of E/K is divisible by p. This is a natural question to consider in view of the fact that the fraction $$c(E/K)/ |E(K)_{\text {tors}}|$$ appears in the second part of the Birch and Swinnerton-Dyer conjecture. We focus on elliptic curves defined over global fields, but we also prove a result for higher dimensional abelian varieties defined over $${\mathbb {Q}}$$ .

Weil polynomials of abelian varieties over finite fields with many rational points

We consider the finite set of isogeny classes of [Formula: see text]-dimensional abelian varieties defined over the finite field [Formula: see text] with endomorphism algebra being a field. We prove that the class within this set whose varieties have the maximal number of rational points is unique, for any prime even power [Formula: see text] big enough and verifying mild conditions. We describe its Weil polynomial and we prove that the class is ordinary and cyclic outside the primes dividing an integer that only depends on [Formula: see text]. In dimension [Formula: see text], we prove that the class is ordinary and cyclic and give explicitly its Weil polynomial, for any prime even power [Formula: see text].

Purely additive reduction of abelian varieties with torsion

Let OK be a discrete valuation ring with fraction field K of characteristic 0 and algebraically closed residue field k of characteristic p>0. Let A/K be an abelian variety of dimension g with a K-rational point of order p. In this article, we are interested in the reduction properties that A/K can have. After discussing the general case, we specialize to g=1, and we study the possible Kodaira types that can occur.

The essential dimension of congruence covers

AbstractConsider the algebraic function $\Phi _{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $\Phi _{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $M$ is proper. As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety.

Complete families of indecomposable non-simple abelian varieties

Given a fixed product of non-isogenous abelian varieties at least one of which is general, we show how to construct complete families of indecomposable abelian varieties whose very general fiber is isogenous to the given product and whose connected monodromy group is a product of symplectic groups or is a unitary group. As a consequence, we show how to realize any product of symplectic groups of total rank g as the connected monodromy group of a complete family of \(g'\)-dimensional abelian varieties for any \(g'\ge g\). These methods also yield a construction of a new Kodaira fibration with fiber genus 4.

Characteristic polynomials of simple non-ordinary abelian varieties

Let p be a prime and let with an odd integer. We describe a method for the construction of characteristic polynomials of simple non-ordinary abelian varieties of dimension g and p-rank g − 1 or g − 2 over the finite field when for some prime and integer

The epsilon constant conjecture for higher dimensional unramified twists of (1)

AbstractLet $N/K$ be a finite Galois extension of p-adic number fields, and let $\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ be an r-dimensional unramified representation of the absolute Galois group $G_K$ , which is the restriction of an unramified representation $\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ . In this paper, we consider the $\mathrm {Gal}(N/K)$ -equivariant local $\varepsilon $ -conjecture for the p-adic representation $T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$ . For example, if A is an abelian variety of dimension r defined over ${{\mathbb Q}_{p}}$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of A is a p-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$ . This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.

On the closure of the Hodge locus of positive period dimension

Given {{mathbb {V}}} a polarizable variation of {{mathbb {Z}}}-Hodge structures on a smooth connected complex quasi-projective variety S, the Hodge locus for {{mathbb {V}}}^otimes is the set of closed points s of S where the fiber {{mathbb {V}}}_s has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for {{mathbb {V}}}^otimes is a countable union of closed irreducible algebraic subvarieties of S, called the special subvarieties of S for {{mathbb {V}}}. Under the assumption that the adjoint group of the generic Mumford–Tate group of {{mathbb {V}}} is simple we prove that the union of the special subvarieties for {{mathbb {V}}} whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S. This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space {{mathcal {A}}}_g of principally polarized Abelian varieties of dimension g, the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of {{mathcal {A}}}_g is either a closed algebraic subvariety of S or is Zariski-dense in S.

On the standard conjecture for projective compactifications of Néron models of -dimensional Abelian varieties

We prove that the Grothendieck standard conjecture of Lefschetz type holds for a smooth complex projective -dimensional variety fibred by Abelian varieties (possibly, with degeneracies) over a smooth projective curve if the endomorphism ring of the generic geometric fibre is not an order of an imaginary quadratic field. This condition holds automatically in the cases when the reduction of the generic scheme fibre at some place of the curve is semistable in the sense of Grothendieck and has odd toric rank or the generic geometric fibre is not a simple Abelian variety.

On the construction of weakly Ulrich bundles

A well-known conjecture of Eisenbud, Schreyer and Weyman suggests that any projective variety carries an Ulrich (and hence also weakly Ulrich) bundle. This is known only in a handful of cases. In this paper I provide weakly Ulrich bundles on a class of surfaces and threefolds. I construct a weakly Ulrich bundle of large rank on any smooth complete surface in P3 over fields of characteristic p>0 and also for some classes of surfaces of general type in Pn. I also construct intrinsic weakly Ulrich bundles on any Frobenius split variety of dimension at most three. The bundles constructed here are in fact ACM and weakly Ulrich bundles and so I call them almost Ulrich bundles. Frobenius split varieties in dimension three include as special cases: (1) smooth hypersurfaces in P4 of degree at most four, (2) more generally, Frobenius split Fano varieties of dimension at most three, (3) Frobenius split, smooth quintics in P4 (4) more generally, Frobenius split Calabi-Yau varieties of dimension at most three (5) Frobenius split (i.e. ordinary) abelian varieties of dimension at most three. These results also imply that Chow form of these varieties is the support of a single intrinsic determinantal equation.

Non‐vanishing theorems for central L‐values of some elliptic curves with complex multiplication

The paper uses Iwasawa theory at the prime $p=2$ to prove non-vanishing theorems for the value at $s=1$ of the complex $L$-series of certain quadratic twists of the Gross family of elliptic curves with complex multiplication by the field $K = \BQ(\sqrt{-q})$, where $q$ is any prime $\equiv 7 \mod 8$. Our results establish some broad generalizations of the non-vanishing theorem first proven by D. Rohrlich using complex analytic methods. Such non-vanishing theorems are important because it is known that they imply the finiteness of the Mordell-Weil group and the Tate-Shafarevich group of the corresponding elliptic curves over the Hilbert class field of $K$. It is essential for the proofs to study the Iwasawa theory of the higher dimensional abelian variety with complex multiplication which is obtained by taking the restriction of scalars to $K$ of the particular elliptic curve with complex multiplication introduced by Gross.

Singularities of Divisors of Low Degree on Simple Abelian Varieties

Abstract It is known by the results of Kollár, Ein, Lazarsfeld, Hacon, and Debarre that effective divisors representing principal and other low-degree polarizations on complex abelian varieties have mild singularities. In this note, we extend these results to all polarizations of degree $<g$ on simple $g$-dimensional abelian varieties, settling a conjecture of Debarre and Hacon.

How large is Ag(Fq)?

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.

Cyclic components of quotients of abelian varieties mod p

Let A an abelian variety of dimension r, defined over Q. For p a rational prime, we denote by Fp the finite field of cardinality p. If A has good reduction at p, let A¯p be the reduction of A at p. Let Γ be a free subgroup of the Mordell–Weil group A(Q), and let Γp be the reduction of Γ at p. In this paper for abelian varieties of type I, II, III, and IV, under Generalized Riemann Hypothesis, Artin's Holomorphy Conjecture, and Pair Correlation Conjecture, we obtain asymptotic formulas for the number of primes p, with p≤x, for which the quotient A¯p(Fp)Γp has at most 2r−1 cyclic components.

The number of monodromy representations of Abelian varieties of low p-rank

Let Ag be an abelian variety of dimension g and p-rank λ≤1 over an algebraically closed field of characteristic p>0. We compute the number of homomorphisms from π1ét(Ag,a) to GLn(Fq), where q is any power of p. We show that for fixed g, λ, and n, the number of such representations is polynomial in q, and give an explicit formula for this polynomial. We show that the set of such homomorphisms forms a constructible set, and use the geometry of this space to deduce information about the coefficients and degree of the polynomial.In the last section we prove a divisibility theorem about the number of homomorphisms from certain semidirect products of profinite groups into finite groups. As a corollary, we deduce that when λ=0,#Hom(π1ét(Ag,a),GLn(Fq))|GLn(Fq)| is a Laurent polynomial in q.

The uniformization of the moduli space of principally polarized abelian 6-folds

Abstract Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space 𝒜 6 \mathcal{A}_{6} of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general principally polarized abelian variety of dimension 6 is a Prym–Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E 6 E_{6} lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E 6 E_{6} -covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge–Hurwitz eigenbundles with respect to the Kanev correspondence and (3) a description of the ramification divisor of the Prym–Tyurin map from the Hurwitz space to 𝒜 6 \mathcal{A}_{6} in the terms of syzygies of the Abel–Prym–Tyurin curve.