Supersingular variety

{{short description|Mathematical concept}} In mathematics, a '''supersingular variety''' is (usually) a smooth projective variety in nonzero characteristic such that for all ''n'' the slopes of the Newton polygon of the ''n''th crystalline cohomology are all ''n''/2.<ref name="dejong">{{citation|last=de Jong|first=Aise Johan|author-link=Aise Johan de Jong|title=Shioda's conjecture|url=http://math.columbia.edu/~dejong/wordpress/?p=3859|year=2014}}</ref> For special classes of varieties such as elliptic curves, it is common to use various ad hoc definitions of "supersingular", which are (usually) equivalent to the one given above. At the opposite extreme, a variety is called '''ordinary''' if its Newton polygon coincides with its Hodge polygon.<ref name="mazur">{{citation|last=Mazur|first=Barry|author-link=Barry Mazur|title=Frobenius and the Hodge filtration|journal=Bulletin of the American Mathematical Society|volume=78|issue=5|year=1972|pages=653–667|doi=10.1090/S0002-9904-1972-12976-8|doi-access=free}}</ref> Despite the terminology, "supersingular" and "singular" do not indicate that the variety has singularities.

== History == The term "singular elliptic curve" (or "singular ''j''-invariant") was originally used to refer to complex elliptic curves whose ring of endomorphisms has rank 2, the maximum possible over the complex numbers. In the 1930s, Helmut Hasse discovered that elliptic curves over finite fields can have even larger endomorphism rings of rank 4, and these were termed "supersingular elliptic curves".<ref name="silverman">{{citation|last=Silverman|first=Joseph H.|author-link=Joseph H. Silverman|title=The Arithmetic of Elliptic Curves|edition=2nd|publisher=Springer-Verlag|series=Graduate Texts in Mathematics|volume=106|year=2009|isbn=978-0-387-09493-9}}</ref>

As the theory of crystalline cohomology was developed by Alexander Grothendieck and Pierre Berthelot in the 1960s and 1970s,<ref name="berthelot-ogus">{{citation|last1=Berthelot|first1=Pierre|author1-link=Pierre Berthelot (mathematician)|last2=Ogus|first2=Arthur|author2-link=Arthur Ogus|title=Notes on Crystalline Cohomology|publisher=Princeton University Press|series=Mathematical Notes|volume=21|year=1978|isbn=0-691-08218-9}}</ref> it became possible to characterize the supersingularity of an elliptic curve in terms of the slopes of the Newton polygon of its first crystalline cohomology group. This cohomological perspective led to the generalization of the concept to other classes of varieties: a variety is supersingular when its Newton polygon has all slopes concentrated at the middle value. The notion was extended to abelian varieties, K3 surfaces, and Enriques surfaces.

== Formal definition == Let ''X'' be a smooth projective variety over a perfect field ''k'' of characteristic ''p'' > 0, and let ''W''(''k'') denote the ring of Witt vectors of ''k''. The crystalline cohomology groups ''H''^''n''_cris(''X''/''W''(''k'')) are finitely generated modules over ''W''(''k''), equipped with a Frobenius-linear endomorphism. The Newton polygon of ''H''^''n''_cris(''X''/''W''(''k'')) encodes the ''p''-adic valuations of the eigenvalues of Frobenius acting on the associated F-isocrystal.<ref name="katz">{{citation|last=Katz|first=Nicholas M.|author-link=Nick Katz|title=Slope filtration of F-crystals|journal=Astérisque|volume=63|year=1979|pages=113–163}}</ref>

The variety ''X'' is '''supersingular''' if, for every non-negative integer ''n'', all slopes of the Newton polygon of ''H''^''n''_cris(''X''/''W''(''k'')) are equal to ''n''/2.<ref name="dejong" /> For a variety over a finite field '''F'''_''q'', this is equivalent to the condition that all eigenvalues of Frobenius on the ''l''-adic cohomology ''H''^''n''(''X'', '''Q'''_''l'') are ''q''^''n''/2 times roots of unity.<ref name="katz-messing">{{citation|last1=Katz|first1=Nicholas M.|author1-link=Nick Katz|last2=Messing|first2=William|title=Some consequences of the Riemann hypothesis for varieties over finite fields|journal=Inventiones Mathematicae|volume=23|year=1974|pages=73–77|doi=10.1007/BF01405203}}</ref>

A fundamental result of Barry Mazur and Nicholas Katz establishes that the Newton polygon always lies on or above the Hodge polygon; a supersingular variety represents the case where the Newton polygon lies as high as possible.<ref name="mazur" /><ref name="katz" />

== Classes of supersingular varieties ==

=== Elliptic curves === {{main|Supersingular elliptic curve}} An elliptic curve ''E'' over a field of characteristic ''p'' > 0 is supersingular if and only if its endomorphism ring is an order in a quaternion algebra, giving it rank 4 rather than the rank 2 typical of ordinary elliptic curves.<ref name="silverman" /> In terms of crystalline cohomology, ''E'' is supersingular if and only if the Newton polygon of ''H''¹_cris(''E''/''W''(''k'')) has a single slope of 1/2, rather than slopes 0 and 1 as in the ordinary case.

For each prime ''p'', there are only finitely many ''j''-invariants corresponding to supersingular elliptic curves over the algebraic closure of '''F'''_''p'', and all such ''j''-invariants lie in '''F'''_''p''². The first example of a supersingular elliptic curve was observed by John Tate over fields of characteristic 3, using the Fermat quartic.<ref name="tate1965">{{citation|last=Tate|first=John|author-link=John Tate (mathematician)|title=Algebraic cycles and poles of zeta functions|work=Arithmetic Algebraic Geometry|publisher=Harper and Row|location=New York|year=1965|pages=93–110}}</ref>

=== Abelian varieties === A supersingular abelian variety of dimension ''g'' over a field of characteristic ''p'' > 0 can be defined either as an abelian variety that is isogenous to a product of supersingular elliptic curves, or as one whose endomorphism algebra has dimension (2''g'')² over '''Q'''.<ref name="oort">{{citation|last=Oort|first=Frans|author-link=Frans Oort|title=Subvarieties of moduli spaces|journal=Inventiones Mathematicae|volume=24|year=1974|pages=95–119}}</ref> The Newton polygon of a supersingular abelian variety of dimension ''g'' has all slopes equal to 1/2. Because the cohomology ring of an abelian variety is the exterior algebra of its first cohomology group, supersingularity of an abelian variety is determined entirely by ''H''¹.

=== K3 surfaces === {{main|Supersingular K3 surface}} For K3 surfaces in characteristic ''p'' > 0, two related notions of supersingularity have been studied. A K3 surface is '''Artin supersingular''' if the slopes of Frobenius on ''H''²_cris are all equal to 1, or equivalently if its formal Brauer group has infinite height.<ref name="artin1974">{{citation|last=Artin|first=Michael|author-link=Michael Artin|title=Supersingular K3 surfaces|journal=Annales Scientifiques de l'École Normale Supérieure|series=4th series|volume=7|issue=4|year=1974|pages=543–567}}</ref> A K3 surface is '''Shioda supersingular''' if the rank of its Néron–Severi group equals its second Betti number, which is 22 for all K3 surfaces.

Michael Artin conjectured that the two notions coincide: that a K3 surface with formal Brauer group of infinite height must have Picard number 22.<ref name="artin1974" /> The Tate conjecture for K3 surfaces of finite height was proved by Niels Nygaard and Arthur Ogus,<ref name="nygaard-ogus">{{citation|last1=Nygaard|first1=Niels O.|last2=Ogus|first2=Arthur|author2-link=Arthur Ogus|title=Tate's conjecture for K3 surfaces of finite height|journal=Annals of Mathematics|series=2nd series|volume=122|issue=3|year=1985|pages=461–507|doi=10.2307/1971327}}</ref> and Artin's conjecture was established in characteristic ''p'' ≥ 3 through work of Maulik, Charles, and Madapusi Pera, which together also completed the proof of the Tate conjecture for K3 surfaces in those characteristics.<ref name="maulik">{{citation|last=Maulik|first=Davesh|title=Supersingular K3 surfaces for large primes|journal=Duke Mathematical Journal|volume=163|issue=13|year=2014|pages=2357–2425|doi=10.1215/00127094-2804783|arxiv=1203.2889}}</ref><ref name="charles">{{citation|last=Charles|first=François|title=The Tate conjecture for K3 surfaces over finite fields|journal=Inventiones Mathematicae|volume=194|issue=1|year=2013|pages=119–145|doi=10.1007/s00222-012-0443-y|arxiv=1206.4002}}</ref>

K3 surfaces with Picard number 22 exist only in positive characteristic; Hodge theory implies that the Picard number of a K3 surface over the complex numbers is at most 20. The first example was given by Tate, who observed that the Fermat quartic surface has Picard number 22 over algebraically closed fields of characteristic ''p'' ≡ 3 (mod 4).<ref name="tate1965" />

=== Enriques surfaces === {{main|Supersingular Enriques surface}} In characteristic other than 2, Enriques surfaces are quotients of K3 surfaces by a free involution and do not exhibit supersingular behavior. In characteristic 2, Enrico Bombieri and David Mumford showed that Enriques surfaces fall into three classes — classical, singular, and supersingular — distinguished by the structure of their Picard scheme.<ref name="bombieri-mumford">{{citation|last1=Bombieri|first1=Enrico|author1-link=Enrico Bombieri|last2=Mumford|first2=David|author2-link=David Mumford|title=Enriques' classification of surfaces in char. ''p'', III|journal=Inventiones Mathematicae|volume=35|year=1976|pages=197–232|doi=10.1007/BF01390138}}</ref> A supersingular Enriques surface in characteristic 2 has ''H''¹(''O''_''X'') of dimension 1 with trivial Frobenius action, trivial canonical bundle, and its Picard scheme Pic^τ is isomorphic to the group scheme α₂. Its canonical double cover is a purely inseparable α₂-cover of a surface with trivial dualizing sheaf.

== Shioda and Artin supersingularity for surfaces == For algebraic surfaces more generally, the notions introduced for K3 surfaces can be formulated as follows: * A surface is '''Shioda supersingular''' if the rank of its Néron–Severi group equals its second Betti number. * A surface is '''Artin supersingular''' if its formal Brauer group has infinite height.

The Tate conjecture implies that for surfaces over algebraically closed fields, Artin supersingularity implies Shioda supersingularity.<ref name="artin1974" /> The converse — whether Shioda supersingularity implies Artin supersingularity — is not known in full generality.

== See also == * Crystalline cohomology * F-crystal * Newton polygon * Formal group

== References == {{reflist|30em}}

Category:Algebraic varieties Category:Algebraic geometry