Generic Newton polygons for curves of given p-rank
aa r X i v : . [ m a t h . N T ] D ec GENERIC NEWTON POLYGONS FOR CURVES OF GIVEN p -RANK JEFF ACHTER AND RACHEL PRIESA
BSTRACT . We survey results and open questions about the p -ranks and Newton polygons of Ja-cobians of curves in positive characteristic p . We prove some geometric results about the p -rankstratification of the moduli space of (hyperelliptic) curves. For example, if 0 ≤ f ≤ g −
1, we provethat every component of the p -rank f + M g contains a component of the p -rank f stra-tum in its closure. We prove that the p -rank f stratum of M g is connected. For all primes p and all g ≥
4, we demonstrate the existence of a Jacobian of a smooth curve, defined over F p , whose New-ton polygon has slopes { , , 1 } . We include partial results about the generic Newton polygons ofcurves of given genus g and p -rank f .
1. I
NTRODUCTION
Suppose C is a smooth projective curve of genus g defined over a finite field F q of characteristic p . Then its zeta function has the form Z C / F q ( T ) = L C / F q ( T )( − T )( − qT ) for some polynomial L C / F q ( T ) ∈ Z [ T ] . The Newton polygon ν of C is that of L C / F q ( T ) ; it is a lower convex polygon in R withendpoints (
0, 0 ) and ( g , g ) . Its slopes encode important information about C and its Jacobian.Given a curve C / F q of genus g , there are methods to compute its Newton polygon. After someexperiments, it becomes clear that the typical Newton polygon has slopes only 0 and 1. For small g and p , the other possible Newton polygons do occur, but rarely, leading us to the followingquestion. Question 1.1.
Does every Newton polygon of height g (satisfying the obvious necessary conditions) occuras the Newton polygon of a smooth curve defined over a finite field of characteristic p for each prime p? The answer to this question is unknown, although one now knows that every integer f suchthat 0 ≤ f ≤ g occurs as the length of the line segment of slope 0 for the Newton polygon of acurve in each characteristic p [12]. As an example, we consider the first open case, when g = ν has slopes and . We confirm in Lemma 5.3 that this Newton polygon occurs for a curveof genus 4 for each prime p using a unitary Shimura variety of type U (
3, 1 ) .The main idea in this paper is that the occurrence of a certain Newton polygon for a curve ofsmall genus can be used to prove the occurrence of new Newton polygons for smooth curves forevery larger genus. As an application, we prove in Corollary 5.6 that the Newton polygon ν g − g having g − and occurs as the Newton polygon of asmooth curve of genus g for all primes p and all g ≥ Mathematics Subject Classification. study the geometry of the corresponding loci in M g , the moduli space of smooth proper curvesof genus g .More precisely, the p -rank f and Newton polygon are invariants of the p -divisible group of aprincipally polarized abelian variety. The stratification of the moduli space A g by these invariantsis well-understood, in large part because of work of Chai and Oort. Let A g be the moduli space ofprincipally polarized abelian varieties of dimension g . The Torelli map τ : M g ֒ → A g , which sendsa curve to its Jacobian, allows us to define the analogous stratifications on M g . For dimensionreasons, this gives a lot of information when 1 ≤ g ≤ g ≥ p -rank f stratum M fg is irreducible.In Section 2, we review the fundamental definitions and properties of the p -rank and Newtonpolygon. In Section 3, we review the p -rank and Newton polygon stratifications of A g . Sincedegeneration is one of the few techniques for studying stratifications in M g , in Section 4.1 werecall the Deligne-Mumford compactification of M g , and explain how it interacts with the p -rankstratification.In Section 4.2, we review a theorem that we proved about the boundary of the p -rank strata M fg of M g in [2]. Using this, we prove that M fg is connected for all g ≥ ≤ f ≤ g (Corollary4.5). For f ≥
1, we also prove that every component of M fg contains a component of M f − g in itsclosure (Corollary 4.4).In Section 5, we consider the finer stratification of M g by Newton polygon. We consider aNewton polygon ν fg which is the most generic Newton polygon of an abelian variety of dimension g and p -rank f . The expectation is that the generic point of every component of M fg representsa curve with Newton polygon ν fg . We prove that this expectation holds in the first non-trivialcase when f = g − f = g − supersingular , meaning that the Newton polygon is a line segment of slope . For example, itis not hard to prove that a curve which achieves the Hasse-Weil bound over a finite field must besupersingular. On the other hand, the p -rank stratification is in some ways “transverse” to otherinteresting loci in M g , illustrated by the fact that a randomly chosen Jacobian of genus g and p -rank f behaves like a randomly selected principally polarized abelian variety of dimension g . InSection 4.4 and Section 5 we discuss open questions and conjectures on these topics.We thank the organizers for the opportunity to participate in the RICAM workshop on algebraiccurves over finite fields. 2. S TRUCTURES IN POSITIVE CHARACTERISTIC
Consider a principally polarized abelian variety X of dimension g defined over a field K ofcharacteristic p >
0. If N ≥ p , then the N -torsion group scheme X [ N ] is´etale, and X [ N ]( K ) ∼ = ( Z / N ) ⊕ g depends only on the dimension of X . In contrast, X [ p ] is never reduced, and there is a range of possibilities for the geometric isomorphism class of X [ p ] K and, afortiori , the p -divisible group X [ p ∞ ] : = lim → n X [ p n ] . In this section, we review some attributes of X [ p ] and X [ p ∞ ] , with special emphasis on the case where X is the Jacobian of a curve over a finitefield.2.1. The p -rank. The p -rank of X is the rank of the “physical” p -torsion of X . More precisely, it isthe integer f such that(2.1) X [ p ]( K ) ∼ = ( Z / p ) ⊕ f . ENERIC NEWTON POLYGONS FOR CURVES OF GIVEN p -RANK 3 We will see below (2.2.3) that 0 ≤ f ≤ g . The abelian variety X is said to be ordinary if its p -rankis maximal, i.e., f = g .Specifying a K -point of X [ p ] is equivalent to specifying a homomorphism X [ p ] → ( Z / p ) ofgroup schemes over K , and thus one may also define f by f = dim F p Hom K ( X [ p ] , ( Z / p )) .Now, X [ p ] is a self-dual group scheme, and the dual of ( Z / p ) is the non-reduced group scheme µµµ p , the kernel of Frobenius on the multiplicative group G m . Consequently, it is equivalent to definethe p -rank of X as f = dim F p Hom K ( µµµ p , X [ p ]) .(This last formulation is convenient for defining the p -rank of semiabelian varieties and semistablecurves.)If X is the Jacobian of a smooth, projective curve C , then the p -rank equals the maximum rankof a p -group which occurs as the Galois group of an unramified cover of C [20, Corollary 4.18].2.2. Newton polygons.
Newton polygon of a curve over a finite field.
Let C / F q be a smooth, projective curve of genus g . Then its zeta function Z C / F q ( T ) = exp ( ∑ k ≥ C ( F q k ) T k / k ) is a rational function of the form Z C / F q = L C / F q ( T )( − T )( − qT ) where L C / F q ( T ) ∈ Z [ T ] is a polynomial of degree 2 g . The L -polynomial factors over Q as L C / F q ( T ) = ∏ ≤ j ≤ g ( − α j T ) where the roots can be ordered so that(2.2) α j α g + j = q for each 1 ≤ j ≤ g .Each α j has archimedean size √ q ; for each ι : Q ֒ → C , one has (cid:12)(cid:12) ι ( α j ) (cid:12)(cid:12) = √ q . In contrast, thereis a range of possibilities for the p -adic valuations of the α j . The Newton polygon of C (or of itsJacobian X ) is a combinatorial device which encodes these valuations.Let K be a field with a discrete valuation v , and let h ( T ) = ∑ a i T i ∈ K [ T ] be a polynomial. TheNewton polygon of h ( T ) is defined in the following way.In the plane, graph the points ( i , v ( a i )) , and form its lower convex hull. This object is called theNewton polygon of h . Equivalently, it suffices to track the multiplicity e ( λ ) with which each slope λ occurs in the diagram. Thus, we will often record a Newton polygon as the function Q / / Z ≥ λ ✤ / / e ( λ ) which, to each λ , assigns the length of the projection of the “slope λ ” part of the Newton polygononto its first coordinate. This function encodes the valuation of the roots of h . More precisely, it isnot hard to check that the number of α ∈ K such that v ( α ) = − λ and h ( α ) = e ( λ ) . JEFF ACHTER AND RACHEL PRIES F IGURE λ = a λ / b λ with gcd ( a λ , b λ ) =
1. Since h ( T ) is defined over K , there is an integralityconstraint(2.3) e ( λ ) λ ∈ Z for each λ ∈ Q ,which implies that the line segments of the Newton polygon break at points with integral coordi-nates. Also,(2.4) ∑ λ e ( λ ) / b λ = deg h .We will often work with the equivalent data m ( λ ) : = e ( λ ) / b λ .Now equip Q with the p -adic valuation, normalized so that v ( q ) =
1. The Newton polygon of C / F q is that of L C / F q ( T ) . The choice of p -adic valuation means that the Newton polygon of C isunchanged by finite extension of the base field. Moreover, the relation (2.2) implies that e ( λ ) = λ Q ∩ [
0, 1 ] (2.5) e ( λ ) = e ( − λ ) .(2.6)A Newton polygon satisfying (2.3), (2.5) and (2.6) will be called an admissible symmetric Newtonpolygon of height ∑ λ e ( λ ) / b λ .2.2.2. Examples.
Let E / F q be an elliptic curve. There is an integer a such that | a | ≤ √ q such that E ( F q ) = − a + q .Then Z E / F q ( T ) = − aT + qT ( − T )( − qT ) .Suppose gcd ( a , p ) =
1. (This is the generic case.) Then the Newton polygon of E is the lowerconvex hull of the points { (
0, 1 ) , (
1, 0 ) , (
2, 1 ) } ,and the slopes of E are {
0, 1 } ; we have m ( λ ) = ( λ ∈ {
0, 1 } ( a , p ) = p . Then the Newton polygon of E is the lower convexhull of points { (
0, 1 ) , ( ≥ ) , (
2, 1 ) } ,and the only slope of E is { } ; m ( ) =
1, and all other multiplicities are zero. Such an ellipticcurve is called supersingular. (See Figure 2.1.)We will use the next example (where g =
4) in the proof of Lemma 5.3.
Example 2.1.
There exists a (hyperelliptic) curve of genus 4 defined over F whose Newton poly-gon has slopes 1/4 and 3/4. ENERIC NEWTON POLYGONS FOR CURVES OF GIVEN p -RANK 5 Proof.
Using Sage, we calculated a list of possibilities for monic polynomials f ( x ) ∈ F [ x ] of de-gree nine such that the hyperelliptic curve C : y = f ( x ) has 3-rank 0. One such possibility is f ( x ) = x + x + x + x + x + x + x + x .For the prime 3, Sage cannot directly compute the L-polynomial of C . Instead, we compute thedegree 4 truncation of the zeta function of C to be:127 T + T + T + T + C has the form Z C / F ( T ) = L C / F ( T ) / ( − T )( − T ) where L C / F ( T ) = + aT + bT + cT + dT + cT + bT + aT + T for some coefficients a , b , c , d . By taking the degree four Taylor polynomial of Z C / F ( T ) , we solve a = b = c = d =
6. Then the slopes of the Newton polygon of L C / F ( T ) = T + T + (cid:3) Newton polygon of an abelian variety over an arbitrary field.
There is an equivalent notion ofNewton polygon which also makes sense for an arbitrary abelian variety over an arbitrary fieldof positive characteristic. For each λ ∈ Q ≥ , write λ = a λ / b λ = a λ / ( a λ + c λ ) with a λ and b λ relatively prime. Manin defines a certain p -divisible group G λ = G a λ , c λ over F p , with dimension a λ , codimension c λ and, thus, height b λ [18]. If G is any p -divisible group over an algebraicallyclosed field k , then there is an isogeny G → ⊕ λ H ⊕ m G ( λ ) λ . The isogeny itself is not canonical, but thecollection of all nonnegative integers { m G ( λ ) } is independent of all choices. Let e G ( λ ) = b λ m G ( λ ) .Now let X / K be an abelian variety of dimension g ; the Newton polygon of X is that of its p -divisible group X [ p ∞ ] K . It is not hard to verify that e X [ p ∞ ] is an admissible symmetric Newtonpolygon of height 2 g . In the special case where C / F q is a smooth projective curve over a finitefield, the Newton polygon of C (as defined in Section 2.2.1) coincides with that of its Jacobian. Ingeneral, we define the Newton polygon of C / K as that of its Jacobian.The Newton polygon of a p -divisible group, and thus of an abelian variety, is invariant underisogeny. (Note, however, that even if X [ p ∞ ] and Y [ p ∞ ] are isogenous, it does not follow that X and Y are isogenous.) Moreover, if G and H are p -divisible groups, it follows immediately from thedefinition that m G ⊕ H ( λ ) = m G ( λ ) + m H ( λ ) .The symmetry condition m ( λ ) = m ( − λ ) forces the inequality 0 ≤ f X ≤ dim X noted inSection 2.1.An abelian variety X / K is ordinary if and only if all its slopes are 0 and 1. The p -rank of X is equal to the multiplicity m ( ) of the slope 0 in the Newton polygon. An abelian varietyis supersingular if all the slopes of its Newton polygon equal 1/2. Thus, if an abelian variety issupersingular, then it has p -rank zero. However, in dimension at least three, the converse is false.For example, there are abelian varieties with p -rank 0 whose Newton polygons have slopes g and g − g and are thus not supersingular for g ≥ Semicontinuity and purity.
We now consider a family of p -divisible groups, such as thatcoming from a family of abelian varieties in characteristic p . It is not too hard to show that the p -rank is a lower semicontinuous function, i.e., that it can only decrease under specialization. Infact, if the p -rank does change, it does so in codimension one: Lemma 2.2. [22, Lemma 1.6]
Let X → S be an abelian variety over an integral scheme in positivecharacteristic, and suppose that X has generic p-rank f . Let S < f ⊂ S denote the locus parametrizing thoses such that X s has p-rank strictly less than f . Then either S < f is empty or S < f is pure of codimension one. JEFF ACHTER AND RACHEL PRIES
There are analogous semicontinuity and purity results for Newton polygons, although the for-mer requires more notation to state, and the proof of the latter is much deeper than that of Lemma2.2.The partial ordering on Newton polygons is defined as follows. Let ν = { λ ⊕ m λ , . . . , λ ⊕ m λ r r } ,where λ i < λ i + . Let Γ ( ν ) ⊂ R be the convex hull of (
0, 0 ) and { ( j ∑ i = m λ i c λ i , j ∑ i = m λ i a λ i ) : 1 ≤ j ≤ r } .If µ and ν are two Newton polygons, we will write µ (cid:22) ν if Γ ( µ ) and Γ ( ν ) have the same endpointsand if all points of Γ ( µ ) lie on or above those of Γ ( ν ) . (This convention may seem a little surprising,but it has the pleasant consequence that “smaller” Newton strata are in the closures of “larger”ones.) See Figure 2.2 for the g = S = Spec ( R ) be the spectrum of a local ring, with geometric generic point η and geometric closedpoint s . If G is a p -divisible group over S , then ν ( G s ) (cid:22) ν ( G η ) . Like the p -rank, the Newtonpolygon is a discrete invariant which changes (if at all) in codimension one. Proposition 2.3. [10]
Let S be an integral, excellent scheme. Let G → S be a p-divisible group over S,and let U ⊂ S be the largest dense set on which the Newton polygon is constant. Then either U = S or codim (( S r U ) , S ) = . Notation on stratifications and Newton polygons.
Let X → S be any family of abelian va-rieties of relative dimension g in positive characteristic. For any Newton polygon ν , let S ν be thereduced subspace such that s ∈ S ν if and only if the Newton polygon of X s is ν . In general, theNewton stratification refines the p -rank stratification S = ⊔ S f , where s ∈ S f if and only if the p -rank of X s is f .If C → S is a family of smooth, projective curves, then the Newton polygon and p -rank strataof S are those corresponding to the relative Jacobian Jac ( C ) → S .Suppose that S is irreducible and X → S is a family of abelian varieties. Since there are onlyfinitely many symmetric admissible Newton polygons of any particular height, by semicontinuitythere is a nonempty open subset U over which the Newton polygon of X is constant. We call theNewton polygon of (any geometric fiber of) X U → U the generic Newton polygon of X → S , orsimply of S if the family of abelian varieties is clear from context. Definition 2.4.
For each 0 ≤ f ≤ g , define a Newton polygon ν fg as follows: ν gg = { ⊕ g , 1 ⊕ g } ν g = ( { ⊕ g } g ≤ { g , g − g } g ≥ < f < g , set ν fg = ν ff ⊕ ν g − f .This is the largest (“most generic”) admissible symmetric Newton polygon of height 2 g and p -rank f . Similarly, let σ g = { ⊕ g } be the supersingular Newton polygon of height 2 g . ENERIC NEWTON POLYGONS FOR CURVES OF GIVEN p -RANK 7 σ ν ⊕ σ ν ν ⊕ σ ν ⊕ ν ν ν ν F IGURE ν to ν ′ if and only if ν ≺ ν ′ .With this notation, we depict the poset of Newton polygons for g = ν of height 2 g , let codim ( ν ) be the length of anypath from ν gg to ν in the poset of all such Newton polygons.3. S TRATIFICATIONS ON THE MODULI SPACE OF ABELIAN VARIETIES
The p -rank and Newton stratifications of A g are well understood. We review some of theirfeatures here, for two different reasons.First, for 1 ≤ g ≤ M g ֒ → A g is an open immersion, and M ∗ g → A g is an isomorphism. Con-sequently, in small genus, the stratifications on A g are perfectly mirrored in M g . This informationis used directly below.Second, as we shall see, A g is in some sense more highly structured than M g . Thus, the resultsdescribed here for A g might elicit some of the most optimistic conjectures one might make aboutsuch stratifications on M g . (Indeed, some “obvious” conjectural statements are too strong to betrue; see Section 7.1.)3.1. The p -ranks of abelian varieties.Theorem 3.1. [21] Let g ≥ and ≤ f ≤ g. Then A fg is nonempty and pure of codimension g − f in A g . The nonemptiness is easy to see; it suffices to take a product of f ordinary elliptic curves and g − f supersingular elliptic curves, equipped with the product principal polarization. Corollary 3.2.
Let g ≥ and ≤ f ≤ g. Let S be an irreducible component of A fg . (a) If f < g, then S is in the closure of A f + g . (b) If f > , then the closure of S contains an irreducible component of A f − g .Proof. Part (a) follows from the dimension count of Theorem 3.1 and purity for p -ranks (Lemma2.2).We do not know an elementary proof of part (b), although it is an immediate consequence ofTheorem 3.3. (cid:3) Note that, unless ( g , f ) = (
1, 0 ) or (
2, 0 ) , Theorem 3.3 implies that A fg is actually irreducible.3.2. Newton polygons of abelian varieties.
Thanks especially to work of Oort and of Chai andOort, the Newton stratification on A g is well-understood. Theorem 3.3. [7]
Let ν be a symmetric admissible Newton polygon of height g. (a) The stratum A ν g is nonempty and has codimension codim ( ν ) in A g . (b) If ν = σ g , then A ν g is irreducible. (c) The supersingular locus A σ g g is connected but reducible. JEFF ACHTER AND RACHEL PRIES
In analyzing Newton strata A ν g , two key tools are the Serre-Tate theorem and the action of Heckeoperators. In concert with Dieudonn´e theory, the first tool allows one to use (semi-)linear algebrato study the deformation space of an abelian variety. The second tool, and in particular the factthat Newton strata are stable under this large group of partial symmetries of A g , allows one todeduce global information about Newton strata.In general, both of these structures are missing from M g , and the state of our knowledge is,correspondingly, much cruder. For a given symmetric admissible Newton polygon ν of height 2 g ,it is typically not even known if M ν g is nonempty, let alone pure or even irreducible. Degeneration (as in Theorem 4.2) and low genus phenomena (e.g., the fact that every principally polarized abelianthreefold is the Jacobian of a stable curve) are among the few tools we have at our disposal. In thesecond half of this paper, we show how these techniques can be combined to yield informationabout stratifications on M g .4. T HE p - RANK STRATIFICATION OF THE MODULI SPACE OF STABLE CURVES
Let M g / F p be the moduli stack of smooth proper curves of genus g . Even if one is intrinsicallyonly interested in smooth curves, one is quickly led, following Deligne and Mumford, to study M g , the moduli stack of stable proper curves of genus g [11].4.1. The moduli space of stable curves.
It turns out that M g is open in M g , which is proper. Theboundary ∂ M g = M g r M g is a union ∂ M g = ∪ ≤ i ≤⌊ g /2 ⌋ ∆ i [ M g ] , whose construction we brieflyrecall.For a natural number r , let M g ; r be the moduli stack of r -labeled stable curves of genus g . Thereare finite clutching morphisms M g ;1 × M g ;1 κ g g / / M g + g M g ;2 κ g / / M g + in which the labeled points are identified; see [17, Section 3] for more details. Here, we simplyrecord the following facts.Suppose that, for i ∈ {
1, 2 } , ( C , P i ) is a smooth, pointed curve with moduli point s i ∈ M g i ;1 ( k ) .Then κ g , g ( s , s ) is the moduli point of the curve C of genus g + g obtained by identifying P and P . One has(4.1) Pic ( C ) ∼ = Pic ( C ) ⊕ Pic ( C ) .Now suppose that ( C , P , Q ) ∈ M g ;2 ( k ) is a smooth, 2-pointed curve. Then κ g ( s ) is the modulipoint of the curve e C obtained from C by identfying P and Q , and there is an exact sequence(4.2) 1 / / G m / / Pic ( e C ) / / Pic ( C ) / / ∆ [ M g ] = κ g − ( M g − ) , and for 1 ≤ i ≤ g − ∆ i [ M g ] = κ i , g − i ( M i ;1 × M g − i ;1 ) . If S isa stack equipped with a morphism S → M g , we let ∆ i [ S ] = S × M g ∆ i [ M g ] . Let M ∗ g = M g r ∆ ;this is precisely the locus of curves whose Picard varieties are actually abelian varieties, and notmerely semiabelian varieties. The Torelli map τ : M ∗ g → A g is a birational morphism onto itsimage, and contracts fibers on the boundary. More precisely, (the Torelli theorem states that) τ isinjective on M g . On the boundary, τ forgets the identification point; if P and Q are points on acurve C , then τ (( C , P ) , ( C , P )) = τ (( C , Q ) , ( C , P )) . ENERIC NEWTON POLYGONS FOR CURVES OF GIVEN p -RANK 9 The p -rank stratification of M g . The notion of p -rank makes sense for stable, and not justsmooth, curves.On ∆ we find, in the notation of (4.2), that f e C = f C + ∆ i with i > f C = f C + f C .(4.4)Thus, the p -rank stratification extends to M g . We emphasize that M fg parametrizes those stablecurves whose p -rank is f , while M fg is the closure of the set of smooth curves of genus g and p -rank f . In particular, if f >
0, one has M fg ( M fg .From (4.3) and (4.4) it immediately follows that κ g , g ( M f g × M f g ) ⊆ M f + f g + g κ g − ( M fg − ) ⊆ M f + g .Faber and van der Geer exploit this structure to show: Theorem 4.1. [12]
Suppose g ≥ and ≤ f ≤ g. Then M fg is nonempty and pure of codimension g − fin M g . In fact, much more is true. First, using relations (4.3)-(4.4) and the fundamental dimensioncount supplied by Theorem 4.1, it is not hard to see that M fg is dense in M fg ; every stable curveof genus g and p -rank f is a limit of smooth curves with the same discrete parameters. Moreover,the recursive structure of the boundary is compatible with the p -rank stratification: Theorem 4.2. [2, Lemma 3.2 and Prop. 3.4]
Suppose g ≥ and ≤ f ≤ g. Let S be an irreduciblecomponent of M fg . (a) If f > , then S contains the image of an irreducible component of M f − g − under κ g − . (b) Suppose ≤ i ≤ g − . Let f and f be nonnegative integers such that ≤ f ≤ i; ≤ f ≤ g − i; and f + f = f . Then S contains the image of an irreducible component of M f i × M f g − i under κ i , g − i . Consequently, closures of components of p -rank strata contain chains of elliptic curves: Corollary 4.3. [2, Corollary 3.6]
Suppose g ≥ , ≤ f ≤ g, and A ⊂ { · · · , g } has cardinality f .Let S be an irreducible component of M fg . Then S contains the moduli point of a chain of elliptic curvesE , · · · , E g , where E j is ordinary if and only if j ∈ A. Connectedness of p -rank strata. We combine Theorem 4.1 with degeneration techniques toprove:
Corollary 4.4.
Suppose g ≥ and ≤ f ≤ g. Let S be an irreducible component of M fg . (a) If f < g, then S is in the closure of M f + g . (b) If f > , then S contains an irreducible component of M f − g .Proof. Part (a) is a direct consequence of purity for p -rank (Lemma 2.2) and the dimension countTheorem 4.1; the proof proceeds by induction on g − f .For part (b), the statement is clearly true for g =
1, or more generally when f = g . Nowsuppose g ≥
2. By Theorem 4.2(b), S contains an irreducible component of M × M f − g − under the image of κ g − . Since M is irreducible, its closure contains points of p -rank zero, and ∆ [ S ] f − isnonempty. Therefore, S < f is nonempty and, by Lemma 2.2, has codimension one in S . For each i between 0 and g , ∆ i [ S ] < f has codimension two in S . Therefore, S < f is the closure of S < f , andthe latter has codimension one in S . The basic dimension count (Theorem 4.1) now shows that S contains an irreducible component of M f − g . (cid:3) In contrast with the theory of p -rank strata of A g , at present we have essentially no nontrivialinformation about irreducibility of various M fg . Still, the method of degeneration detects a smallamount of the topology of these strata. Recall that M fg denotes the p -rank f stratum of the modulispace of stable curves of genus g . Corollary 4.5.
If g ≥ and ≤ f ≤ g, then M fg is connected.Proof. Since p -rank strata in M ∗ coincide with those of A , and since p -rank strata in A g areirreducible (Theorem 3.3), each M f is irreducible. Similarly, M and M are irreducible, while M is connected.We next prove by induction on g that, for each g ≥ ≤ f ≤ g , M fg is connected.Fix integers f and f with 0 ≤ f ≤
2, 0 ≤ f ≤ g −
2, and f + f = f . Let S and S be irre-ducible components of M fg . By Theorem 4.2(b), each closure S i contains an irreducible componentof κ g − ( M f × M f − g − ) . By the inductive hypothesis, M f × M f − g − is connected. Consequently, S ∪ S is connected. The theorem now follows by considering each of the (finitely many) irre-ducible components of M fg . (cid:3) Open questions about the p -rank stratification. The results above indicate that the p -rankstratification is in some ways “transverse” to other interesting loci in M g . For example, as long as ( g , f ) = (
1, 0 ) or (
2, 0 ) , then there exists a smooth projective curve over F p of genus g and p -rank f whose automorphism group is trivial [1]. More generally, there is a precise sense in which arandomly chosen Jacobian of genus g and p -rank f behaves like a randomly selected principallypolarized abelian variety of dimension g [2].On the other hand, there is a non-trivial interplay between the p -rank and the automorphismgroup of a curve. For example, there are constraints on the p -rank of a cyclic tame cover of P [5].There are even stronger constraints in the case of a wildly ramified Galois action. The Deuring-Shafarevich formula places severe limitations on the p -rank for a p -group cover of curves [9, 27].The p -rank stratification of the moduli space of Artin-Schreier curves is discovered in [25]. Seealso [13, 14].Here are several other open questions about the p -ranks of Jacobians of curves. Question 4.6.
Suppose g ≥ and ≤ f ≤ g − . Is M fg irreducible? Question 4.7.
Suppose g ≥ and ≤ f ≤ g − . Does there exist a curve of genus g and p-rank fdefined over F p ?
5. S
TRATIFICATION BY N EWTON POLYGON
In this section, we explain how Theorem 4.2 yields information about generic Newton polygons,beginning with the cases g ≤
4, and extending to arbitrary g . Recall (Definition 2.4) the Newtonpolygons ν fg . ENERIC NEWTON POLYGONS FOR CURVES OF GIVEN p -RANK 11 Newton polygons of curves of small genus.Lemma 5.1.
The p-rank zero locus M is irreducible, with generic Newton polygon ν = { } .Proof. The only symmetric Newton polygons of height 6 and p -rank zero are ν and σ = { ⊕ } ;but the supersingular locus in A is pure of dimension 2 (3.3), while M is pure of dimension 3(4.1). The result now follows from the Torelli theorem and Theorem 3.3. (cid:3) Lemma 5.2.
Let S be an irreducible component of M . Then the generic Newton polygon of S lies on orbelow σ ⊕ ν = { } .Proof. By Theorem 4.2, S contains a component of κ ( M × M ) , which has generic Newtonpolygon { } ⊕ { } (Lemma 5.1). The result for S follows from the semicontinuity ofNewton polygons. (cid:3) Lemma 5.3. In M , (a) there is at least one irreducible component with generic Newton polygon ν ; and (b) there is at most one irreducible component with generic Newton polygon ν ⊕ σ .Proof. Let S be an irreducible component of M ; then S has dimension 5 (Theorem 4.1). Supposethe generic Newton polygon of S is ν ⊕ σ . The locus in A of abelian fourfolds with Newtonpolygon ν ⊕ σ is irreducible of dimension 5 (Theorem 3.3). Thus S , or rather its image τ ( S ) under the Torelli map, must coincide with this locus. In particular, such an S , if it exists, is unique;this proves (b).For part (a), it suffices to show that there exists some curve whose Jacobian has Newton polygon ν . Consider the moduli space Z of principally polarized abelian fourfolds equipped with anaction by Z [ ζ ] of signature (
3, 1 ) . Then Z is contained in the (compactified) Torelli locus.This has been known for some time, but we provide a sketch here. For a squarefree polynomial f ( x ) of degree 6, let C f denote the curve with affine equation y = f ( x ) . Then Jac ( C f ) has an actionby Z [ ζ ] of signature (
3, 1 ) . On one hand, the parameter space for such curves has dimension6 − dim PGL =
3. On the other hand, Z itself is irreducible (since Q ( ζ ) has class number one)of dimension 3 · =
3. Consequently, τ ( M ) contains an open, dense subspace of U , and τ ( M ∗ ) ,which is equal to the closure of M in A , contains all of Z .It now suffices to show that there exists a point on Z parametrizing an abelian variety withNewton polygon ν . If p splits in Q ( ζ ) , this follows from [19, Section 2.2]. (In the notation of [19], ( q , h − q ) = (
1, 3 ) , and α is the Newton polygon of slope 1/4.) If p is inert in Q ( ζ ) , this followsfrom [6, Section 5.4]. (In the notation of [6], ( n − s , s ) = (
3, 1 ) , ρ = p =
3, this follows fromExample 2.1. (cid:3)
Generic Newton polygons.Proposition 5.4.
Let g , f ∈ N and let g = g + f . (a) If the generic Newton polygon of every component of M g is ν g , then the generic Newton polygonof every component of M fg is ν fg . (b) If the generic Newton polygon of at least one component of M g is ν g then the generic Newtonpolygon of at least one component of M fg is ν fg .Proof. For (a), let S be a component of M fg and consider its closure in M fg . By Theorem 4.2(b), S contains the image of an irreducible component of M ff ;1 × M g ;1 . Now M ff ;1 is irreduciblewith generic Newton polygon ν gg . By hypothesis, every component of M g ;1 has generic Newtonpolygon ν g . By semicontinuity, the generic Newton polygon ν of S lies on or below ν fg . On the other hand, the p -rank f condition implies that ν has f slopes of 0 and 1. This is only possible if ν = ν fg .The proof of (b) is by induction on f . The base case is the hypothesis that there exists a com-ponent S g of M g with generic Newton polygon ν g . Now suppose, as inductive hypothesis, that S g − is a component of M f − g − which has generic Newton polygon ν f − g − . We add a labeling of twopoints to each curve represented by a point of S g − by letting T g − = S g − × M g − M f − g − . Thenconsider the image under the clutching morphism Z g = κ g − ( T g − ) which is contained in ∆ [ M fg ] .By [2, Lemma 3.2], there exists an irreducible component S g of M fg such that S g contains Z g . Bysemicontinuity, the generic Newton polygon ν of S g lies on or below ν fg . Then ν = ν fg by the p -rank f condition. (cid:3) If g ≤
2, then the p -rank of an abelian variety determines its Newton polygon. More generally,if f ∈ { g − g − g } , then there is a unique symmetric admissible Newton polygon of height2 g and p -rank f . In contrast, if f < g − p -rank constrains, but does not determine, theNewton polygon. Corollary 5.5.
Let g ≥ . If S is a component of M g − g , then S has generic Newton polygon ν g − g = { ⊕ g − , 1/3, 2/3, 1 ⊕ g − } .Proof. By Lemma 5.1, the generic Newton polygon of every component of M is ν . The resultfollows from Proposition 5.4(a). (cid:3) We obtain partial information in the next case when f = g − Corollary 5.6.
Let g ≥ . There exists a component of M g − g with generic Newton polygon ν g − g . Inparticular, there is a smooth projective curve whose Jacobian has Newton polygon ν g − g .Proof. By Lemma 5.3(a), there exists a component S of M with generic Newton polygon ν . Theresult follows from Proposition 5.4(b). (cid:3) A better understanding of M would allow one to prove results for arbitrary g when f = g − YPERELLIPTIC CURVES
Recall that a hyperelliptic curve is a smooth projective curve C which can be realized as a doublecover C → P of the projective line. Among all curves, hyperelliptic curves have enjoyed specialattention. On the practical side, algorithmic methods for handling hyperelliptic curves over finitefields are much more highly developed than they are for arbitrary curves. On the theoretical side,hyperelliptic curves over finite fields are a natural function-field analogue of quadratic numberfields, and thus an attractive site for investigation of conjectures. In this section, we briefly sketchthe extent to which the ideas and results surveyed here for M g extend to the moduli space ofhyperelliptic curves. Throughout, we assume that the characteristic p is odd .The boundary of H g . Let H g be the moduli space of hyperelliptic curves. Any given curve admitsat most one hyperelliptic involution ι , and thus there is an inclusion H g ֒ → M g . Let H g be theclosure of H g in M g . The boundary ∂ H g necessarily admits a decomposition ∂ H g = ∪ ∆ i [ H g ] , butconstructing the full boundary is somewhat delicate. Briefly, if two hyperelliptic curves ( C , P ) and ( C , P ) , with hyperelliptic involutions ι and ι , are clutched, then the resulting curve is hy-perelliptic if and only if each P i is fixed by ι i . Consequently, for 0 < i < g , ∆ i [ H g ] is described asthe image of e H i × e H g − i / / e H g / / H g ENERIC NEWTON POLYGONS FOR CURVES OF GIVEN p -RANK 13 where e H g is the moduli space of hyperelliptic curves equipped with a labeling of the 2 g + e H g → H g is the (finite-to-one) forgetful map.The description of irreducible components of ∆ [ H g ] is somewhat more intricate, since the curveobtained by identifying points P and Q of the hyperelliptic curve C is hyperelliptic if and only if ι ( P ) = Q . In fact, there is a decomposition of ∆ [ H g ] as a union ∆ [ H g ] = κ g − ( H g − ; 1 ) [ ≤ i ≤⌊ g ⌋ λ i , g − − i ( H i ;1 × H g − − i ;1 ) of irreducible divisors. We refer to [3, 30] for more details.The fact that, for i > ∆ i [ H g ] is parametrized by hyperelliptic curves with a discrete choice ofpoint makes it much more difficult to unravel the p -rank stratification ∂ H g .As in Theorem 4.1, it turns out that if 0 ≤ f ≤ g , then H fg is nonempty and pure of codimension g − f [15].This is used in proving the currently optimal hyperelliptic analogue to Theorem 4.2: Theorem 6.1. [3, Lemma 3.4]
Suppose g ≥ and ≤ f ≤ g. Let S be an irreducible component of H fg . (a) If f > , then each irreducible component of ∆ [ S ] contains either an irreducible component of κ g − ( H f − g − ) or of some λ i , g − − i ( H f i ;1 × H f g − − i ;1 ) with ≤ f ≤ i, ≤ f ≤ g − − i,f + f = f . (b) If f = , then S contains the image of an irreducible component of e H i × e H g − i for some ≤ i ≤ g − . In contrast to the case of general curves (Corollary 4.3), at present one only knows that hyper-elliptic curves degenerate to trees of elliptic curves:
Corollary 6.2. [3, Theorem 3.11(c)]
Suppose g ≥ and ≤ f ≤ g. Let S be an irreducible componentof H fg . Then S contains the moduli point of some tree of elliptic curves, of which f are ordinary and g − fare supersingular. Stratification by p -rank. To some extent, the p -rank stratification of H g is known to be much likethat of M g .Corollary 4.4 holds, mutatis mutandis , for H g . (Part (a) relies only on dimension counting; part(b), whose proof relies on degeneration, already appears as [3, Corollary 3.15].)However, even for g =
3, we start losing information. It is not known if H is irreducible.While H is connected, it is not clear if e H is connected, making it more difficult to prove a resultanalogous to Corollary 4.5 for hyperelliptic curves.Stratification by Newton polygon. Less is known about the Newton stratification of H g than thatof M g . However, it is known that, for each irreducible component of H , the generic Newtonpolygon is ν [23].The analogue of Proposition 5.4 is valid for H g , too. In part (a), even though one has less controlover degenerations, one still knows that, for an irreducible component S of H fg , S contains anirreducible component of κ g − ( e H × e H f − g − ) . Part (b) is valid for H g , as well. The key observationis that, if S is an irreducible component of H f − g − , then κ g − ( S × H g − H g − ) is in the boundary of some irreducible component of H fg .We conclude that the generic Newton polygon of each component of H g − g is ν g − g .7. S OME CONJECTURES ABOUT N EWTON POLYGONS OF CURVES
In this section, we discuss variations of the following question.
Question 7.1.
Given g and p, does every symmetric admissible Newton polygon of height g occur for theJacobian of a smooth projective curve of genus g? The first open case of this question is when g =
4, for the Newton polygons σ , ν ⊕ σ , and ν ⊕ σ . Using Theorem 4.2, it is not hard to see that each of these Newton polygons is realized bya singular curve of compact type. It is unlikely that appeal to Shimura varieties, as in the proof ofLemma 5.3, will resolve this question for all p . For example, if p is inert in Q ( ζ ) , then no abelianvariety with p -rank one has an action by Z [ ζ ] . In particular, ν ⊕ σ is not the Newton polygonof any abelian variety with moduli point in Z .7.1. Non-existence philosophy.
In [24], Oort explains why the answer to Question 7.1 could be no . Consider the partial ordering of Newton polygons. Suppose that: : (i) codim ( ξ ) ≥ g −
3, i.e., the length of the longest chain of Newton polygons connecting ξ and ν gg is larger than 3 g −
3; and : (ii) the denominators of ξ are “large”.Then [24, Expectation 8.5.4] states that one expects that there is no curve of compact type whoseJacobian has Newton polygon ξ .By this reasoning, one expects that there is no curve of genus 11 whose Jacobian has Newtonpolygon ξ = G ⊕ G , with slopes 5/11 and 6/11. On the other hand, in characteristic p = F , namely y + y = x + x + x + x + x , which doeshave slopes 5/11 and 6/11.The dimension of A g is g ( g + ) /2 and the dimension of its supersingular locus is ⌊ g /4 ⌋ . Thelength of the longest chain of Newton polygons connecting the supersingular Newton polygon σ g and the ordinary Newton polygon ν gg is the difference between these, which is greater than 3 g − g ≥
9. It is still possible that every Newton polygon in the chain occurs for a Jacobian, but ifso, then there are two Newton polygons ξ and ξ , such that A ξ g is in the closure of A ξ g but M ξ g is not in the closure of M ξ g . In other words, under Condition (i), M g does not admit a perfectstratification by Newton polygon.The Newton polygon of a curve of compact type is the join of the Newton polygons of itscomponents. If the denominators of ξ are all less than g , then one can try to construct a singularcurve with Newton polygon ξ from curves of smaller genus. For example, it is easy to see that σ g is the Newton polygon of a tree of supersingular elliptic curves. If ξ is indecomposable as asymmetric Newton polygon, then it cannot occur for the Jacobian of a singular curve of compacttype. As a means of making Condition (ii) more precise, one could restrict to the case that ξ isindecomposable.Another variation is to restrict to the Jacobians of smooth curves. In fact, Oort conjectures thatif ξ i is the Newton polygon of a point of M g i for i =
1, 2, then the join of ξ and ξ occurs as theNewton polygon of a point of M g + g , in other words, as the Newton polygon of the Jacobian ofa smooth curve [24, Conjecture 8.5.7]. Remark 7.2.
The original motivation for this non-existence expectation was the following. Let Y ( cu ) ( K , g ) denote the statement: There exists an abelian variety A of dimension g defined over K which is not isogenous to the Jacobian of any curve of compact type; (here, this is an isogeny ofabelian varieties without polarization). There is an expectation that Y ( cu ) ( Q , g ) is true for every g ≥
4. To prove Y ( cu ) ( Q , g ) , it suffices to find a prime p and a Newton polygon ξ of height 2 g suchthat ξ does not occur as the Newton polygon of a Jacobian of a curve [24, (8.5.1)]. It turns out that Y ( cu ) ( Q , g ) was proven by other methods [8, 28]. attributed to Katz and to Oort by, respectively, Oort and Katz ENERIC NEWTON POLYGONS FOR CURVES OF GIVEN p -RANK 15 Supersingular curves.
Recall that σ g = { ⊕ g } . The supersingular locus M σ g g has been stud-ied extensively. When p =
2, Van der Geer and Van der Vlugt proved that there is a smoothcurve of every genus which is supersingular. More generally, the same methods show there existsupersingular curves of arbitrarily large genus defined over F p for all primes p . Example 7.3.
Let R ( x ) ∈ F p [ x ] be an additive polynomial, i.e., a polynomial of the form R [ x ] = a + a x p + · · · + a d x p d . Consider the Artin-Schreier curve Y with affine equation y p − y = xR ( x ) ;it has genus ( p − )( p d + ) /2. Then Y is supersingular by [29, Theorem 13.7].7.3. Other non-existence results.
To this date, the only non-existence results about Newton poly-gons are for Jacobians of curves with automorphisms. Newton polygons of degree p covers of P have been studied using techniques for exponential sums and Dwork cohomology. For example[26], when p =
2, there are no hyperelliptic curves of genus 2 n − p cover of theprojective line [4]. R EFERENCES [1] Jeffrey D. Achter, Darren Glass, and Rachel Pries,
Curves of given p-rank with trivial automorphism group , MichiganMath. J. (2008), no. 3, 583–592.[2] Jeffrey D. Achter and Rachel Pries, Monodromy of the p-rank strata of the moduli space of curves , Int. Math. Res. Not.IMRN (2008), no. 15, Art. ID rnn053, 25.[3] ,
The p-rank strata of the moduli space of hyperelliptic curves , Adv. Math. (2011), no. 5, 1846–1872,10.1016/j.aim.2011.04.004.[4] R´egis Blache,
Valuation of exponential sums and the generic first slope for Artin-Schreier curves , J. Number Theory (2012), no. 10, 2336–2352, 10.1016/j.jnt.2012.04.017.[5] Irene I. Bouw,
The p-rank of ramified covers of curves , Compositio Math. (2001), no. 3, 295–322,10.1023/A:1017513122376.[6] Oliver B ¨ultel and Torsten Wedhorn,
Congruence relations for Shimura varieties associated to some unitary groups , J. Inst.Math. Jussieu (2006), no. 2, 229–261.[7] Ching-Li Chai and Frans Oort, Monodromy and irreducibility of leaves , Ann. of Math. (2) (2011), no. 3, 1359–1396,10.4007/annals.2011.173.3.3.[8] ,
Abelian varieties isogenous to a Jacobian , Ann. of Math. (2) (2012), no. 1, 589–635, 10.4007/an-nals.2012.176.1.11.[9] Richard M. Crew,
Etale p-covers in characteristic p , Compositio Math. (1984), no. 1, 31–45.[10] A. J. de Jong and F. Oort, Purity of the stratification by Newton polygons , J. Amer. Math. Soc. (2000), no. 1, 209–241.[11] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus , Inst. Hautes ´Etudes Sci. Publ. Math.(1969), no. 36, 75–109.[12] Carel Faber and Gerard van der Geer,
Complete subvarieties of moduli spaces and the Prym map , J. Reine Angew. Math. (2004), 117–137.[13] Massimo Giulietti and G´abor Korchm´aros,
Automorphism groups of algebraic curves with p-rank zero , J. Lond. Math.Soc. (2) (2010), no. 2, 277–296, 10.1112/jlms/jdp066.[14] Darren Glass, The 2-ranks of hyperelliptic curves with extra automorphisms , Int. J. Number Theory (2009), no. 5,897–910, 10.1142/S1793042109002468.[15] Darren Glass and Rachel Pries, Hyperelliptic curves with prescribed p-torsion , Manuscripta Math. (2005), no. 3,299–317.[16] Nicholas M. Katz,
Slope filtration of F-crystals , Journ´ees de G´eom´etrie Alg´ebrique de Rennes (Rennes, 1978), Vol. I,Soc. Math. France, Paris, 1979, pp. 113–163.[17] Finn F. Knudsen,
The projectivity of the moduli space of stable curves. II. The stacks M g , n , Math. Scand. (1983), no. 2,161–199.[18] Yuri I. Manin, Theory of commutative formal groups over fields of finite characteristic , Uspehi Mat. Nauk (1963), no. 6(114), 3–90.[19] Elena Mantovan, On certain unitary group Shimura varieties , Ast´erisque (2004), no. 291, 201–331, Vari´et´es deShimura, espaces de Rapoport-Zink et correspondances de Langlands locales.[20] James S. Milne, ´Etale cohomology , Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton,N.J., 1980.[21] Peter Norman and Frans Oort,
Moduli of abelian varieties , Ann. of Math. (2) (1980), no. 3, 413–439. [22] Frans Oort,
Subvarieties of moduli spaces , Invent. Math. (1974), 95–119.[23] , Hyperelliptic supersingular curves , Arithmetic algebraic geometry (Texel, 1989), Progr. Math., vol. 89,Birkh¨auser Boston, Boston, MA, 1991, pp. 247–284.[24] , in problems from the Workshop on Automorphisms of Curves , Rend. Sem. Mat. Univ. Padova (2005), 129–177.[25] Rachel Pries and Hui June Zhu,
The p-rank stratification of Artin-Schreier curves , Ann. Inst. Fourier (Grenoble) (2012), no. 2, 707–726, 10.5802/aif.2692.[26] Jasper Scholten and Hui June Zhu, Hyperelliptic curves in characteristic 2 , Int. Math. Res. Not. (2002), no. 17, 905–917,10.1155/S1073792802111160.[27] Dor´e Subrao,
The p-rank of Artin-Schreier curves , Manuscripta Math. (1975), no. 2, 169–193.[28] Jacob Tsimerman, The existence of an abelian variety over Q isogenous to no Jacobian , Ann. of Math. (2) (2012), no. 1,637–650, 10.4007/annals.2012.176.1.12.[29] Gerard van der Geer and Marcel van der Vlugt, Reed-Muller codes and supersingular curves. I , Compositio Math. (1992), no. 3, 333–367.[30] Kazuhiko Yamaki, Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristic , Asian J. Math. (2004), no. 3, 409–426.C OLORADO S TATE U NIVERSITY , F
ORT C OLLINS , CO 80523
E-mail address : [email protected] URL : C OLORADO S TATE U NIVERSITY , F
ORT C OLLINS , CO 80523
E-mail address : [email protected] URL ::