Semistable abelian variety#Semistable elliptic curve

In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field.

For an abelian variety $A$ defined over a field $F$ with ring of integers $R$ , consider the Néron model of $A$ , which is a 'best possible' model of $A$ defined over $R$ . This model may be represented as a scheme over $\mathrm{Spec}(R)$ (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism

$\mathrm{Spec}(F) \to \mathrm{Spec}(R)$

gives back $A$ . The Néron model is a smooth group scheme, so we can consider $A^0$ , the connected component of the Néron model which contains the identity for the group law. This is an open subgroup scheme of the Néron model. For a residue field $k$ , $A^0_k$ is a group variety over $k$ , hence an extension of an abelian variety by a linear group. If this linear group is an algebraic torus, so that $A^0_k$ is a semiabelian variety, then $A$ has semistable reduction at the prime corresponding to $k$ . If $F$ is a global field, then $A$ is semistable if it has good or semistable reduction at all primes.

The fundamental semistable reduction theorem of Alexander Grothendieck states that an abelian variety acquires semistable reduction over a finite extension of $F$ .Grothendieck (1972) Théorème 3.6, p. 351

Semistable elliptic curve

A semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type.Husemöller (1987) pp.116-117 Suppose {{math| E}} is an elliptic curve defined over the rational number field $\mathbb{Q}$ . It is known that there is a finite, non-empty set S of prime numbers {{math| p}} for which {{math| E}} has bad reduction modulo {{math| p}}. The latter means that the curve $E_p$ obtained by reduction of {{math| E}} to the prime field with {{math| p}} elements has a singular point. Roughly speaking, the condition of multiplicative reduction amounts to saying that the singular point is a double point, rather than a cusp.Husemoller (1987) pp.116-117 Deciding whether this condition holds is effectively computable by Tate's algorithm.Husemöller (1987) pp.266-269 Therefore in a given case it is decidable whether or not the reduction is semistable, namely multiplicative reduction at worst.

The semistable reduction theorem for {{math| E}} may also be made explicit: {{math| E}} acquires semistable reduction over the extension of {{math| F}} generated by the coordinates of the points of order 12.This is implicit in Husemöller (1987) pp.117-118{{citation|chapter=Algorithm for determining the type of a singular fiber in an elliptic pencil

|last=Tate|first=John | authorlink=John Tate (mathematician)

|series=Lecture Notes in Mathematics

|publisher=Springer|publication-place= Berlin / Heidelberg

|issn=1617-9692

|volume=476

|editor1-last=Birch | editor1-first=B.J. | editor1-link=Bryan John Birch

| editor2-last=Kuyk | editor2-first=W.

|title=Modular Functions of One Variable IV

|doi=10.1007/BFb0097582

|year=1975

|isbn=978-3-540-07392-5

|pages=33–52

|mr=0393039

|zbl=1214.14020

}}

References

{{cite book

| last = Grothendieck

| first = Alexandre

| author-link = Alexandre Grothendieck

| title = Séminaire de Géométrie Algébrique du Bois Marie - 1967-69 - Groupes de monodromie en géométrie algébrique - (SGA 7) - vol. 1

|series=Lecture Notes in Mathematics |volume=288

| year = 1972

| publisher = Springer-Verlag

| location = Berlin; New York

| language = fr

| pages = viii+523

| no-pp = true

|doi= 10.1007/BFb0068688

|isbn=978-3-540-05987-5

| mr = 0354656

}}

{{cite book | last=Husemöller | first=Dale H. | title=Elliptic curves | others=With an appendix by Ruth Lawrence | series=Graduate Texts in Mathematics | volume=111 |publisher=Springer-Verlag | year=1987 | isbn=0-387-96371-5 | zbl=0605.14032 }}
{{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | url=https://archive.org/details/surveydiophantin00lang_347 | url-access=limited | publisher=Springer-Verlag | year=1997 | isbn=3-540-61223-8 | zbl=0869.11051 | page=[https://archive.org/details/surveydiophantin00lang_347/page/n83 70] }}

Category:Abelian varieties

Category:Diophantine geometry

Category:Elliptic curves