arXiv subjects

Steve Thakur

Publications and source records attributed to Steve Thakur.

4 recordsLinked to original sources

arXiv2023

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 which is vital in pairing-based cryptography. In particular, we show that for a prescribed CM field $L$ of degree $\geq 4$, prescribed integers $m$, $n$ and any prime $\ell\equiv 1 \mod{mn}$ 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 $\ell$ and the field extension generated by the $\ell$-torsion points of degree $mn$ over the field of definition. We also study a class of absolutely simple higher dimensional abelian varieties whose endomorphism algebras are central over imaginary quadratic fields.

math.NT↗

arXiv2020

Isogenies of certain abelian varieties over finite fields with p-ranks zero

We study the isogenies of certain abelian varieties over finite fields with non-commutative endomorphism algebras with a view to potential use in isogeny-based cryptography. In particular, we show that any two such abelian varieties with endomorphism rings maximal orders in the endomorphism algebra are linked by a cyclic isogeny of prime degree.

math.NT↗

arXiv2018

On some abelian varieties of type IV

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).

math.NT↗

arXiv2017

Reductions of abelian varieties of generalized Mumford type

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.

math.NT↗