Abelian surfaces and the non-Archimedean Hodge D-conjecture -- the semi-stable case

If $X$ is a smooth projective variety over ${\mathbb R}$, the Hodge ${\mathcal D}$-conjecture of Beilinson asserts the surjectivity of the regulator map to Deligne cohomology with real coefficients. It is known to be false in general but is true in some special cases like Abelian surfaces and $K3$-surfaces - and still expected to be true when the variety is defined over a number field. We prove an analogue of this for Abelian surfaces at a non-Archimedean place where the surface has bad reduction. Here the Deligne cohomology is replaced by a certain Chow group of the special fibre. The case of good reduction is harder and was first studied by Spiess in the case of products of elliptic curve and by me in general.


Introduction 1.The Hodge-D-conjecture for Abelian surfaces
Let A be an Abelian surface over a p-adic local field K and A be a semi-stable model over the ring of integers O = O K .Let A v be the special fibre over the closed point v -which we assume is semi-stable, namely a union of divisors with normal crossings whose components are smooth.The aim of this paper is to prove a non-Archimedean analogue of the Hodge-D-conjecture for such Abelian surfaces.This conjecture states that the map Here PCH 1 (A v ) is a certain sub-quotient of the Chow group of the special fibre.This group has the property that where L p (H 2 (A), s) is the local L-factor at p.This group can hence be viewed as a p-adic version of the Real Deligne cohomology -which has that property with respect to the Archimedean factor -and hence the map ∂ can be viewed as a p-adic version of the regulator map.
When A v is smooth, that is, p is a prime of good reduction, the group PCH 1 (A v ) is simply CH 1 (A v ).This case was studied in [Spi99] and [Sre14].When A is a product of (non-isogenous) elliptic curves and p is a prime of semi-stable reduction for both this was studied in [Sre08].This paper essentially closes the chapter -proving it in the remaining case of semi-stable reduction of simple Abelian surfaces.
Note that the group CH 2 (A, 1) ⊗ Q has many diffrent avatars -it is the same as the K-cohomology group H 1 (A, K 2 ) and the motivic cohomology group H 2 M (A, Q(2)).
Acknowledgements: This work was largely done when the author was visiting CRM in Montreal some years ago.I would like to thank CRM for their hospitality and ISI for their support while this work was done.
2 The target space of the boundary map Let X be a smooth proper variety over a local field K and O the ring of integers of K with closed point v and generic point η.
By a model X of X we mean a flat proper scheme X → Spec(O) together with an isomorphism of the generic fibre X η with X.Let Y be the special fibre X v = X × Spec(k(v)) and let i : Y → X denote the inclusion map.We will always also make the assumption that the model is strictly semi-stable, which means that it is a regular model and the fibre Y is a divisor with normal crossings whose irreducible components are smooth, have multiplicity one and intersect transversally.

Consani's Double Complex
In [Con98], Consani defined a double complex of Chow groups of the components of the special fibre with a monodromy operator N, following the work of Steenbrink [Ste76] and Bloch-Gillet-Soulé [BGS95].Using this complex she was able to relate the higher Chow group of the special fibre at a semi-stable prime to the regular Chow groups of the components.This relation is what is used in defining the group PCH.
Let Y = t i=1 Y i be the special fibre of dim n with Y i its irreducible components.For I ⊂ {1, . . ., t}, define For u and t with 1 u t < r define the map Putting these together induces the map δ(u).Let δ(u) * and δ(u) * denote the corresponding maps on Chow homology and cohomology respectively.They further induce the Gysin and restriction maps on the Chow groups. Define These maps have the properties that

The group PCH
Let a, q be two integers with q − 2a > 0.
Here n is the dimension of Y.Note that if q − 2a > 1 and Y is non-singular, this group is 0, while if Y is singular and semi-stable, the Parshin-Soulé conjecture implies that this group is Our interest is in the remaining case, namely when q − 2a = 1 and Y is singular.
The 'Real' Deligne cohomology has the property that its dimension is the order of the pole of the Archimedean factor of the L-function at a certain point on the left of the critical point.The group PCH 1 (Y) is expected to have a similar property.Let F * be the geometric Frobenius and N(v) the number of elements of k(v).The local L-factor of the (q − 1) st -cohomology group is then Theorem 2.1 (Consani).Let v be a place of semistable reduction.Assuming the weight-monodromy conjecture, the Tate conjecture for the components and the injectivity of the cycle class map on the components Y I , the Parshin-Soulé conjecture and that F * acts semisimply on H * ( X, Q ) I .we have From this point of view the group PCH q−a−1 (Y, q − 2a − 1) can be viewed as a non-Archimedean analogue of the 'Real' Deligne cohomology.Since the L-factor at a prime of good reduction does not have a pole at s = a when q − 2a > 1, the Parshin-Soulé conjecture can be interpreted as the statement that this non-Archimedean Deligne cohomology has the correct dimension, namely 0, even at a prime of good reduction.
As is clear from the definition, the group PCH depends on the choice of the semi-stable model of X.
However, Consani's theorem says that the dimension does not.So to a large extent one can work with any semi-stable model.Perhaps the correct definition is one obtained by taking a limit of semi-stable models as in the work of Bloch, Gillete and Soulé [BGS95] on non-Archimedean Arakelov theory.

Elements of the higher Chow group
From this point on we specialize to the case when X is a surface and further n = 2, q = 3 and a = 1.We will be interested in group CH 2 (X, 1) and the map to PCH 1 (Y) := PCH 1 (Y, 0).This is related to the order of the pole of the L-function of H 2 (X) at s = 1.Soon we will further specialize to the case when X is an Abelian surface.
Let X be a surface over a field K.The group CH 2 (X, 1) has the following presentation [Ram89].It is generated by formal sums of the type where C i are curves on X and f i are K-valued functions on the C i satisfying the cocycle condition Relations in this group are give by the tame symbol of pairs of functions on X.
There are some elements of this group coming from the product structure Here Nm L K is the norm map from a finite extension L of K.The image of this group as L runs through all finite extensions of K is called the subgroup of decomposable elements, CH 2 dec (X, 1) A theorem of Bloch [Blo86] says that CH 1 (X L , 1) is simply L * where L is the field of definition of X L so such an element looks like a sum of elements of the type (C, a) where C is a curve on X L and a is in L * .The group of indecomposable elements is the quotient group In general, it is hard to find elements in this group.
The group CH 2 (X, 1) ⊗ Q has several avatars -it is the same as the K-cohomology group H 1 Zar (X, K 2 ) ⊗ Q and the motivic cohomology group H 3 M (X, Q(2)).

The boundary map
The usual Beilinson regulator maps the higher Chow group to the Real Deligne cohomology.In the non-Archimedean context, it appears that the boundary map plays a similar role.It is defined as follows where fi is the function f i on the closure Ci of C i in the semi-stable model X of X.By the cocycle condition, the 'horizontal divisor', namely, the closure i div , is 0 and so the boundary is supported on the special fibre.Further, since the boundary ∂ of an element is the sum of divisors of functions, it lies in Ker(i * i * ).
For a decomposable element of the form (C, a) the boundary map is particularly simple to compute, Where C v is the special fibre of a model C. In particular, a cycle in the special fibre which is not the restriction of the closure of a cycle in the generic fibre cannot appear in the boundary of the subgroup of decomposable elements.
3 Semi-stable reduction of Abelian surfaces

Types of semi-stable reductions
If A is an abelian surface over a local field, Kulikov and Persson-Pinkham classified the possible semi-stable degenerations.For a surface X the dual graph of its special fibre is defined as follows.It is the simplicial complex with one vertex v i for every component If A is the Neron model of an Abelian surface -so that K A = 0 -the possible semi-stable special fibres are • Type 1 -A p is smooth and the dual graph is a point.
All the elliptic surfaces are isomorphic.The dual graph is S 1 .
• Type 3 -A p is a cycle of rational surfaces, each isomorphic to P 1 × P 1 such that the dual graph is topologically S 1 × S 1 .The double curves are '−1 hexagons' -there are six components in every double curve and each component is a −1 curve.

The group PCH 1 (A p ).
We want to study the boundary of map In this case the target space is We have to study each case separately.

Type 1 degenerations of Abelian surfaces
This is the case when the special fibre is a smooth abelian surface and was studied in [Sre14].Here the target space of the boundary map is simply CH 1 (A p ) ⊗ Q and the rank of this space is at least 2. We showed that there exists a new element of the higher Chow group of A for each new element of CH 1 (A p ).The argument here is quite subtle and uses deformation theory.This includes the case when the special fibre is a product of two elliptic curves.

Type 2 degenerations of Abelian surfaces
In this case the special fibre is a cycle of elliptic ruled surfaces Y i .These surfaces intersect at elliptic curves which are the bases of the elliptic ruled surfaces.All the elliptic curves are isomorphic.Here we have As the conjectures upon which it is conditional hold in this case, we can apply Consani's theorem.Hence the dimension of the group PCH 1 (A p ) ⊗ Q is the order of vanishing of the local L-factor at v of the L-function of H 2 (A).This dimension can be computed using the analogue of the Clemens-Schmidt exact sequence due to [BGS95] and turns out to be 2. Hence we need to construct 2 higher Chow cycles.
One of them can be constructed as follows.
so D ∩ div(π) is the divisor of the function π restricted to D hence is 0 in CH 2 (A) and so maps to 0 in CH 2 (Y (1) ).Hence the restriction of a generic cycle always lies in the group PCH 1 (A p ).This bounds one of the generators of the group PCH 1 (A p ).The conjecture predicts that there is second element of the higher Chow group which bounds the other generator of this group.We will construct this cycle in the next section.

Type 3 degenerations of Abelian surfaces
In this case the individual components are P 2 blown up at the vertices of a triangle -which we will denote by P2 .This results in a '-1 hexagon' where three of the six sides are the strict transforms of the edges of the triangle and the other three are the exceptional fibres.They are all (-1)-curves on P2 and if two components intersect, they intersect along one of these curves.
Here one knows from the analogue of the Clemens-Schmidt exact sequence that the dimension is 3. Hence the conjecture predicts that there are three elements of the higher Chow group.One of them is the boundary of a decomposable element coming from the genus two curve on the generic fibre.We will show that there are at least two other elements which are linearly independent.
hence is an element of the higher Chow group CH 2 (A, 1).

Surjectivity
Our conjecture states that the boundary map in the localization sequence is surjective.We show that the element of Collino's described above, with suitable choices of points P and Q, suffices to show surjectivity in the cases when the special fibre of the Abelian surface is singular, as well as in the case when the special fibre is a product of elliptic curves.
We do this by computing the boundary of the element in terms of the components of the regular minimal model of the curve C. For that we need the theorems of Parshin [Par72] on minimal models of genus 2 curves.
In all the cases, the computation of the boundary is done as follows.Suppose the special fibre where f is the function f on the closure C and H is the horizontal divisor div(f).To compute the a j we do the following.We know that the decomposable element (C, p k ) has boundary div and in particular, X j is not in the support.The degree of a divisor of a function on a curve on an algebraic surface which is not contained in the support is 0. Hence restricting this to X j gives us an equation Using that and what we know about the intersection numbers (X i .X j ) gives us a linear equation among the a i not including a j However, we can simplify our calculations using the following observation.If X is a component of the special fibre C p then (X.C p ) = 0.So equivalently we have the equation though here we have to use what we know abou the self intersection (X j .X j ).
Repeating this with the different components gives us a system of simultaneous equations in the a i which we can solve quite easily.We get as many equations as components this way and so the space of solutions is one dimensional.Sometimes it is convenient to make a choice of the coefficient of one of the components in order to get a 'nice' description of the boundary.
From the Néron mapping property the map ι x : C → A extends to a map, which we will also denote by ι x , where C ns is the curve C with the singular points removed and A 0 is the Néron model of the Jacobian.The special fibre of the Néron model of the Jacobian is the group the special fibre of a minimal regular model with components Y i .Each component of the special fibre is an extension of an abelian variety by a power of G m -so in our case it is either an abelian surface or an extension of an elliptic curve E by G m or G m × G m .We chose a particular component where the closure of the zero section lies and define that to be the identity component.The set of components has the structure of a finite abelian group.
The element in the higher Chow group that we consider is Ξ P,Q .The boundary of the element in the special fibre A p .Since the horizontal cycles cancel, one has The curves ι P (X i ) and ι Q (X i ) are linearly equivalent in PCH 1 (Y) hence the boundary is Hence what we have to show is that for a suitable choice of P and Q one obtains a new cycle in PCH 1 (Y) and in the case of Type 3, we show that for different choices of P and Q we can get two new cycles.
We now do a case by case analysis.There are seven cases of minimal regular models of genus two curves.
In all that follows let C be in the minimal regular model of a genus 2 curve C and C p the special fibre.We use the notation of [Par72].In the pictures, the bold lines correspond to the curves and the thin lines indicate where the Weierstrass points lie.
Case Type of Jacobian Rank of

Case I
In this case the curve C reduces to a smooth genus 2 curve and the Jacobian is of Type 1.Since the special fibre has only one component one has div( fP ) = H + aC p Computing the intersection with C p shows that a = 0. Hence Collino's cycle has no boundary here -but the decomposable element can be used to bound C p .
However, the dimension of PCH 1 (Y) ⊗ Q = CH 1 (Y) ⊗ Q is at least 2 owing to the existance of the Frobenius endomorphism.The conjecture predicts there are at least two higher Chow cycles.Further, it is usually the case that the Picard number of the special fibre is strictly larger than that of the generic fibre.In those cases decomposable cycles will not suffice to prove surjectivity.Collino's cycle has boundary 0 so does not work.
Hence we have to find new indecomposable cycles.This is the content of [Sre14].The idea there was to 'deform' a rational curve corresponding to the extra cycle to construct a new element.Curiously this is the hardest case.

Case II
In this case the stable model of the curve is a genus 1 curve with a node and the Jacobian is of Type 2. The special fibre C p of the regular minimal model consists of a genus 1 curve E and a chain of (2n − 1) rational curves X i meeting E at two points α and β, where n 2 is an integer.One has X 2 i = −2 = E 2 .Finally, the closure of four of the Weierstrass points meets E and the remaining two meet the middle component X n .
Choose P and Q such that P meets E and Q meets X n .One has div( fP To compute a and b i we modify by a decomposable element and intersect with E and X i .As we remarked we can simply consider the restriction to E or X i .For reasons of symmetry one has and by symmetry so combining these two, the boundary of the element Ξ P,Q in the Néron special fibre is A different choice of Weierstrass points will either change sign, if the roles of P and Q are reversed, or have boundary 0, if P and Q lie on the same component. Each component of the Néron special fibre is a non-split extension of E by G m and the group of components is isomorphic to Z/(2n−1)Z.Each X i is isomorphic to G m and its closure in the special fibre of the degenerate Abelian surface is a P 1 .The special fibre of the closure of the curve C P is a copy of E in one component with a chain of P 1 s meeting E at two different points which are translates of each other.Hence the boundary of Adding twice this to our computation of the boundary of Ξ P,Q gives that, up to a decomposable element, the boundary of This can be seen to be non-zero by intersecting with E, for instance.In particular, it is not a multiple of C p .
Hence the two elements C p and ∂(Ξ P,Q ) are linearly independent and therefore generate PCH 1 (Y) ⊗ Q.

Case III
In this case the stable model is a genus 0 curve with two nodes and the Jacobian is of Type 3.This can be viewed as the case when the elliptic curve in Case II degenerates to a nodal curve.Here the special fibre C p of the regular minimal model consists of a genus 0 curve B and two chains of rational curves X i , 1 i 2n − 1 and Y j , 1 j 2m − 1.The closures of two of the Weierstrass points meet B as well as X n and Y m .One has X 2 i = Y 2 j = −2 and B 2 = −4.In this case there are essentially three different elements we can construct.Suppose P, Q and R are three Weierstrass points whose closures lie on B, X n and Y m respectively, then one has the elements Ξ P,Q , Ξ P,R and Ξ Q,R .However, it is easy to see that Ξ P,Q − Ξ P,R = Ξ Q,R in CH 2 (X, 1) as they differ by the tame symbol of a pair of functions.
Suppose P lies on B and Q lies on X n .Then an analysis similar to what is done above shows, up to the boundary of a decomposable element, Similarly, if P lies on B and R lies on Y m one has The boundary of a decomposable element (C, p a ) is, like before, Intersecting with X n and Y m , for instance, shows that the boundaries are linearly independent and hence they generate the group PCH 1 (Y) ⊗ Q.So once again, the boundary map is surjective.

Case IV
In this case the stable model is a union of two elliptic curves meeting at a point and the closures of three of the Weierstrass points lie on each elliptic curve.Here the Jacobian is smooth, hence is of Type 1.The regular minimal model consists of the two elliptic curves along with a chain of rational curves joining them.
The elliptic curves E i satisfy E 2 i = −1 while the rational curves X j , 1 j r satisfy X 2 j = −2.This was studied by Spiess [Spi99] when the generic fibre product of elliptic curves as well.While he constructed a particular element using an irreducible genus 2 curve and two elliptic curves in the generic fibre, in fact one can use the element Ξ P,Q constructed above, with P and Q being chosen such that their closures lie on different components.Then if the divisor of fP is div A convenient choice of b 0 is r + 1 as in that case we have b i = (r + 1 − 2i) = −b r+1−i and the boundary is where a denotes the greatest integer less than or equal to a.Under the map to the Jacobian the X i s map to a point and so the cycle maps to 2(r + 1)(E 1 − E 2 ) which is clearly not a multiple of E 1 + E 2 .Hence the boundary of the decomposable cycle, along with this cycle, generate the group PCH 1 (C p ) ⊗ Q.

Case V
In this case the stable model is a union on an elliptic curve with a nodal rational curve and the Jacobian is of Type 2. The minimal regular model consists of the elliptic curve E along with a chain of rational curves X i satisfying X 2 i = −2 linking E with a rational curve B. One has B 2 = −3 and E 2 = −1.Further, there is a chain of rational curves Y j , 1 j 2m − 1 linking two points of B with Y 2 j = −2.This is essentially the case when one the elliptic curves in Case IV degenerates to a nodal rational curve and corresponds to the case when the extension class of the elliptic surface is trivial -the surface is a product E × P 1 .
Here three of the closures of the Weierstrass points lie on E, one the points lies on B and finally two lie on Y m .We chose P and Q such that the closure of P lies on E and the closure of Q lies on B. Calculating as before suppose div( fP 1) and d j = a − 2(r + 1) for all j where we use the fact that by symmetry d j = d s+1−j .So the boundary is There is another element we can consider -when Q meets the component Y m instead of B -but a similar calculation shows that the boundary is the same.If P lies on B and Q lies on Y m or any such combination, the boundary can be seen to be 0.

Case VI
In this case the stable model is the union of two nodal rational curves meeting at a point.The minimal regular model consists of two rational curves B 1 and B 2 each with a chain of rational curves Y j , 1 j 2n − 1 and Z k , 1 k 2m − 1.The curves B i are also linked by a chain of rational curves X i , 1 i s.Finally, This is the case when both the elliptic curves in Case IV degenerate to nodal rational curves.We studied this in [Sre08] -when the generic fibre was assumed to be a product of two non-isogenous elliptic curves.
However, one can use the element above to prove surjectivity in more generality.Here the Jacobian is of Type 3 so one expects 2 new elements.
To get the first we choose P and Q such that their closures lie on B 1 and B 2 respectively.As before one

4
Higher Chow cycles 4.1 Collino's construction Collino [Col97] constructed a higher Chow cycle on a principally polarized Abelian surface A as follows.Since A is principally polarized, A = Jac(C) where C be a genus 2 curve.Let P and Q be two ramification points on C.There is a function f on C with divisor div(f) = 2P − 2Q Let C P and C Q be the images of the curve C under the maps ι P and ι Q where ι x (y) = y − x and let f P and f Q be the function f being thought of as a function on C P and C Q respectively.Then div(f P ) = 2(0) − 2(Q − P) and div(f Q ) = 2(P − Q) − 2(0) Since P − Q is a two torsion point on the Abelian surface, P − Q = Q − P. Hence the element using a calculation similar to that that above shows b i = b 0 − 2i.
2 and B 2 1 = B 2 2 = −3.A pair of the closures of the Weierstrass points meet the curves Y n and Z m each and the remaining two meet B 1 and B 2 .