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Fix a prime number $ell$. Graphs of isogenies of degree a power of $ell$ are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called $mathfrak l$-isogenies, resolving that they are (almost) volcanoes in any dimension. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as $(ell, ell)$-isogenies: those whose kernels are maximal isotropic subgroups of the $ell$-torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.
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We construct non-isogenous simple ordinary abelian varieties over an algebraic closure of a finite field with isomorphic endomorphism algebras.
The Shafarevich-Tate group $W (mathscr{A})$ measures the failure of the Hasse principle for an abelian variety $mathscr{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove that $W (mathscr{A})cong Cl~(Lambda)oplus Cl~(Lambda)$ or $W (mathscr{A})cong left(mathbf{Z}/2^kmathbf{Z}right) oplus Cl_{~mathbf{odd}}~(Lambda)oplus Cl_{~mathbf{odd}}~(Lambda)$, where $Cl~(Lambda)$ is the ideal class group of a ring $Lambda$ associated to the K-theory of the non-commutative tori and $2^k $ divides the order of $Cl~(Lambda)$. The case of elliptic curves with complex multiplication is considered in detail.
Computing endomorphism rings of supersingular elliptic curves is an important problem in computational number theory, and it is also closely connected to the security of some of the recently proposed isogeny-based cryptosystems. In this paper we give a new algorithm for computing the endomorphism ring of a supersingular elliptic curve $E$ that runs, under certain heuristics, in time $O((log p)^2p^{1/2})$. The algorithm works by first finding two cycles of a certain form in the supersingular $ell$-isogeny graph $G(p,ell)$, generating an order $Lambda subseteq operatorname{End}(E)$. Then all maximal orders containing $Lambda$ are computed, extending work of Voight. The final step is to determine which of these maximal orders is the endomorphism ring. As part of the cycle finding algorithm, we give a lower bound on the set of all $j$-invariants $j$ that are adjacent to $j^p$ in $G(p,ell)$, answering a question in arXiv:1909.07779.
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.
Faltings proved that there are finitely many abelian varieties of genus $g$ of a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many smooth hypersurfaces in $A$, with good reduction outside $S$, representing a given ample class in the Neron-Severi group of $A$, up to translation, as long as the dimension of $A$ is at least $4$. Our approach builds on the approach of arXiv:1807.02721 which studies $p$-adic variations of Hodge structure to turn finiteness results for $p$-adic Galois representations into geometric finiteness statements. A key new ingredient is an approach to proving big monodromy for the variations of Hodge structure arising from the middle cohomology of these hypersurfaces using the Tannakian theory of sheaf convolution on abelian varieties.
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