The universal vector extension of an abeloid variety

Let $A$ be an abelian variety over a complete non-Archimedean field $K$. The universal cover of the Berkovich space attached to $A$ reflects the reduction behaviour of $A$. In this paper the universal cover of the universal vector extension $E(A)$ of $A$ is described. In a forthcoming paper ( arXiv:2007.04659), this will be one of the crucial tools to show that rigid analytic functions on $E(A)$ are all constant.


Introduction 1.Background
Let K be an algebraically closed non-trivially valued complete non-Archimedean field.The driving force behind Tate's foundation [Tat71] of rigid analysis was the uniformization of elliptic curves over a p-adic field: given an elliptic curve E over K, there is an isomorphism EpKq -K ˆ{q Z of rigid analytic spaces for some |q| ă 1 if and only if the j-invariant of E is not integral.Within the framework of Berkovich spaces, analytic spaces are locally path connected and locally contractible (in contrast with the total disconnectedness of K), so that the usual theory of universal covers and fundamental groups can be applied.Tate's result then can be restated as follows.The topological space underlying the Berkovich space E an attached to E is contractible if and only if E has good reduction; if this is not the case, the universal cover of E an is G an m , its fundamental group is identified with q Z Ă K ˆfor some |q| ă 1, and E an " G an m {q Z .Mumford generalized Tate's theorem both for higher-genus curves, see [Mum72b], and higher-dimensional abelian varieties, see [Mum72a].Given an abelian variety A over K, the universal cover of A an is of the form E an for a semi-abelian variety (1.1) where T is a K-torus and B an abelian variety with good reduction.The topological space E an is again seen to be contractible, and the fundamental group of A an is seen to be a free abelian group Λ Ă EpKq of rank dim T .Later on Lütkebohmert [Lüt16, Corollary 7.6.5]obtained such a uniformization result for all abeloid varieties, that is, proper smooth connected analytic groups over K -the rigid analytic analogue of a complex torus.

Motivation
In this paper the universal cover of the universal vector extension of an abelian variety is made explicit.Recall that a vector extension of an abelian variety A over a field k is a short exact sequence of algebraic groups where F is a finite-dimensional k-vector space and V pFq the vector group attached to it.Such an extension corresponds (up to isomorphism) to elements of the cohomology group H 1 pA, O A q b K F. Taking F " H 1 pA, O A q _ , the vector extension with isomorphism class id P End H 1 pA, O A q, 0 ÝÑ V `H1 pA, O A q _ ˘ÝÑ EpAq ÝÑ A ÝÑ 0, is said to be the universal one; it has the property that any other vector extension G is obtained as the push-out of EpAq along the linear map H 1 pA, O A q _ Ñ F given by the isomorphism class of G.
When k " C, the uniformization of EpAq admits a quite explicit description, which should hopefully clarify the statements in the rigid context.The exponential map V :" Lie A Ñ ApCq is a universal cover of ApCq.We identify the fundamental group π 1 pApCq, 0q with the kernel of the exponential map Λ Ď Lie A. Hodge theory permits one to see H 1 pA, O A q _ as the complex vector space V conjugate to V ; consequently, we have an inclusion θ Λ : Λ Ñ V .Then, EpAqpCq " pV ˆV q{Λ with Λ embedded diagonally.The choice of a basis of Λ induces a biholomorphism EpAqpCq -pC ˆq2g .In particular, the complex manifold EpAqpCq is Stein, meaning that it can be holomorphically embedded in C n as a closed subspace for some n ě 0. Such an embedding cannot be algebraic, as all algebraic functions on EpAq are constant.In other words, EpAq is an example of a non-affine algebraic variety whose associated complex space is Stein.
The motivation of the present paper comes from the study of the analogous question over K; see [Mac22].

Results
From now on let us work over K. To ease notation, the superscript 'an' is dropped, and all algebraic varieties are treated as analytic spaces.Let 0 ÝÑ T ÝÑ E ÝÑ B ÝÑ 0 be the semi-abelian variety which is the universal cover of A and Λ Ď EpKq the fundamental group.In order to state the main result, it is necessary to also have at hand the uniformization of the dual abelian variety Ǎ. See B as the dual of its dual B; then the group morphism Λ Ñ BpKq defines a semi-abelian variety where Ť is the K-torus with group of characters Λ.Let Λ be the group of characters of the K-torus T .The datum of Λ Ď EpKq induces an embedding of Λ Ď ĚpKq, and Ǎan " Ěan { Λ.
For an algebraic group G, let ω G " pLie Gq _ denote the dual of its Lie algebra.Since the quotient map Ěan Ñ Ǎan is étale, the spaces of invariant 1-forms ω Ě and ω Ǎ are identified.The above short exact sequence of algebraic groups gives the following one: where d p is the pull-back of 1-forms along p.The first result concerns structure.
Theorem A. The universal cover ẼpAq of EpAq is contractible, is the pull-back to Ã of EpAq and is the push-out of EpBq ˆB E along d p; i.e. there is a commutative and exact diagram In the statement the isomorphisms ω Ǎ -H 1 pA, O A q _ and ω B -H 1 pB, O B q _ are understood.The fundamental group π 1 pEpAq, 0q of EpAq with base-point 0 can be identified with a subgroup of (the Krational points of) its universal cover ẼpAq.The projection ẼpAq Ñ E given by Theorem A then induces a map ϕ : π 1 pEpAq, 0q ÝÑ Λ.
In order to understand how π 1 pEpAq, 0q sits inside ẼpAq, note that Theorem A gives an isomorphism CokerpEpBq ˆB E Ñ ẼpAqq -V pω Ť q.Let pr u : ẼpAq ÝÑ V pω Ť q be the induced projection.The image of π 1 pEpAq, 0q is described by means of the universal vector hull of the group Λ, that is, the K-linear map θ Λ : Λ Ñ ω Ť defined as follows.By definition, Ť is the K-torus with group of characters Λ. Seeing χ P Λ as a character χ : Ť Ñ G m , set θ Λ pχq :" χ ˚dz z P ω Ť , where z is the coordinate function on G m .Theorem B. The map ϕ : π 1 pEpAq, 0q Ñ Λ is an isomorphism, and Concretely, when the abelian variety A has good reduction, the above results simply say that EpAq is contractible, so that it coincides with its universal cover.In the extremely opposite situation, when A " T {Λ has totally degenerate reduction, the result is more interesting and gives the following description of EpAq: EpAq " pT ˆV pω Ť qq {tpχ, θ Λ pχqq : χ P Λu.
In the framework of 1-motives, Theorems A and B say that ẼpAq is the universal vector extension of the 1-motive M " rΛ Ñ Es; see [BVS01, Section 1.4] and [Ber09, Section 2.2].

Content of the paper
In order to show Theorems A and B, the natural approach would be to compare ẼpAq with the universal vector extension M 6 of the 1-motive M " rΛ Ñ Es.However, ẼpAq is an analytic vector extension of E, and since E is not proper, it is not clear a priori why it should be an algebraic one.Therefore, we cannot apply the universal property of M 6 directly.Instead, we will proceed by making an explicit construction of EpAq giving the main results above as a byproduct.Unfortunately, the universal property of the universal vector extension does not say much about how EpAq is constructed.It is instead more insightful to look at the moduli space A 6 of translation-invariant line bundles on A endowed with a (necessarily integrable) connection -such a moduli space is canonically isomorphic to the universal vector extension.The construction of A 6 presented here (see Section 2), although quite natural, seemingly does not appear in the literature; it is inspired from [BHR11], even though the Hodge-theoretical reasoning therein had to be circumvented.Such a definition for A 6 has several advantages.First, it translates right away to the rigid analytic framework for abeloid varieties.Second, its explicit nature permits one to perform the necessary computations (see notably Proposition 4.2, Theorem 4.6 and Theorem 4.11).Third, it allows one to determine the canonical linear isomorphism through which A 6 is obtained by push-out from EpAq (see Theorem 2.20). (1)The 'universal cover' of A 6 is then defined by hand (see Section 4.3) and only ultimately shown to be contractible (see Proposition 4.15), so that it is literally the universal cover of A 6 .

Conventions
Let X be a locally ringed space and a sequence of O X -modules indexed by integers.For morphisms of locally ringed spaces f : Y Ñ X and g : X Ñ Z, and an O X -module M, let f ˚pFq, g ˚pFq, pFq b M denote the sequences obtained from pFq by, respectively, pulling back along f , pushing forward along g and taking the tensor product with M.
Let K be a complete non-trivially valued non-Archimedean field.In this paper K-analytic spaces are considered in the sense of Berkovich (see [Ber93]).By abuse of notation, given a K-analytic space X, an O X -module here is what is called an O X G -module in op.cit.As soon as the K-analytic space X is good (that is, every point admits an affinoid neighbourhood), the two notions coincide (op.cit., Proposition 1.3.4).For a K-analytic space S, an S-analytic space in groups will be called simply an S-analytic group.An abeloid variety over S is a proper, smooth S-analytic group with connected fibers.

Moduli of rank 1 connections and universal extension of an abelian scheme
Let S be a scheme, α : A Ñ S an abelian scheme, e : S Ñ A the zero section and ω A " e ˚Ω1 A{S .Let µ, pr 1 , pr 2 : A ˆS A Ñ A be, respectively, the group law, the first and the second projections.

Connections on homogeneous line bundles
A homogeneous line bundle on A is the datum of a line bundle on A together with an isomorphism of O A -modules ϕ : pr 1 L b pr 2 L Ñ µ ˚L.The isomorphism ϕ is called rigidification.
Remark 2.1.The isomorphism pe, eq ˚ϕ : e ˚L b e ˚L Ñ e ˚L induces a trivialization u : O S Ñ e ˚L of the line bundle e ˚L on S.
Let ∆ 1 be the first-order thickening of A ˆS A along the diagonal, p i : ∆ 1 Ñ A the i th projection for i " 1, 2, A 1 the first-order thickening of A along e, ι : A 1 Ñ A the closed immersion, π : A 1 Ñ S the structural morphism and τ : A 1 Ñ ∆ 1 the morphism defined by p 1 ˝τ " x ˝π and p 2 ˝τ " ι.Following Grothendieck, a connection on L is an isomorphism of O ∆ 1 -modules ∇ : p 1 L Ñ p 2 L whose restriction to A is the identity.Similarly, an infinitesimal rigidification (2) at e is an isomorphism of O A 1 -modules ρ : π ˚e˚Ñ ι ˚L whose restriction to the zero section is the identity.(Refer to Sections A.4 and A.5 for basics on connections and infinitesimal rigidifications.)In particular, for a connection ∇ on L, τ ˚∇ is an infinitesimal rigidification of L at e. Proposition 2.2.For a homogeneous line bundle L on A, we have a bijection Proof.The argument is borrowed from (2) and (3) in the proof of [MM74, Proposition 3.2.3,pp.39-40].
Injectivity.Let ∇, ∇ 1 : p 1 L Ñ p 2 L be connections on L. Since both isomorphisms ∇ and ∇ 1 are the identity when restricted to the diagonal, they differ by an homomorphism of O X -modules L Ñ Ω 1 α b L. The latter is equivalent to the datum of a global section ω of Ω 1 α as L is a line bundle.On the other hand, the evaluation at e homomorphism ε : α ˚Ω1 α Ñ ω A is also an isomorphism.Therefore, the isomorphisms τ ˚∇ and τ ˚∇1 differ by εpωq, which is 0 if and only if ω is.Surjectivity.Let ρ : π ˚e˚L Ñ ι ˚L be an infinitesimal rigidification of L. The morphism p 2 ´p1 : ∆ 1 Ñ A factors through the closed immersion ι : A 1 Ñ A. Indeed, when restricted to the diagonal, the map p 2 ´p1 has constant value e.As A 1 (resp.∆ 1 ) is the first-order thickening of A along e (resp. of X ˆS X along the diagonal ∆), the morphism p 2 ´p1 induces a morphism η : ∆ 1 Ñ A 1 between first-order thickenings (2) Mazur-Messing call an 'infinitesimal rigidification' simply a 'rigidification'.Here, the adjective 'infinitesimal' is added in order to distinguish the concept from that of a rigidification of a homogeneous line bundle.such that ι ˝η " p 2 ´p1 .The pull-back of the rigidification ϕ along the morphisms of S-schemes pe ˝π ˝η, p 1 q, pp 2 ´p1 , p 1 q : ∆ 1 Ñ A ˆA furnishes the following isomorphisms of line bundles over ∆ 1 : On the other hand, taking the tensor product of η ˚ρ with p 1 L gives rise the isomorphism of O ∆ 1 -modules Consider the unique homomorphism of O ∆ 1 -modules ∇ : p 1 L Ñ p 2 L making the following diagram commutative: (2.1) To conclude, one has to show that the infinitesimal rigidification τ ˚∇ is ρ.Notice that the endomorphism η ˝τ of the S-scheme A 1 is the identity.Indeed, since ι is a closed immersion (thus a monomorphism of schemes), it suffices to show the equality ι ˝η ˝τ " ι.But ι ˝η ˝τ " pp 2 ´p1 q ˝τ " ι ´e ˝π " ι as A 1 -valued points of A because e ˝π is the neutral element of the group ApA 1 q.Now, pulling back the diagram (2.1) along τ gives the following commutative diagram of O A 1 -modules: where u : O S Ñ e ˚L is the trivialization introduced in Remark 2.1.(Here we used the equalities p 1 ˝τ " x ˝π and p 2 ˝τ " ι holding by the definition of τ.)The equality τ ˚∇ " ρ follows.This concludes the proof.□ For a homogeneous line bundle L on A, let (At A{S pLq) 0 ÝÑ Ω 1 A{S ÝÑ At A{S pLq ÝÑ O A ÝÑ 0 be its Atiyah extension (see Section A.4). Recall that connections on L correspond to splittings of the Atiyah extension of L. Proposition 2.3.Let pL, ϕq be a homogeneous line bundle on A.
(1) If the cohomology group H 1 pS, ω A q vanishes, then the line bundle L admits a connection.
(2) The following sequence of O S -modules is short exact: (3) The homomorphism α ˚α˚p At A{S pLqq Ñ pAt A{S pLqq of short exact sequences of O A -modules obtained by adjunction is an isomorphism. Proof.
(1) According to Proposition 2.2, it suffices to show that the line bundle L admits an infinitesimal rigidification at e.With the notation of Proposition A.6, an infinitesimal rigidification corresponds to a splitting of the short exact sequence of O S -modules The latter is an extension of the line bundle e ˚L on S by the vector bundle ω A b e ˚L; therefore, its isomorphism class lies in H 1 pS, H ompe ˚L, ω A b e ˚Lqq " H 1 pS, ω A q.By hypothesis, the cohomology group H 1 pS, ω A q vanishes; hence the short exact sequence π ˚pι ˚Lq splits, and the line bundle L admits a connection.
(2) The push-forward along α is a left-exact functor; therefore, it suffices to show that the natural map p : α ˚At A{S pLq Ñ α ˚OA is surjective.The statement is local on S; thus the scheme S may be supposed to be affine.Under this assumption, the cohomology group H 1 pS, ω A q vanishes, and because of (1), the line bundle L admits a connection.In other words, by Proposition A.4, the Atiyah extension pAt A{S pLqq of L admits a splitting s : O A Ñ At A{S pLq.The homomorphism of O S -modules α ˚s : α ˚OA Ñ α ˚At A{S pLq is a section of p; that is, it satisfies p ˝α˚s " id.In particular, the homomorphism p is surjective.
(3) The homomorphism of short exact sequences of O A -modules in question is the commutative diagram where the three vertical arrows are given by adjunction.The leftmost and the rightmost vertical arrows are isomorphisms; thus the central vertical arrow must be so by the five lemma.□ Remark 2.4.The Atiyah extension pAt A{S pLqq is obtained as the tensor product with L _ of the short exact sequence where J 1 A{S pLq is the O X -module of first-order jets of L. Proposition A.7 furnishes an isomorphism ϕ : e ˚pJ 1 A{S pLqq Ñ π ˚pι ˚Lq of short exact sequences of O S -modules.Taking the tensor product with e ˚L_ induces an isomorphism e ˚pAt A{S pLqq ÝÑ π ˚pι ˚Lq b e ˚L_ of short exact sequences of O S -modules.Also note that, by Remark 2.1, the line bundle e ˚L on S is trivial, whence we have an isomorphism e ˚pAt A{S pLqq -π ˚pι ˚Lq.

The canonical extension of the trivial line bundle
The functor associating with an S-scheme S 1 the set of isomorphism classes of homogeneous line bundles on A ˆS S 1 is representable by an abelian scheme α : Ǎ Ñ S, called the dual abelian scheme (this is [FC90, Theorem 1.9]; to see that the definition in loc.cit. is equivalent to the one here, see [Oor66,Proposition 18.4]).Let L be the Poincaré bundle (that is, the universal homogeneous line bundle on A ˆS Ǎ), q : A ˆS Ǎ Ñ Ǎ the projection onto the second factor and pAt q pLqq the Atiyah extension relative to q of the Poincaré bundle L. Set U Ǎ :" q ˚At q pLq.The sequence of O Ǎ-modules obtained by pushing forward the short exact sequence pAt q pLqq along q, is short exact by Proposition 2.3(2) applied to the abelian scheme A ˆS Ǎ over Ǎ.
Definition 2.5.The extension p U Ǎq is called the canonical extension of O Ǎ.
Remark 2.6.The pull-back of the Atiyah extension of L along the unramified morphism pid A , ěq is the Atiyah extension of the trivial line bundle O A .Therefore, the canonical derivation d A{S : O A Ñ Ω 1 A{S , being a connection on the trivial bundle, defines a splitting of ě˚p U Ǎq, called the canonical splitting.Remark 2.7.Let A 1 be the first-order thickening of A along the zero section e, ι : A 1 ˆS Ǎ Ñ A ˆS Ǎ the morphism obtained from the closed immersion A 1 Ñ A by base change along α and π : A 1 ˆS Ǎ Ñ Ǎ the second projection.Remark 2.4 (applied to the abelian scheme A ˆS Ǎ over Ǎ) furnishes an isomorphism of sequences of O Ǎ-modules pe, id Ǎq ˚pAt q pLqq " ÝÑ π ˚pι ˚Lq.
On the other hand, by Proposition 2.3(3), the evaluation at the zero section p U Ǎq " q ˚pAt q pLqq ÝÑ pe, id Ǎq ˚pAt q pLqq is an isomorphism of sequences of O Ǎ-modules.Composing these isomorphisms yields an isomorphism of sequences of O Ǎ-modules p U Ǎq -π ˚pι ˚Lq.

Moduli space of connections
Definition 2.8.The S-scheme Ǎ6 " Pp U Ǎq ∖ Pp α˚ω A q is called the moduli space of rank 1 connections on A.
Theorem 2.9.The S-scheme Ǎ6 represents the functor associating with an S-scheme S 1 the set of isomorphism classes of triples pL, ϕ, ∇q made up of a homogeneous line bundle pL, ϕq on the abelian scheme A S 1 and a connection Proof.For an S-scheme S 1 , an S 1 -valued point of Ǎ6 consists of the datum of a morphism of S-schemes f : S 1 Ñ Ǎ and a splitting s : Let pL, ϕq be the homogeneous line bundle on the abelian scheme A 1 :" A S 1 obtained as the pull-back of the Poincaré bundle L along the morphism id A ˆf : A 1 :" AˆS S 1 Ñ AˆS Ǎ.Let α 1 : A 1 Ñ S 1 be the morphism obtained from α by base change.Then, by Proposition 2.3(3), the short exact sequence of O A 1 -modules α 1˚f ˚p U Ǎq is the Atiyah extension pAt α 1 pLqq of the homogeneous line bundle L. By Proposition A.4, the splitting α 1˚s : O A 1 Ñ At α 1 pLq corresponds to a connection on L (notice the equality End L " O S , due to L being a line bundle).□ The tensor product of line bundles together with a connection endows Ǎ6 with the structure of a group S-scheme.The natural projection π : Ǎ6 Ñ Ǎ, pL, ϕ, ∇q Þ Ñ pL, ϕq is a faithfully flat morphism of group S-schemes.The kernel of π is by definition made of connections on the trivial bundle O A .Now, a connection on the trivial line bundle is nothing but d A{S `ω for a global differential form on A, where d A{S is the canonical α ´1O S -linear derivation.The isomorphism α ˚Ω1 A{S -e ˚Ω1 A{S ": ω A thus yields a short exact sequence of group S-schemes The following remark will only be needed once, quite further in text (see the proof of Theorem 4.6).The reader may harmlessly skip it at first.Remark 2.10.As any affine bundle on an abelian scheme, the group structure on Ǎ6 is defined by a unique isomorphism of O Ǎ-modules where μ, pr 1 , pr 2 : Ǎ ˆS Ǎ Ñ Ǎ are, respectively, the group law, the first and the second projection, and `B is the Baer sum of extensions (see Examples 2.15 and 2.17 below).According to Example 2.18, the isomorphism ψ is the push-forward along the morphism pA 1 ˆS Ǎq ˆA1 pA 1 ˆS Ǎq Ñ Ǎ ˆS Ǎ of the rigidification φ : pid, pr 1 q ˚L b pid, pr 2 q ˚L ÝÑ pid, μq ˚L of the Poincaré bundle L on the abelian scheme A ˆS Ǎ over A.

The Lie algebra of the dual abelian scheme (redux)
Let Ǎ1 be the first-order thickening of Ǎ along its zero section ě, π : A ˆǍ 1 Ñ A the projection onto the first factor, ι : A ˆS Ǎ1 Ñ A ˆS Ǎ the morphism obtained from the closed embedding Ǎ1 Ñ Ǎ by base change with respect to α : A Ñ S and α 1 : AˆS Ǎ1 Ñ S the structural morphism.Consider the isomorphism class , where L is the Poincaré bundle on A ˆS Ǎ.Let c 1 pι ˚Lq be the global section of the sheaf of abelian groups R 1 α 1˚O ÂˆǍ 1 on S defined as the image of rι ˚Ls via the homomorphism of abelian groups by the Grothendieck-Leray spectral sequence The first projection π : A ˆS Ǎ1 Ñ A induces a homeomorphism on the underlying topological spaces, whence an isomorphism R 1 α ˚pπ ˚OÂ is the truncated exponential.We therefore have an exact sequence of sheaves of abelian groups on S The pull-back of the line bundle ι ˚L along the morphism pid, ěq : A Ñ A ˆS Ǎ1 is trivial; thus c 1 pι ˚Lq is (the image via the truncated exponential of) a section of O S -module R 1 α ˚α˚ω Ǎ still written c 1 pι ˚Lq.The projection formula yields an isomorphisms R 1 α ˚α˚ω Ǎ -R 1 α ˚OA b ω Ǎ " H ompLie Ǎ, R 1 α ˚OA q through which the class c 1 pι ˚Lq corresponds to an isomorphism (see [BLR90, Section 8.4, Theorem 1])

Universal property of the canonical extension
We borrow notation from Section 2.4.For a vector bundle F on S, the Grothendieck-Leray spectral sequence H p pS, R q α ˚α˚F q ñ H p`q pA, α ˚Fq yields an exact sequence of Γ pS, O S q-modules For an extension of O A by α ˚F, let rFs P H 1 pA, α ˚Fq be its isomorphism class.The global section cpFq :" p F prFsq of the O S -module R 1 α ˚α˚F can also be seen as the connecting homomorphism in the long exact sequence of O S -modules The abelian scheme A coincides with the dual abelian scheme of Ǎ.By means of this identification, consider the canonical extension p U A q on A and its isomorphism The truncated exponential map exp : On the other hand, the morphism π is a homeomorphism on the underlying topological spaces, thus an isomorphism of abelian groups The image of rι ˚Ls via the preceding isomorphism is the isomorphism class rπ ˚ι˚L s of the invertible π ˚OAˆǍ 1 -module π ˚ι˚L .Proposition 2.11.With the notation above, The second statement can be reformulated by saying that, via R 1 α ˚α˚ω Ǎ -H ompLie Ǎ, R 1 α ˚OA q, the following equality holds: Proof.
(1) Remark 2.7 furnishes an isomorphism of short exact sequence of O A -modules p U A q -π ˚pι ˚Lq.
For an affine open cover tA i u iPI of A, the isomorphism class r U A s is represented by a 1-cocycle, say For a homomorphism ϕ : ω Ǎ Ñ F with F a vector bundle on S, let pF ϕ q be the short exact sequence obtained by push-out of p U A q along the homomorphism α ˚ϕ.Let γ F pϕq denote its isomorphism class rF ϕ s P H 1 pA, α ˚Fq; then this construction defines a map γ F : Hompω Ǎ, Fq ÝÑ H 1 pA, α ˚Fq.
Theorem 2.12.The map γ F is injective and Γ pS, O S q-linear, and its image is the set of isomorphism classes of extensions This statement is an immediate consequence of the following more precise fact.To state it, consider the homomorphism e ˚: H 1 pA, α ˚Fq Ñ H 1 pS, Fq given by the pull-back of an extension along the morphism e.Since e is a section of α, the composite map e ˚˝α ˚is the identity of H 1 pS, Fq.Also, note that an extension pFq as above splits if and only if e ˚rFs " 0. Lemma 2.13.For a vector bundle F on S, (1) p F ˝γF is the isomorphism induced by Φ A b id F : H ompω Ǎ, Fq Ñ R 1 α ˚α˚F on global sections, (2) the following sequence of Γ pS, O S q-modules is short exact: is commutative.It follows from Proposition 2.11(2) that the composite map p F ˝γF is the one induced on global sections by (2) First of all, notice that the map γ F is injective and Γ pS, O S q-linear.Second, the composite map e ˚˝γ F vanishes: for a homomorphism ϕ : ω Ǎ Ñ F, the splitting of the short exact sequence of O S -module e ˚p U A q induced by the derivation d Ǎ{S (see Remark 2.6) induces a splitting of e ˚pF ϕ q.Therefore, it remains to show that the image of γ F is the whole Kerpe ˚q.Set r F :" pp F ˝γF q ´1 ˝pF .Then r F ˝γF is the identity of Hompω Ǎ, Fq, and the sequence of Γ pS, O S q-modules ÝÑ Hompω Ǎ, Fq ÝÑ 0 is short exact by (2.2).Since γ F and e ˚are sections of, respectively, r F and α ˚, the result follows.□

Preliminaries on extensions of an abelian scheme
Let G be an affine, commutative, faithfully flat and finitely presented group S-scheme.

Principal bundles.
A principal G-bundle on A is a faithfully flat A-scheme P endowed with an action of G such that the morphism pσ , pr P q : GˆS P Ñ P ˆAP is an isomorphism, where σ , pr P : GˆS P Ñ P are, respectively, the morphism defining the action and the projection onto A. Let H 1 fppf pA, Gq denote the set of isomorphism classes of principal G-bundles on A.
Let ρ : G Ñ G 1 be a morphism of group S-schemes, where G 1 has the same properties of G, and let P be a principal G-bundle on A. The quotient ρ ˚P of S-scheme G 1 ˆS P via the action gpg 1 , xq " pρpgqg 1 , gxq of G exists and is a principal G 1 -bundle on A, called the push-out (see [Ols16, Propositions 4.5.6 and 12.1.2]).For principal G-bundles P and P 1 on A, the push-out P ^P 1 :" µ G˚p P ˆA P 1 q of the principal G ˆS G-bundle P ˆA P 1 along the sum map µ G : G ˆS G Ñ G is called the sum.This operation endows the set H 1 fppf pA, Gq with the structure of an abelian group with neutral element G ˆS A and inverse P Þ Ñ r´1s ˚P , where r´1s is the inverse map on G. Furthermore, the push-out of an endomorphism of G as a group S-scheme induces an endomorphism of the abelian group H 1 fppf pA, Gq.This equips H 1 fppf pA, Gq with the structure of a module on the ring EndpGq of endomorphisms of G.For instance, if G " V pFq for some vector bundle F on S, the set H 1 fppf pA, Gq is naturally a Γ pS, O S q-module.Example 2.14.Suppose G " G m .The principal G m -bundle associated with a line bundle L on A is the total space of L deprived of its zero section V pLq ˆ.The so-defined map PicpAq Ñ H 1 fppf pA, G m q, L Þ Ñ V pLq îs an isomorphism of abelian groups (see [SGA1, Exposé XI, Proposition 5.1]), that is, a sum of principal G m -bundles corresponds to a tensor product of line bundles.
Example 2.15.Suppose G " V pFq for some vector bundle F on S, and let In particular, the sum of such principal V pFq-bundles is the V pFq-bundle associated with the Baer sum of the corresponding extensions.

Law groups on principal bundles.
An extension of A by G is the datum of a short exact sequence of commutative group S-schemes where the morphism p E is faithfully flat.Note that an extension E of A by G is naturally a principal G-bundle over A. Since the natural homomorphism O S Ñ f ˚OA is an isomorphism, the commutativity of E is automatic.Morphisms of extensions are defined in the evident way.An isomorphism of extensions is a morphism inducing the identity on A and G. Let ExtpA, Gq be set of isomorphism classes of extensions of A by G.For a morphism of S-group schemes ρ : G Ñ G 1 , with G 1 having the same properties as G, and an extension pEq, the cokernel ϕ ˚pEq of the morphism pi E , ρq : G Ñ E ˆS G 1 is called the push-out of pEq.
The Baer sum of extensions pEq and pE 1 q of A by G is the push-out of E ˆA E along the sum morphism G ˆS G Ñ G.The Baer sum endows ExtpA, Gq with the structure of an abelian group.Similarly to the case of principal G-bundles, the abelian group ExtpA, Gq is endowed with the structure of an EndpGq-module.
Seeing an extension as a principal bundle gives rise to a homomorphism of EndpGq-modules The map λ G is injective, and its image is the set of isomorphism classes of principal G-bundles admitting a rigidification (see [Ser59, Théorème 15.5]).
Example 2.16.Suppose G " G m and identify principal G m -bundles with line bundles as in Example 2.14.A rigidification of a line bundle Example 2.17.Suppose G " V pFq for some vector bundle F on S. Identify principal V pFq-bundles with extensions of O A by f ˚F as in Example 2.15.For a short exact sequence of O A -modules where pr 1 pFq `B pr 2 pFq is the Baer sum of the extensions pr 1 pFq and pr 2 pFq.
Example 2.18.For a vector bundle F on S, let S 1 be the first-order thickening of V pF _ q along its zero section.
Let L be a line bundle on A 1 :" A ˆS S 1 endowed with a rigidification ϕ : pr 1 L b pr 2 L Ñ µ Å1 L, with the obvious notation.Let s : A Ñ A 1 be the closed immersion induced by the zero section of F _ and π : A 1 Ñ A the projection onto A. The O A -module F:" π ˚L sits into the following short exact sequence of O A -modules: According to Proposition A.2, the push-forward of the O A 1 ˆA1 -module pr 1 L b pr 2 L along the morphism π ˆπ : A 1 ˆS1 A 1 Ñ A ˆS A is the Baer sum of the extensions pr 1 Fb pr 2 s ˚L and pr 1 s ˚L b pr 2 F. Assume further that the line bundle s ˚L is isomorphic to O A .Then, the vector bundle F is an extension of O A by f ˚F, and the isomorphism pπ ˆπq ˚ϕ : pr 1 F`B pr 2 FÑ µ Å F is a rigidification like the one considered in Example 2.17.

The universal vector extension
A vector extension of the abelian scheme A is an extension of A by V pFq for some vector bundle F on S called its vector part.Recall the injective homomorphism of Γ pS, O S q-modules λ F :" λ V pFq : ExtpA, V pFqq ÝÑ H 1 fppf pA, V pFqq " H 1 pA, α ˚Fq defined in Section 2.6.2 and the map p F : H 1 pA, α ˚Fq Ñ H 0 pS, R 1 α ˚α˚F q introduced in (2.2).According to [MM74, Proposition 1.10], the composite homomorphism λ F{S :" p F ˝λF : ExtpA, Fq ÝÑ H 0 pS, R 1 α ˚α˚F q (3) The definition of a rigidification in [SGA7-I, Exposé VII, Section 1] involves the commutativity of two diagrams which is automatic over abelian schemes. is bijective.For the vector bundle F " pR 1 α ˚OA q _ on S, this gives an isomorphism Definition 2.19.The universal vector extension is the one corresponding to the identity via the above isomorphism: (EpAq) 0 ÝÑ V ppR 1 α ˚OA q _ q ÝÑ EpAq ÝÑ A ÝÑ 0.
The extension EpAq deserves the 'universal' title because of the following property.First notice that, for vector bundles F and F 1 on S and a homomorphism of O S -modules ϕ : is commutative, where the leftmost vertical arrow is given by push-out along ϕ.Moreover, for a vector extension G, the global section λ F{S pGq of R 1 α ˚α˚F defines a homomorphism of O S -modules λ F{S pGq : pR 1 α ˚OA q _ Ñ F. The extension G is isomorphic (in a unique way!) to the push-out of pEpAqq along λ F{S pGq.The moduli space A 6 of rank 1 connections on Ǎ is a vector extension of the abelian scheme A with vector part ω Ǎ. Recall the isomorphism of O S -modules Φ A : Lie Ǎ Ñ R 1 α ˚OA introduced in Section 2.4.Proposition 2.11(2) implies the following.
Theorem 2.20.The vector extension A 6 is the push-out of EpAq along As a consequence, one obtains a down-to-earth description of how vector extensions are constructed.For a homomorphism of O S -modules ϕ : ω Ǎ Ñ F, let E ϕ be the push-out of A 6 along ϕ.
Corollary 2.21.The map Hompω Ǎ, Fq Ñ ExtpA, Fq, ϕ Þ Ñ rE ϕ s is the inverse of pΦ _ A b id F q ˝λF{S .Observe that the affine bundle underlying E ϕ is PpF ϕ q ∖ Ppα ˚Fq, where pF ϕ q is the push-out of p U A q along ϕ.

Preliminaries on Tate-Raynaud uniformization
Let K be a complete non-trivially valued non-Archimedean field and R its ring of integers.

Raynaud's generic fiber of a formal abelian scheme
Let S be an admissible formal R-scheme, B a formal abelian scheme over the formal R-scheme S, B its dual and L B the Poincaré bundle on BˆS B. As customary, Raynaud's generic fibers of formal schemes are referred to by straight letters (as opposed to curly ones for formal schemes).More explicitly, let S, B and B be Raynaud

Datum of a toric extension
Let Λ be a free abelian group of rank equal to that of Λ, ΛS the constant S-analytic group with value Λ and T the split S-torus with group of characters Λ.From now on suppose that S is connected, so that sections of the morphism ΛS Ñ S are naturally in one-to-one correspondence with elements Λ and will be henceforth identified with those.Let č : ΛS Ñ B be a morphism of S-analytic groups, and consider the extension ε : E Ñ S of the proper K-analytic group B by the torus T determined by č: The extension E is described as follows.For an S-analytic space S 1 , an S 1 -valued point g of the S-analytic space E is the datum of ‚ an S 1 -valued point b " ppgq of B and, ‚ for χ P Λ, a trivialization xg, χy E of the line bundle L B,pb, čp χqq on S 1 .Moreover, the trivializations are required to satisfy the following compatibility: for χ, χ1 P Λ, where the equality is meant to be understood via the isomorphism

Dual datum
Let Λ be a free abelian group of finite rank, Λ S the constant S-analytic group with value Λ, Ť the split S-torus with group of characters Λ and i : Λ S Ñ E a morphism of S-analytic groups which is a closed immersion.See the abeloid variety B as the dual of B; then the group morphism c " p ˝i : M S Ñ B determines an extension Ě of the proper S-analytic group B by the torus Ť : Just to fix notation, for an S-analytic space S 1 , an S 1 -valued point ǧ of Ě corresponds to the datum of ‚ an S 1 -valued point b " pp ǧq of B and, ‚ for χ P Λ, a trivialization xχ, ǧy Ě of the line bundle L B,pcpχq, bq on S 1 .As before, the trivializations are subsumed to the relation, for χ, χ 1 P Λ, xχ `χ1 , ǧy Ě " xχ, ǧy Ě b xχ 1 , ǧy Ě .Now, for χ P Λ, by the symmetry of the Poincaré bundle, there is a unique S-valued point ǐp χq of Ě such that, for χ P Λ, xχ, ǐp χqy Ě " xipχq, χy E .This defines an injective morphism of S-analytic groups ǐ : ΛS Ñ Ě.
In an attempt of unburdening the (already overwhelming) notation, in what follows Λ S is identified with the image of i, and the subscript E is dropped from the pairing x´, ´yE , and similarly for ΛS , ǐ and x´, ´y Ě .

Quotient
Since the subgroup Λ S is closed (hence fiberwise discrete) in E, the (topological) quotient A :" E{Λ S exists.Moreover, assume that the structural morphism α : A Ñ S is proper.Under these working hypotheses, the bilinear pairing x´, ´y on Λ ˆΛ is non-degenerate, the subgroup ΛS is closed in Ě, the quotient Ǎ :" Ě{ ΛS exists, and the structural morphism α : Ǎ Ñ S is proper (see [BL91, Proposition 3.4]).Let u : E Ñ A and ǔ : Ě Ñ Ǎ be the quotient maps.The situation is summarized in the following diagrams:

Coherent sheaves on the quotient
Descent of modules along the morphism u can be restated in terms of Λ S -linearizations.More precisely, the datum of a coherent O A -module is equivalent to that of a coherent O E -module V endowed with a Λ S -linearization λ : pr E V ÝÑ σ ˚V , where σ , pr E : Λ S ˆS E Ñ E are, respectively, the restriction to Λ S ˆS E of the group law of E and the projection onto the second factor.Quite concretely, the group Λ S being constant, the datum of a Λ Slinearization of a coherent O E -module V boils down to that, for χ P Λ, of an isomorphism of O E -modules where tr χ is the translation by χ on E. Additionally, the collection of isomorphisms λ χ is required to fulfil the following compatibility, for χ, χ 1 P Λ:

Homogeneous line bundles
Owing to [BL91, Theorem 6.7], given a homogeneous line bundle L on A, its pull-back u ˚L on E is isomorphic to p ˚M for some line bundle M on B.Moreover, the natural rigidification where µ E is the group law on E, is equivariant with respect to the Λ S -linearization λ on p ˚M induced by the isomorphism u ˚L -p ˚M.It follows that the Λ S -linearization λ can be expressed as the datum of isomorphisms, for χ P Λ, λ χ : p ˚M ÝÑ tr χ p ˚M, s Þ ÝÑ s b rpχq, where the isomorphism tr χ p ˚M -p ˚M b ε ˚cpχq ˚M coming from the homogeneity of M is taken into account (recall that ε is the structural morphism of E) and r is a trivialization of the line bundle c ˚M on Λ S .What is more, the trivialization r must satisfy, for χ, χ 1 P Λ, the relation rpχq b rpχ 1 q " rpχ `χ1 q, where, as is customary at this stage, the above equality is meant to be understood via the isomorphism cpχq ˚M b cpχ 1 q ˚M -cpχ `χ1 q ˚M given by the rigidification of M.

Duality
By [BL91, Theorem 6.8] the S-analytic group Ǎ represents the functor associating with an S-analytic space S 1 the set of isomorphism classes of homogeneous line bundles on A ˆS S 1 .Furthermore, let L A be the Poincaré bundle on A ˆS Ǎ.According to loc.cit., there is a (necessarily unique) isomorphism of line bundles ξ : pu, ǔq ˚LA " ÝÑ L E :" pp, pq ˚LB on E ˆS Ě compatible with the implied rigidifications of the homomogeneous line bundles L A and L B .The isomorphism ξ endows the line bundle L E with a pΛ ˆΛq S -linearization λ, which can be described as follows.For characters χ P Λ and χ P Λ, an S-analytic space S 1 and S 1 -valued points x of E and x of Ě, the isomorphism of O S 1 -modules λ pχ,χ 1 q,px,x 1 q : L E,px, xq " ÝÑ L E,px`χ, x`χq induced by the linearization λ is In order to make sense of (3.2), observe that xx, χy E is by definition a section of the line bundle L E,px, χq , while xχ, χy E is a section of the line bundle L E,pχ, χq and xx, χy E b xχ, χy E is seen as a section of the line bundle L E,px`χ, χq via the isomorphism of O S 1 -modules induced by the rigidification of the homogeneous line bundle L B .Arguing similarly, xχ, xy Ě b v is a section of line bundle L E,px`χ,x 1 q , so that the right-hand side of (3.2) is the section of the line bundle L E,px`χ, x`χq via the isomorphism of O S 1 -modules

The universal vector extension of an abeloid variety
Let K be a non-trivially valued complete non-Archimedean field and S a K-analytic space.

Definition.
Let A be an abeloid variety over S, that is, a proper and smooth S-analytic group with (geometrically) connected fibers.Suppose that the functor associating with an S-analytic space S 1 the set of isomorphism classes of (fiberwise) homogeneous line bundles on A S 1 is represented by an abeloid variety Ǎ over S. For instance, this is the case if S is a K-rational point (see [Lüt16, Corollary 7.6.5])or if Ǎ admits a uniformization as the one described in Section 3 (see [BL91, Theorem 6.8]).Under this assumption, translating the arguments of Section 2 into the rigid analytic framework permits one to define the canonical extension p U A q on the abeloid variety A and the moduli space A 6 of rank 1 connections on A.

Canonical extension on the universal cover.
From now on, and up until Section 4.6, the abeloid variety A is supposed to admit a uniformization as the one described in Section 3. We borrow notation introduced therein and consider the canonical extension p U B q on (Raynaud's generic fiber of) the formal abelian scheme B.
obtained as the push-out of short exact sequence of O E -modules p ˚p U B q along d p is called the canonical extension of O E .

Alternative description.
To perform 'explicit' computations, it is often useful to have at hand a down-to-earth expression for U E , similar to that in Remark 2.7.Let X " A, B, E and, respectively, X " Ǎ, B, Ě.Let X1 " Ǎ1 , B1 , Ě1 be the first-order thickenings of X, and ι X : X ˆS X1 ÝÑ X ˆS X, π X : X ˆS X1 ÝÑ X, respectively, the closed immersion and the projection onto the first factor.With this notation, the considerations in Remark 2.7 furnish isomorphisms where L A and L B are, respectively, the Poincaré bundles on A ˆS Ǎ and B ˆS B. Recall the isomorphism ξ : pu, ǔq ˚LA " ÝÑ L E :" pp, pq ˚LB of line bundles on E ˆS Ě considered in Section 3.7.By definition, the canonical extension U E is the push-forward along the morphism π E of the line bundle ι E L E on E ˆS Ě1 : Pushing forward the isomorphism ι E ξ along the map π E yields an isomorphism of O E -modules

Linearization of the canonical extension
By means of the isomorphism u ˚UA -U E described above, the canonical extension U E acquires a natural Λ S -linearization.The task undertaken here is to give an explicit expression for it.Unfortunately, this point is as crucial as dreadfully technical.

4.2.1.
Describing the Λ S -linearization of the line bundle ι E L E is easily achieved.Indeed, such a linearization is the pull-back along the morphism ι E of the pΛ ˆΛq S -linearization of the line bundle L E .Let ȷ : Ě1 Ñ Ě denote the closed immersion and, for χ P Λ, consider the trivialization xχ, ȷy of the line bundle L E,pχ,ȷ q " ´χ, id Ě1 ¯˚ι E L E on Ě1 .Evaluating (3.2) at χ " 0, x " id E and x " ȷ shows that the Λ S -linearization of the line bundle ι E L E is given, for χ P Λ, by the isomorphism (4.1) where tr χ is the translation by χ on E. To make sense of the formula, notice that the homogeneity of the line bundle ι E L E furnishes an isomorphism where ε 1 : E ˆĚ 1 Ñ Ě1 is the projection onto the second factor.

4.2.2.
The Λ S -linearization of the unipotent bundle U E is somewhat trickier to come by.Rather formally, for χ P Λ, the isomorphism U E Ñ tr χ U E is just the push-forward of the isomorphism (4.1) along the Λ S -equivariant morphism π E .The key observation (see Proposition A.2) is that the isomorphism (4.2) becomes, after pushing forward along π E , an isomorphism of O E -modules (Notice that both the vector bundles U E and ε ˚χ˚U E are extensions of O E by ε ˚ω Ě ; thus their Baer sum is well defined.) Accordingly, the tensor product in (4.1) is now replaced by a 'sum'.To be more precise about what this possibly means, recall how the Baer sum of the extensions p U E q and pε ˚χ˚U E q is constructed.First, consider the O E -submodule V Ď U E ' ε ˚χ˚U E made of pairs pv, wq whose components have same projection in O E .The O E -module V is an extension of O E by ε ˚pω Ě ' ω Ě q, and the Baer sum in question is its push-out along the sum map ω Ě ' ω Ě Ñ ω Ě .For a pair pv, wq, the aforementioned 'sum' v `w is the image of pv, wq in U E `B ε ˚χ˚U E .Summing up, we proved the following.Proposition 4.2.With the notation above, the Λ S -linearization of the unipotent bundle U E is given, for χ P Λ, by the isomorphism where q : U E Ñ O E is the projection in the datum of the extension p U E q.
The meaning of the preceding formula is unveiled by the following.
Remark 4.3.The equality where ε1 : Ě1 Ñ S is the structural morphism.By means of the preceding, the trivialization xχ, ȷ y of the line bundle L E,pχ,ȷ q can be seen as a section of the vector bundle χ ˚UE on S.Moreover, the sum in expression (4.3) does make sense for the following reason: the projection in O S " L E,pχ, ěq of the section xχ, ȷ y of the extension χ ˚UE is xχ, ěy " 1 (where ě is the neutral section of Ě); thus the sections v and qpvq.xχ,ȷ y have equal projection in O S .

Universal cover of the universal vector extension
Definition 4.4.The universal cover of the universal vector extension A 6 is Remark 4.5.When S is a point, the name 'universal cover' is well deserved as the topological space E 6 is contractible (see Proposition 4.15) and comes with a covering map E 6 Ñ A 6 (see Theorem 4.6).
Let Ě1 be the first-order thickening of Ě along the neutral section and ȷ : Ě1 Ñ Ě the closed immersion.As explained in Remark 4.3, for χ P Λ, the trivialization xχ, ȷ y of the line bundle L E,pχ,ȷ q on Ě1 defines a section of the vector bundle χ ˚UE on S.Moreover, the projection of xχ, ȷ y in O S is 1; thus the section xχ, ȷ y defines a splitting of the short exact sequence χ ˚p U E q.Therefore, xχ, ȷ y defines an S-valued point χ 6 of the universal cover E 6 .Let i 6 : Λ S ÝÑ E 6 , χ Þ ÝÑ χ 6 :" xχ, ȷ y be the so-defined morphism of S-analytic spaces.For χ P Λ, the projection of χ 6 in E is by definition χ; thus the morphism i 6 is a closed immersion.The isomorphism of O E -modules u ˚UA -U E introduced in Section 4.1.3induces an isomorphism of E-analytic spaces A 6 ˆA E -E 6 .It follows that E 6 is naturally endowed with a structure of S-analytic space in groups and the natural morphism of S-analytic spaces is a group morphism.
Theorem 4.6.The map i 6 is a group morphism with image the kernel of u 6 ; that is, the following sequence of S-analytic groups is short exact: Proof.The group law on the affine bundle A 6 " Pp U A q∖Ppα ˚ω Ǎq is given by an isomorphism of O A -modules where µ A , pr 1 , pr 2 : A ˆS A Ñ A are, respectively, the group law, the first and the second projection on A.
According to Remark 2.10, or better its rigid analytic analogue, such an isomorphism ψ A is the push-forward along the projection of the isomorphism pid A ˆȷ, id A ˆȷq ˚ϕA , where is the rigidification of the Poincaré bundle L A and pid A ˆȷ, id A ˆȷ q : `A ˆS Ǎ1 ˘ˆǍ 1 `A ˆS Ǎ1 ˘ÝÑ `A ˆS Ǎ˘ˆǍ `A ˆS Ǎt he closed immersion.By design, the group law of E 6 is constructed from that of A 6 .That is, the group law on the affine bundle E 6 " Pp U E q ∖ Ppε ˚ω Ě q is defined by the isomorphism of O E -modules the isomorphism u ˚UA -U E (and with the obvious notation) being allowed for.Thanks to (4.3) and with the notation therein, the Λ S -linearization of the unipotent bundle U E is the datum of the isomorphisms, for χ P Λ, where q : U E Ñ O E is the projection in the datum of the extension p U E q.Given χ, χ 1 P Λ, via the isomorphism ψ E , the wanted additivity χ 6 `χ16 " pχ `χ1 q 6 is seen to be nothing but the compatibility λ χ`χ 1 " tr χ λ χ 1 ˝λχ to which is subsumed the Λ S -linearization of U E .
This substantially also proves the remaining assertions.Indeed, the affine bundle A 6 is the quotient of E 6 by the natural action of Λ S on it induced by the Λ S -linearization of U E .Moreover, from this point of view, the map u 6 is just the quotient morphism, and the above considerations show that the action of Λ S on E 6 is described, for χ P Λ, as the translation by χ 6 .This amounts to saying that the sequence of S-analytic groups is short exact, thus concluding the proof.□

A computation on toric bundles
The next task is acquiring a better understanding of the map i 6 .This is done via a computation on toric bundles whose proof is clearer when stated in a broader generality.So let us momentarily reset our notation, and let X be a separated S-analytic space, Λ a free abelian group of finite rank n, Λ Ñ PicpXq, χ Þ Ñ L χ a group homomorphism and P the X-analytic space whose points s with values on a X-analytic space f : X 1 Ñ X form the set of data, for χ P Λ, of a trivialization xχ, sy of the line bundle f ˚Lχ .Moreover, the trivializations above satisfy, for χ, χ 1 P Λ, the relation (4) (4.4) xχ, sy b xχ, sy " xχ `χ1 , sy.
Let p : P Ñ X be the projection.The split torus T over S with group of characters Λ acts naturally on P by the rule defined, for an S-analytic space S 1 , S 1 -valued points s of P and t of T and a character χ P Λ, by xχ, tsy " χptqxχ, sy.
Let u : S Ñ P be a morphism of S-analytic spaces, P 1 the first-order thickening of P along the section u and j : P 1 Ñ P the closed immersion.To make the notation more flexible, given a finite morphism of K-analytic spaces Z Ñ S and a coherent O Z -module F, let us denote again by F its push-forward onto (4) Several abuses of notation have been perpetrated here.Rather than isomorphism classes of line bundles, one should fix, for χ P Λ, a line bundle L χ and, for χ, χ 1 P Λ, isomorphisms L χ b L χ 1 -L χ`χ 1 through which the formula xχ, sy b xχ, sy " xχ `χ1 , sy ought to be understood.

S.
Consider the S-valued point x " ppuq of X, the first-order thickening X 1 of X along the section x, the closed immersion i : X 1 Ñ X and the following commutative and exact diagram of O S -modules: (4.5) The map i ˚Lχ Ñ j ˚p˚L χ is given by adjunction with respect to the morphism p 1 : P 1 Ñ X 1 induced by p.The isomorphism T ˆS P Ñ P ˆX P given by the action of T yields an isomorphism of O S -modules u ˚Ω1 P {X -e ˚Ω1 T {S ": ω T , where e is the neutral section of T .Consider the homomorphism q : j ˚p˚L χ ÝÑ ω T b x ˚Lχ defined via the isomorphisms u ˚Ω1 P {X b x ˚Lχ -j ˚p˚L χ {i ˚Lχ and u ˚Ω1 P {X -ω T .Proposition 4.7.With the notation above, qpxχ, jyq " χ ˚dz z b u, where z is the coordinate function on G m,S and χ is seen as a character T Ñ G m,S .
Proof.The argument is just a tedious dévissage until reaching the case Λ " Z, χ " 1, X " S, L 1 " O S , u " 1, for which the result is substantially trivial.
First step.To begin with, reduce to the case Λ " Z and χ " 1.Consider the X-analytic space P χ parametrizing trivializations of the line bundle L χ and the morphism pr χ : P Ñ P χ sending a point s of P with values in an X-analytic space X 1 to the trivialization xχ, sy.Then, by design, the trivialization xχ, jy can be seen as the composite morphism pr χ ˝j : P 1 Ñ P χ .The latter factors through the first-order thickening P χ,1 of P χ along the section xχ, uy " pr χ puq, giving rise to the commutative diagram where j χ is the closed immersion and pr χ,1 the factorization of pr χ .Upon letting p χ : P χ Ñ X be the projection, the homomorphism q χ : j χp χL χ Ñ ω G m bx ˚Lχ , defined analogously to q, fits in the commutative diagram of O S -modules where the leftmost vertical arrow is given by adjunction with respect to the map pr χ,1 and the rightmost one by pull-back of differential forms along the character χ.In particular, it suffices to show that the trivialization of the line bundle L χ given by the P χ,1 -valued point j χ of P χ is mapped to dz z b u by q χ .Second step.Suppose Λ " Z and χ " 1.To simplify notation, simply write L instead of L χ .The aim of this second step is to reduce to the case X " S, L " O S and u " 1.In order to do this, consider the fiber P x of P at x, which can also be seen as the principal G m -bundle over S associated with the line bundle x ˚L on S. In this case, the identity map of S plays the role of the section x, and the line bundle j ˚p˚L on P 1 is replaced by the line bundle π ˚x˚L on P x,1 , where P x,1 is the first-order thickening of P x at u and π : P x,1 Ñ S is the structural morphism.Diagram (4.5) for the principal bundle P x reads more simply as the following: where x ˚L Ñ π ˚x˚L is natural map.The restriction map from P 1 to P x,1 furnishes a homomorphism of commutative diagrams of O S -modules from (4.5) to the above, matching the entries in the obvious manner.In particular, the projection q x : π ˚x˚L Ñ x ˚L b ω G m defined similarly to q sits in the following diagram: Strictly speaking, the above argument permits one to reduce to the case X " S and to a line bundle L on S.However, by means of the isomorphism O S -L induced by the given trivialization u, one is finally led back to the case of the trivial line bundle on S and u " 1. Third step.Suppose X " S, L " O S and u " 1.Let z be the coordinate function on P " G m .The O S -algebra O P 1 is then identified with O S rzs{pz ´1q 2 and its ideal generated by z ´1 with the O S -module ω G m , so that In these terms, the map q is just the projection onto ω G m .Moreover, and tautologically enough, the closed immersion j : P 1 Ñ P corresponds to the invertible function z on P 1 .Writing z " 1 `pz ´1q shows that the image of z in ω G m coincides with that of z ´1, which in turn corresponds to the invariant differential dz z .□

Relation with the universal vector hull of the fundamental group
In this section the map i 6 is related to the universal vector hull of the group Λ.To define the latter, recall that the group Λ is by definition the lattice of characters of the torus Ť .Given χ P Λ, let χ : Ť Ñ G an m,S again denote the corresponding character.
Definition 4.8.The universal vector hull of Λ S is the S-analytic group morphism z , where z is the coordinate function on G m and dz z the corresponding invariant differential form.
The map θ Λ deserves the name 'universal vector hull' because of the following property: given a vector bundle E over S and a morphism of S-analytic groups f : Λ S Ñ V pEq, there exists a unique homomorphism of O S -modules ϕ : ω Ť Ñ E such that f " ϕ ˝θΛ ; see [MM74, Example 1.3 a) and Proposition 1.4].Now, the canonical extension p U E q is by definition the push-out along the injective map d p : ω B Ñ ω Ě of the extension p ˚p U B q.The quotient U E {p ˚UB therefore sits in the following commutative and exact diagram of O E -modules: The isomorphism ε ˚ω Ť -U E {p ˚UB allows one to define a projection q : U E Ñ ε ˚ω Ť .In particular, for χ P Λ, this defines a homomorphism of O S -modules q : χ ˚UE Ñ ω Ť .Recall that χ 6 " xχ, ȷ y can be seen as a section of χ ˚UE (see Remark 4.3).
Proof.Up to transliteration, the statement is a special case of Proposition 4.7.Needless to say, ‚ the S-analytic space therein X plays the role of B, ‚ the principal bundle P that of Ě, ‚ the line bundle L χ that of L B,pcpχq,idq and ‚ the section u that of xχ, ěy, where ě is the neutral section of Ě.For χ P Λ the section xχ, jy corresponds via this dictionary to the trivialization xχ, ȷ y.Nonetheless, the reader might still be lost in translation while trying to see why diagram (4.5) reads as (4.6).To remedy that, first, notice that in the current framework, the line bundle ě˚L B,pcpχq,idq is always understood to be trivialized via xχ, ěy.This should elucidate the omnipresence of the line bundle x ˚Lχ as opposed to the absence of the corresponding line bundle ě˚L B,pcpχq,idq .Second, the statement of Proposition 4.10 revolves around the vector bundles U E and p ˚UB on E (rather, their fibers at χ), whereas in Proposition 4.7 the line bundles j ˚p˚L χ and i ˚Lχ are considered (better, their push-forward onto S).However, the line bundle j ˚p˚L χ translates to L E,pχ,ȷ q and, as already observed in Remark 4.3, the push-forward of L E,pχ,ȷ q onto S coincides with χ ˚UE .Along a similar line, the vector bundle i ˚Lχ on S plays the role of χ ˚p˚U B .Third, in diagram (4.6), there is nothing whatsoever like u ˚Ω1 P {X .This is because the projection q in the statement of Proposition 4.7 is constructed by further taking into account the isomorphism ω T -u ˚Ω1 P {X already implied here.□ Diagram (4.6) permits one to define a morphism of S-analytic groups Unwinding the definitions, Proposition 4.10 is rephrased as follows.
Example 4.12.Theorem 4.11 is more eloquent when A is an abeloid variety over K with totally degenerate reduction, that is, A " T {Λ.If Ť is the torus with group of characters Λ and θ Λ : Λ Ñ V pω Ť q the universal vector hull of Λ, then A 6 " pT ˆV pω Ť qq{tpχ, θ Λ pχqq : χ P Λu.
4.6.Universal cover of affine bundles 4.6.1.Extensions.Let F be a vector bundle over S, ϕ A : ω Ǎ Ñ F a homomorphism of O S -modules and pF A q the short exact sequence of O S -modules obtained by push-out of the canonical extension p U A q along ϕ A .Via the isomorphism U E -u ˚UA obtained in Section 4.1.3,the short exact sequence of O E -modules pF E q :" u ˚pF A q is seen to be the push-out of p U E q along the homomorphism of O S -modules where d ǔ : ω Ǎ Ñ ω Ě is the isomorphism given by pull-back of differential forms along the étale morphism ǔ.
On the other hand, by its very definition, the short exact sequence p U E q is itself the push-out of p ˚p U B q along the K-linear homomorphism d p : ω B Ñ ω Ě .Thus, the preceding considerations furnish an isomorphism pF E q -p ˚pF B q of short exact sequences of O E -modules, where pF B q is the push-out of the canonical extension p U B q along the homomorphism of O S -modules

Affine bundles.
Consider the affine bundle π A : V pF A q Ñ A and the morphism of S-analytic spaces Φ A : A 6 ÝÑ ApF A q induced by the homomorphism p U A q Ñ pF A q of short exact sequences of O A -modules given by the definition of pF A q as a push-out.The affine bundle ApF A q carries a unique S-analytic group structure for which the morphism Φ A is a group morphism.The pull-back V pF A q ˆA E to E is by definition the affine bundle π E : ApF E q Ñ E associated with the extension pF E q.By transport of structure, the E-analytic space ApF E q is an S-analytic group such that the natural morphism deduced from Φ A is a group morphism.The natural action of Λ S on V pF E q given by the natural Λ Slinearization of F E is described, for χ P Λ, as the translation by the point Φ E pχ 6 q, where χ 6 is the S-point of the universal vector extension E 6 considered in Section 4.3.Now, the isomorphism of short exact sequences of O E -modules pF E q -p ˚pF B q permits one to identify the affine bundle ApF E q with the fibered product ApF B q ˆB E, where π B : ApF B q Ñ B is the affine bundle associated with pF B q.This identification respects the natural S-analytic group structures involved and will be implied in what follows.Let q : ApF E q Ñ ApF B q be the projection, so that the following square of K-analytic space is Cartesian: Let C B be the cokernel of ϕ B .Arguing as for the map pr u : E 6 Ñ V pω Ť q permits one to define a morphism of S-analytic groups pr u,B : ApF B q Ñ V pC B q. Let ϕ T : ω Ť Ñ C B the unique map fitting in the following commutative and exact diagram of homomorphism of O S -modules: (4.7) The construction of pF E q as a push-out of p U E q along ϕ E implies that the diagram of S-analytic spaces is commutative, where the upper horizontal arrow is the projection considered in Theorem 4.11 and pr u,E :" pr u,B ˝q : ApF E q Ñ V pC B q.These considerations together with Theorem 4.11 prove the following.

Contractibility of the universal cover.
In this final section the contractibility of the space ApF E q above is addressed when S is a K-rational point.
Lemma 4.14.Let X be a smooth connected admissible formal R-scheme with Raynaud's generic fiber X :" X η .Then, for any closed analytic subspace of Z Ĺ X, the open subset X ∖ Z is contractible.
Proof.Let h : X ˆr0, 1s Ñ X be the deformation retraction onto the skeleton SkpXq of X given by [Ber99, Theorem 5.2].Since the formal scheme X is smooth connected, the skeleton SkpXq is a singleton, namely the unique preimage of the generic point of X under the reduction map X Ñ X.According to item (v) of loc.cit., for 0 ă t ď 1 and x P X, the local ring at the point hpx, tq is a field.Thus the only closed analytic subspace of X containing hpx, tq is X itself.In particular, the point hpx, tq belongs to X ∖ Z, and the statement follows.□ Proposition 4.15.The topological space ApF E q is contractible and is a universal cover of ApF A q.
Proof.Let F 0 be the image of the R-module ω B via the map ϕ B : ω B Ñ F and F 0 the push-out of the canonical extension p U B q on B along ω B Ñ F 0 .For χ P Λ, the line bundle L B, χ " L B|Bˆt čp χqu extends to a line bundle L B, χ on B. Consider a basis χ1 , . . ., χn of Λ and the smooth connected formal R-scheme X :" P pF 0 q ˆB P `OB ' L χ1 ˘ˆB ¨¨¨ˆB P `OB ' L χn ˘.
Now ApF E q is the complement of a Cartier divisor in X :" X η ; thus by Lemma 4.14 it is contractible.Since ApF A q is the quotient of ApF E q by the (free) action of Λ, the statement follows.□ Applying this with F " ω Ǎ and ϕ A " id gives that E 6 is contractible and E 6 Ñ A 6 is a universal cover, which justifies the name universal cover for E 6 .

A.1. Vector bundles on first-order thickenings
Let X 0 and X 1 be schemes endowed with a closed immersion s : X 0 Ñ X 1 and a morphism f : X 1 Ñ X 0 such that f ˝s " id X 0 .Suppose that the sheaf of ideals I :" KerpO X 1 Ñ O X 0 q is of square zero.For a quasi-coherent O X 1 -module F, consider the short exact sequence (F) 0 ÝÑ IF ÝÑ F ÝÑ F{IF ÝÑ 0.
The sequence of O X 0 -modules obtained by pushing forward pFq along f is short exact because the morphism f is affine (affineness only depends on the underlying reduced structure).Pushing forward a homomorphism of O X 1 -modules ϕ : F Ñ F 1 yields a homomorphism of short exact sequences of O X 0 -modules f ˚ϕ : f ˚pFq Ñ f ˚pF 1 q.The so-defined functor Proof.The only thing to show is that, for a splitting ϕ : s ˚F Ñ f ˚F of the short exact sequence f ˚pFq, the homomorphism Φ : f ˚s˚F Ñ F obtained by extending ϕ O X 1 -linearly is an isomorphism.To check this, one may reason locally on X 0 and choose a splitting f ˚F -s ˚F ' s ˚F b f ˚I of the short exact sequence f ˚pFq.This allows for the identities IF " I b F and f ˚pI Fq " f ˚I b s ˚F, which hold, respectively, because F is flat and because the ideal I is of square zero.Via these identifications, the splitting ϕ is of the form v Þ Ñ pv, εpvqq for a homomorphism of O X -modules ε : s ˚F Ñ s ˚F b f ˚I .Write a section of f ˚s˚F as pv, v 1 q for sections v of s ˚F and v 1 of s ˚F b f ˚I ; then the map Φ is defined as pv, v 1 q Þ Ñ pv, εpvq `v1 q, where the term εpv 1 q vanished because the ideal I is of square zero.Such an expression clearly defines an isomorphism, which concludes the proof.□

A.2. Tensor product and Baer sums
For simplicity, assume X 1 to be the first-order thickening of V pE _ q along its zero section s, where E is a vector bundle on X 0 and V pE _ q the total space of its dual.Concretely, the scheme X 1 is the spectrum of the O X 0 -module O X 0 ' E endowed with an O X 0 -algebra structure defined by the formula pa, vq ¨pb, wq " pab, aw `bvq.In what follows, it will be important to distinguish whether tensor products are taken with respect to O X 0 or O X 1 .To mark the difference but at the same time make the notation lighter, for i " 0, 1 and O For a vector bundle V on X 1 , set V 0 :" s ˚V , so that the vector bundle f ˚V on X 0 is an extension of V 0 by V 0 b 0 E. In order to make sense of the following statement, observe that, for vector bundles Unfortunately, the argument is quite clumsy and goes through the explicit construction of the Baer sum in question.To recall it, let p : f ˚V Ñ V 0 and p 1 : f ˚V 1 Ñ V 1 0 denote the homomorphisms given by restriction to X 0 , and consider the O X 0 -submodule W Ď pV 0 b 0 f ˚V 1 q ' pf ˚V b 0 V 1 0 q made of pairs whose components have the same image in V 0 b 0 V 1 0 via, respectively, p b id V 1 0 and id V 0 bp 1 .The vector bundle W on X 0 fits into the following short exact sequence of O X 0 -modules: The Baer sum mentioned in the statement is the push-out of the short exact sequence pW q along the sum map pE b 0 given by universal property of tensor product.The vector bundle E, seen as a sheaf of ideals of f ˚OX 1 , is of square zero; hence Ker ϕ " pE b 0 V 0 q b 0 pE b 0 V 1 0 q.The right-hand side of the previous equality is seen to also be the intersection of the kernels of the homomorphisms p b id f ˚V 1 and id f ˚V 1 bp 1 .That is, upon setting ψ " pp b id f ˚V 1 , id f ˚V 1 bp 1 q : f ˚V b 0 f ˚V 1 ÝÑ pV 0 b 0 f ˚V 1 q ' pf ˚V b 0 V 1 0 q, the homomorphisms ϕ and ψ share the same kernel.Moreover, the identity pp b id V 1 0 q ˝pid f ˚V bp 1 q " p b p 1 " pid V 0 bp 1 q ˝ppb id f ˚V 1 q of maps f ˚V b 0 f ˚V 1 Ñ V 0 b 0 V 1 0 implies that the image of ψ is W . Consequently, the homomorphism ϕ factors uniquely through an epimorphism φ : W Ñ f ˚pV b 1 V 1 q of O X 0 -modules.The kernel of ϕ is contained in that of p b p 1 , and the homomorphism φ acts on the quotient as the sum map, where K and K 1 are the kernels of the homomorphisms p b id f ˚V 1 and id f ˚V bp 1 , respectively.In other words, the homomorphism φ fits into the following commutative and exact diagram of O X 0 -modules: where the upper row is the short exact sequence pW q.To put it differently, the lower row is the push-out of the upper one along the sum map; that is, the extension f ˚pV b 1 V 1 q is the desired Baer sum.□

A.3. Connections
Let S be a scheme, f : X Ñ S a separated morphism of schemes, ∆ X : X Ñ XˆS X the diagonal morphism and I :" KerpO XˆS X Ñ ∆ X˚OX q its augmentation ideal.With this notation, the sheaf of differentials (denoted by Ω 1 f or Ω 1 X{S ) relative to f is the O X -module ∆ X I. Let ∆ X,1 be the first infinitesimal neighbourhood of the diagonal, that is, the closed subscheme of X ˆS X defined by the sheaf of ideals I 2 .For i " 1, 2, let p i : ∆ X,1 Ñ X be the morphism induced by the i th projection.The O X -module J 1 f :" p 1˚O∆ X,1 is called the sheaf of first-order jets.Each p i induces a homomorphism of f ´1O S -algebras j i : O X Ñ J 1 f .The homomorphism of f ´1O S -modules d f :" j 2 ´j1 : O X Ñ Ω 1 f is called the canonical derivation.
Definition A.3.A connection on a vector bundle F on X is an isomorphism ∇ : p 1 F Ñ p 2 F of vector bundles over ∆ X,1 whose restriction to the diagonal ∆ X ∇ is the identity of F.

A.4. Atiyah extension
The kernel of the restriction map O ∆ X,1 Ñ i ˚OX , where i : X Ñ ∆ X,1 is the closed immersion induced by the diagonal, is by definition of square zero.This allows one to adopt the notation introduced in Section A.1 with X 0 " X, X 1 " ∆ X,1 , s " i and f " p 1 .Pushing forward the O ∆ X,1 -module p 2 F along p 1 yields the short exact sequence of O S -modules, where J 1 f pFq :" p 1˚p 2 F is the O X -module of first-order jets of F. Applied to the O ∆ X,1 -module p 2 F, Proposition A.1 implies the following.

A.5. Infinitesimal rigidifications
Let pr 1 , pr 2 : X ˆS X Ñ X be, respectively, the first and the second projection and x : S Ñ X a section of the structural morphism f : X Ñ S. Let X 1 be the first-order thickening of X along x, ι : X 1 Ñ X the closed immersion and π " f ˝ι : X 1 Ñ S the structural morphism.Definition A.5.An infinitesimal rigidification of F at x is an isomorphism of O X 1 -modules ρ : π ˚x˚F Ñ ι ˚F such that x ˚ρ is the identity.
Note that if ∇ : p 1 F Ñ p 2 F is a connection on F, then the homomorphism of O X 1 -modules τ ˚∇ is an infinitesimal rigidification of F at x. Now, it is possible to give a characterization of infinitesimal rigidifications similar to that of connections.For, the augmentation ideal of the closed immersion x 1 : S Ñ X 1 induced by x is of square zero.This permits one to employ the conventions introduced in Section A.1 with X 0 " S, s " x 1 and f " π.For instance, if F is a vector bundle, then pushing forward the O X 1 -module ι ˚F along π yields the short exact sequence π ˚pι ˚Fq 0 ÝÑ x ˚Ω1 X{S b x ˚F ÝÑ π ˚ι˚F ÝÑ x ˚F ÝÑ 0 of O S -modules.Applied to O X 1 -module ι ˚F, Proposition A.1 reads as follows.
Let τ : X 1 Ñ ∆ X,1 be the morphism determined by p 1 ˝τ " x ˝π and p 2 ˝τ " ι.Consider the homomorphism ψ : p 2 F Ñ τ ˚ι˚F of O ∆ X,1 -modules given by adjunction (note the equality ι ˚F " τ ˚p2 F).Pushing it forward along the morphism p 1 gives a homomorphism π ˚ψ : pJ 1 X{S pFqq ÝÑ p 1˚τ˚p ι ˚Fq of short exact sequences of O X -modules.Now, by the definition of τ, the following square is commutative: Therefore, the short exact sequence p 1˚τ˚p ι ˚Fq of O X -modules is nothing but the push-forward along the closed immersion x of the short exact sequence π ˚pι ˚Fq of O S -modules.
Proposition A.7.The homomorphism ϕ : x ˚pJ 1 X{S pFqq Ñ π ˚pι ˚Fq of short exact sequence of O S -modules adjoint to π ˚ψ is an isomorphism.Proof.The homomorphism ψ : p 2 F Ñ τ ˚ι˚F restricted to the diagonal is the evaluation at x. Therefore, the homomorphism π ˚ψ of short exact sequence of O X -modules is the following commutative diagram: where ev x is the evaluation at x.The homomorphism ϕ, which is the adjoint to π ˚ψ, reads as the commutative diagram of O S -modules.The five lemma implies that ϕ is an isomorphism.□ 's generic fibers of S, B and B. Let L B be the line bundle on B ˆS B deduced from L B .The K-analytic space B represents the functor associating with a S-analytic space the group of isomorphism classes of homogeneous line bundles on B ˆS S 1 .Moreover, the universal object is the line bundle L B on B ˆS B (see [BL91, Proposition 6.2]).Let β : B Ñ S and β : B Ñ S be the structural morphisms.For an S-analytic space S 1 Ñ S and morphisms b : S 1 Ñ B, b : S 1 Ñ B of S-analytic spaces, let L B,pb, bq :" pb, bq ˚LB be the line bundle on S 1 obtained by pulling back the Poincaré bundle L B on B ˆS B along the morphism pb, bq : S 1 Ñ B ˆS B.

Definition 4. 1 .
The pull-back of differential forms along the morphism p : Ě Ñ B induces a homomorphism of O S -modules d p : ω B Ñ ω Ě .The short exact sequence of O E -modules

Proposition A. 4 .
The bijection Hompp 1 F, p 2 Fq Ñ HompF, J 1 X{S pFqq given by adjunction induces a bijection␣ connections on F ( -# splittings of the short exact sequence pJ 1 X{S pFqq + .Set At f pFq :" H ompE, J 1 f pFqq.The short exact sequence of O X -modules (At f pFq) 0 ÝÑ Ω 1 f b End F ÝÑ At f pFq ÝÑ End F ÝÑ 0obtained as the tensor product of pJ 1 f pFqq with F _ is called the relative Atiyah extension of F. When F is a line bundle, End F -O X , and splittings of the Atiyah extension pAt f pFqq are in bijection with those of pJ 1 f pFqq, thus with connections on F.
a splitting of the short exact f ˚pFq.The bijection Hompf ˚s˚F , Fq Ñ Homps ˚F, f ˚Fq given by adjunction for a vector bundle F on X 1 induces a bijection " isomorphisms ρ : f ˚s˚F Ñ F such that s ˚ρ " id s *.