7,374 research outputs found
Stratifications of Newton polygon strata and Traverso's conjectures for p-divisible groups
The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over
an algebraically closed field is the least positive integer m such that D[p^m]
determines D up to isomorphism (resp. up to isogeny). We show that these
invariants are lower semicontinuous in families of p-divisible groups of
constant Newton polygon. Thus they allow refinements of Newton polygon strata.
In each isogeny class of p-divisible groups, we determine the maximal value of
isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown
to be optimal in the isoclinic case. In particular, the latter disproves a
conjecture of Traverso. As an application, we answer a question of Zink on the
liftability of an endomorphism of D[p^m] to D.Comment: 50 pages, to appear in Annals of Mathematic
On compatibility between isogenies and polarisations of abelian varieties
We discuss the notion of polarised isogenies of abelian varieties, that is,
isogenies which are compatible with given principal polarisations. This is
motivated by problems of unlikely intersections in Shimura varieties. Our aim
is to show that certain questions about polarised isogenies can be reduced to
questions about unpolarised isogenies or vice versa.
Our main theorem concerns abelian varieties B which are isogenous to a fixed
abelian variety A. It establishes the existence of a polarised isogeny A to B
whose degree is polynomially bounded in n, if there exist both an unpolarised
isogeny A to B of degree n and a polarised isogeny A to B of unknown degree. As
a further result, we prove that given any two principally polarised abelian
varieties related by an unpolarised isogeny, there exists a polarised isogeny
between their fourth powers.
The proofs of both theorems involve calculations in the endomorphism algebras
of the abelian varieties, using the Albert classification of these endomorphism
algebras and the classification of Hermitian forms over division algebras
On Jacquet-Langlands isogeny over function fields
We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic
Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of
-elliptic sheaves. The kernel of the isogeny is a subgroup of the
cuspidal divisor group constructed by examining the canonical maps from the
cuspidal divisor group into the component groups.Comment: 29 page
Isogeny graphs with maximal real multiplication
An isogeny graph is a graph whose vertices are principally polarized abelian
varieties and whose edges are isogenies between these varieties. In his thesis,
Kohel described the structure of isogeny graphs for elliptic curves and showed
that one may compute the endomorphism ring of an elliptic curve defined over a
finite field by using a depth first search algorithm in the graph. In dimension
2, the structure of isogeny graphs is less understood and existing algorithms
for computing endomorphism rings are very expensive. Our setting considers
genus 2 jacobians with complex multiplication, with the assumptions that the
real multiplication subring is maximal and has class number one. We fully
describe the isogeny graphs in that case. Over finite fields, we derive a depth
first search algorithm for computing endomorphism rings locally at prime
numbers, if the real multiplication is maximal. To the best of our knowledge,
this is the first DFS-based algorithm in genus 2
On isogeny classes of Edwards curves over finite fields
We count the number of isogeny classes of Edwards curves over finite fields,
answering a question recently posed by Rezaeian and Shparlinski. We also show
that each isogeny class contains a {\em complete} Edwards curve, and that an
Edwards curve is isogenous to an {\em original} Edwards curve over \F_q if
and only if its group order is divisible by 8 if , and 16
if . Furthermore, we give formulae for the proportion of
d \in \F_q \setminus \{0,1\} for which the Edwards curve is complete or
original, relative to the total number of in each isogeny class.Comment: 27 page
Weil numbers generated by other Weil numbers and torsion fields of abelian varieties
Using properties of the Frobenius eigenvalues, we show that, in a precise
sense, ``most'' isomorphism classes of (principally polarized) simple abelian
varieties over a finite field are characterized up to isogeny by the sequence
of their division fields, and a similar result for ``most'' isogeny classes.
Some global cases are also treated.Comment: 13 page
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