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S. Srimathy

Publications and source records attributed to S. Srimathy.

6 recordsLinked to original sources
arXiv2024

Artin-Schreier-Witt lifts of purely inseparable extensions

Given a discrete valued field $K$ of positive characteristic, we study the cyclic lifting problem of purely inseparable extensions of the residue field. We prove that unlike the mixed characteristic case, cyclic lifts of any finite purely inseparable modular extension exist and show how to explicitly construct them. Moreover, given such a residual extension, we prove the existence of Artin-Schreier-Witt lifts of any finite degree. This follows from a more general construction based on the notion of $\mathcal{G}$-weaves and $\mathcal{G}$-cyclic extensions where $\mathcal{G}$ is an arbitrary gene over $K$. In particular, this gives an affirmative answer to a question in \cite{ramification_survey} as well as implies that there is no cap on the wild ramification index unlike the mixed characteristic case. We also show some interesting applications such as constructing cyclic lifts of fields that are isomorphic to the tensor product of a purely inseparable modular extension and an Artin-Schreier-Witt extension. Finally, we prove a structure theorem of Artin-Schreier-Witt extensions over a discrete valued fields which restricts the ramification type of intermediate field extensions complementing the above results.

math.NT↗
arXiv2024

The genus of division algebras over discrete valued fields

Given a field with a set of discrete valuations $V$, we show how the genus of a division algebra over the field is related to the genus of the residue algebras at various valuations in $V$ and the ramification data. When the division algebra is a quaternion, we show the triviality of genus over many fields which include higher local fields, function fields of curves over higher local fields and function fields of curves over real closed fields. We also consider function fields of curves over global fields with a rational point and show how the genus problem is related to the $2$-torsion of the Tate-Shafarevich group of its Jacobian. As a special case, we show how the methods developed yield better bounds on the size of the genus over function fields of elliptic curves and demonstrate how they can be computed directly using arithmetic data of the elliptic curve with a number of examples.

math.NT↗
arXiv2024

Totally ramified subfields of $p$-Algebras

We conjecture that a $p$-algebra over a complete discrete valued field $K$ contains a totally ramified purely inseparable subfield if and only if it contains a totally ramified cyclic maximal subfield. We prove the conjecture in several cases.

math.RA↗
arXiv2024

Azumaya algebras with involution and classical semisimple group schemes

Let $S$ be a non-empty scheme with 2 invertible. In this paper we present a functor $F: AZ_*^{n'} \rightarrow GS_*^n$ where $AZ_*^{n'}$ and $GS_*^n$ are fibered categories over $Sch_S$ given respectively by degree-$n'$ Azumaya algebras with an involution of type $*$ and rank-$n$ adjoint group schemes of classical type $*$ with absolutely simple fibers. Here $n'$ is a function of $n$. We show that this functor is an equivalence of fibered categories using étale descent, thus giving a classification of adjoint (as well as simply connected) group schemes over $S$, generalizing the well known case when the base scheme is the spectrum of a field. In particular, this implies that every adjoint group scheme of classical type with absolutely simple fibers is isomorphic to the neutral component of the automorphism group scheme of a unique (up to isomorphism) Azumaya algebra with involution. We also show interesting applications of this classification such as specialization theorem for isomorphism classes of Azumaya algebra with involution over Henselian local rings, uniqueness of integral model for groups with good reduction over discrete valued fields and discuss its implications on the Grothendieck-Serre conjecture over certain domains.

math.AG↗