##
**Analytical methods for abelian varieties in positive characteristic. Chapters 3–7.
(Metodi analitici per varietà abeliane in caratteristica positiva. Capitolo 3–7.)**
*(Italian)*
Zbl 0168.18601

Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 19, 277-330 (1965); 19, 481-512 (1965); 20, 101-137 (1966); 20, 331-365 (1966).

(For chapter 3, 4 cf. also Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 19, 277–330 (1965; Zbl 0132.41302.)

A useful preliminary summary appeared in [Centre Belge Rech. Math., Colloque Théor. Groupes Algébr., Bruxelles 1962, 77–85 (1962; Zbl 0163.15301)]. Chapters I and II have been reviewed already [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 18, 1–25 (1964; Zbl 0121.16104)] as has what is in effect Chapter 0 [ibid. 13, 303–372 (1960; Zbl 0092.27203)].

These papers form a single work aimed at studying inseparability phenomena for abelian varieties over an algebraically closed field \(k\) of characteristic \(p> 0\). The usual \(l\)-adic homology modules are, of course, inadequate for this purpose; one needs modules involving \(p\) itself. The author uses finite-dimensional modules over \(K\), the ring of infinite Witt vectors over \(k\) (i. e., the complete unramified \(p\)-adic ring having \(k\) as residue class field).

Chapters I–V study these modules and their relation to analytic groups; the last two chapters give the applications to abelian varieties. The bulk of the work dates from 1961–62; group schemes are not used explicitly and functors minimally. One feature of the method is that it handles the inseparable and separable groups together a good part of the time. Let \(R\) be a complete commutative \(k\)-algebra with coproduct \(\mathbf P: B \to R \otimes R\) satisfying the usual “co”-laws. \(R\) is called a hyperalgebra over \(k\). (Here \(\otimes\) denotes the completed tensor product; in general, topological conditions are omitted from what follows, and there are other mild inaccuracies, too.) With \(R\) is associated functorially \(\mathcal C R\), a \(K\)-module called its “canonical module”. This is the set of those Witt “covectors” \(x = (\ldots, x_{-1}, x_0)\) over \(R\) (see Chapter I) which are “canonical”, i. e., satisfy \(\mathbf P x = 1 \otimes x + x \otimes 1\). Canonical modules are intrinsically treated in “Chapter 0”.

A hyperalgebra \(R\) is a “hyper-domain” if \(\mathcal C R\) is a finite free \(K\)-module and the components of the covectors in \(\mathcal C R\) generate a dense subalgebra of \(R\). The correspondence \(R \to \mathcal C R\) is an isomorphism of the categories of hyperdomains and canonical modules, this being one of the main results of Chapter III. Every hyperdomain is isogenous to a product of certain elementary hyperdomains \(R_{i,j}\). In particular, the \(R_{0,n}\), are the separable ones (the sum of \(\mathbb Q_p/\mathbb Z_p)\), the others are inseparable (formal groups).

Chapter IV introduces the “bidomain” \(\mathcal R\) functorially associated with a given hyperdomain \(R\), defined to be the completion of the direct limit \(R \rightarrow R\rightarrow R\rightarrow \cdots\), the maps being \(p\iota\) in each case \((\iota = \) identity). This \(\mathcal R\) is the smallest hyperalgebra containing \(R\) and on which \(p\iota\) is an isomorphism The set of “canonical” bivectors with components in \(\mathcal R\) is now denoted by \(\mathcal C'R\); it is a vector space over \(K'\), the quotient field of \(K\). This vector space is the basic one used in the applications to abelian varieties. In addition, there is a dual theory, involving “codomains”. It produces duals to the hyperdomain, to the bidomain and to the space \(\mathcal C'R\); these duals will be denoted by \(\tilde R\), \(\tilde{\mathcal C'}R\), etc. The relations between all these things and the Witt operations on the modules occupy Chapter IV. In particular, the dual of \(R_{i,j}\) is observed to be \(R_{j,i}\).

The aim in the rather technical Chapter V is apparently to make the duality between \(\tilde{\mathcal C'}R\) and \(\mathcal C' R\) quite explicit. This is done by interpreting (in the inseparable case) an element \(d\in C' R\) as a derivation on the submodule of “differentiable bivectors” in \(\mathcal R\) – these are ones which satisfy a certain convergence condition. The resulting pairing \((d, x)\) establishes the duality. All this depends on explicit Witt vector calculations.

These algebraic results are now applied in Chapters VI and VII to an abelian variety \(A\) over \(k\). The hyperdomain \(\mathcal R = \mathcal R A\) is taken to be the completion (with respect to its natural topology) of the ring of functions on \(A\) regular at all the points of order \(p^i\) \((i\ge 0)\). \(R\) splits into \(R_{0,f} \otimes S\), \(R_{0,f}\) being the separable part \((f = \) separable dimension of \(A)\) and \(S\) the inseparable part.

The principal result now is the construction, for each divisor \(Y\) on \(A\), of a homomorphism \(\varphi_Y\colon \tilde{\mathcal C'}(\mathcal R A) \to \mathcal C'(\mathcal R A)\) of the canonical \(K'\)-spaces, and the proof that \(\varphi_Y\) has its usual properties. (The usual map denoted by this symbol would be defined only on the separable part \(R_{0,f}\), i.e., on the module of \(p\)-adic Tate vectors.)

Consequences of this are (1) the duality theorem \((\operatorname{Pic}_0 A = \tilde A\) and \(A\) are dual abelian varieties), (2) \(\varphi_Y = 0\) if and only if \(Y\equiv 0\) and (3) the symmetry theorem (proved independently by Yu. I. Manin [Russ. Math. Surv. 18, No. 6, 1–83 (1963); translation from Usp. Mat. Nauk 18, No. 6(114), 3–90 (1963; Zbl 0128.15603)]): The factorization of \(R\) into elementary factors contains \(R_{i,j}\) and \(R_{j,i}\), to equal powers. This last is an immediate consequence of the fact that \(\varphi_Y\) is an isogeny if \(Y\) is nondegenerate, so that \(\mathcal R \tilde A = \tilde{\mathcal R} A\) and \(\mathcal R A\) are isogenous; thus \(\mathcal R A\) is self-dual up to isogeny, and the result follows from the duality of \(R_{i,j}\) and \(R_{j,i}\) noted above. The construction of \(\varphi_Y\) is done separately on the separable and inseparable components of \(\mathcal C'(\mathcal R A)\).

An essential intermediate step uses the “hyperclasses” on \(A\) – these are like Chevalley’s repartitions (adeles), except that to each prime divisor \(X\) one assigns a Witt vector with components in \(k(A)\). The module \(\mathfrak M(A)\) of closed/exact hyperclasses is a canonical module of dimension \(= \dim A\), and is presumably some \(H^1(A, ?)\). A Riemann form can be introduced by using \(\varphi_Y\). The possibility of going on to a theory of \(p\)-adic “period matrices” and “abelian functions” (generalizing H. Morikawa’s results [Nagoya Math. J. 20, 1–27 (1962; Zbl 0115.39001); 21, 231–250 (1962; Zbl 0115.39002)]) is alluded to in a sentence near the end, which is otherwise devoted to tying up these results with previous statements and conjectures of the author, true and false.

A useful preliminary summary appeared in [Centre Belge Rech. Math., Colloque Théor. Groupes Algébr., Bruxelles 1962, 77–85 (1962; Zbl 0163.15301)]. Chapters I and II have been reviewed already [Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 18, 1–25 (1964; Zbl 0121.16104)] as has what is in effect Chapter 0 [ibid. 13, 303–372 (1960; Zbl 0092.27203)].

These papers form a single work aimed at studying inseparability phenomena for abelian varieties over an algebraically closed field \(k\) of characteristic \(p> 0\). The usual \(l\)-adic homology modules are, of course, inadequate for this purpose; one needs modules involving \(p\) itself. The author uses finite-dimensional modules over \(K\), the ring of infinite Witt vectors over \(k\) (i. e., the complete unramified \(p\)-adic ring having \(k\) as residue class field).

Chapters I–V study these modules and their relation to analytic groups; the last two chapters give the applications to abelian varieties. The bulk of the work dates from 1961–62; group schemes are not used explicitly and functors minimally. One feature of the method is that it handles the inseparable and separable groups together a good part of the time. Let \(R\) be a complete commutative \(k\)-algebra with coproduct \(\mathbf P: B \to R \otimes R\) satisfying the usual “co”-laws. \(R\) is called a hyperalgebra over \(k\). (Here \(\otimes\) denotes the completed tensor product; in general, topological conditions are omitted from what follows, and there are other mild inaccuracies, too.) With \(R\) is associated functorially \(\mathcal C R\), a \(K\)-module called its “canonical module”. This is the set of those Witt “covectors” \(x = (\ldots, x_{-1}, x_0)\) over \(R\) (see Chapter I) which are “canonical”, i. e., satisfy \(\mathbf P x = 1 \otimes x + x \otimes 1\). Canonical modules are intrinsically treated in “Chapter 0”.

A hyperalgebra \(R\) is a “hyper-domain” if \(\mathcal C R\) is a finite free \(K\)-module and the components of the covectors in \(\mathcal C R\) generate a dense subalgebra of \(R\). The correspondence \(R \to \mathcal C R\) is an isomorphism of the categories of hyperdomains and canonical modules, this being one of the main results of Chapter III. Every hyperdomain is isogenous to a product of certain elementary hyperdomains \(R_{i,j}\). In particular, the \(R_{0,n}\), are the separable ones (the sum of \(\mathbb Q_p/\mathbb Z_p)\), the others are inseparable (formal groups).

Chapter IV introduces the “bidomain” \(\mathcal R\) functorially associated with a given hyperdomain \(R\), defined to be the completion of the direct limit \(R \rightarrow R\rightarrow R\rightarrow \cdots\), the maps being \(p\iota\) in each case \((\iota = \) identity). This \(\mathcal R\) is the smallest hyperalgebra containing \(R\) and on which \(p\iota\) is an isomorphism The set of “canonical” bivectors with components in \(\mathcal R\) is now denoted by \(\mathcal C'R\); it is a vector space over \(K'\), the quotient field of \(K\). This vector space is the basic one used in the applications to abelian varieties. In addition, there is a dual theory, involving “codomains”. It produces duals to the hyperdomain, to the bidomain and to the space \(\mathcal C'R\); these duals will be denoted by \(\tilde R\), \(\tilde{\mathcal C'}R\), etc. The relations between all these things and the Witt operations on the modules occupy Chapter IV. In particular, the dual of \(R_{i,j}\) is observed to be \(R_{j,i}\).

The aim in the rather technical Chapter V is apparently to make the duality between \(\tilde{\mathcal C'}R\) and \(\mathcal C' R\) quite explicit. This is done by interpreting (in the inseparable case) an element \(d\in C' R\) as a derivation on the submodule of “differentiable bivectors” in \(\mathcal R\) – these are ones which satisfy a certain convergence condition. The resulting pairing \((d, x)\) establishes the duality. All this depends on explicit Witt vector calculations.

These algebraic results are now applied in Chapters VI and VII to an abelian variety \(A\) over \(k\). The hyperdomain \(\mathcal R = \mathcal R A\) is taken to be the completion (with respect to its natural topology) of the ring of functions on \(A\) regular at all the points of order \(p^i\) \((i\ge 0)\). \(R\) splits into \(R_{0,f} \otimes S\), \(R_{0,f}\) being the separable part \((f = \) separable dimension of \(A)\) and \(S\) the inseparable part.

The principal result now is the construction, for each divisor \(Y\) on \(A\), of a homomorphism \(\varphi_Y\colon \tilde{\mathcal C'}(\mathcal R A) \to \mathcal C'(\mathcal R A)\) of the canonical \(K'\)-spaces, and the proof that \(\varphi_Y\) has its usual properties. (The usual map denoted by this symbol would be defined only on the separable part \(R_{0,f}\), i.e., on the module of \(p\)-adic Tate vectors.)

Consequences of this are (1) the duality theorem \((\operatorname{Pic}_0 A = \tilde A\) and \(A\) are dual abelian varieties), (2) \(\varphi_Y = 0\) if and only if \(Y\equiv 0\) and (3) the symmetry theorem (proved independently by Yu. I. Manin [Russ. Math. Surv. 18, No. 6, 1–83 (1963); translation from Usp. Mat. Nauk 18, No. 6(114), 3–90 (1963; Zbl 0128.15603)]): The factorization of \(R\) into elementary factors contains \(R_{i,j}\) and \(R_{j,i}\), to equal powers. This last is an immediate consequence of the fact that \(\varphi_Y\) is an isogeny if \(Y\) is nondegenerate, so that \(\mathcal R \tilde A = \tilde{\mathcal R} A\) and \(\mathcal R A\) are isogenous; thus \(\mathcal R A\) is self-dual up to isogeny, and the result follows from the duality of \(R_{i,j}\) and \(R_{j,i}\) noted above. The construction of \(\varphi_Y\) is done separately on the separable and inseparable components of \(\mathcal C'(\mathcal R A)\).

An essential intermediate step uses the “hyperclasses” on \(A\) – these are like Chevalley’s repartitions (adeles), except that to each prime divisor \(X\) one assigns a Witt vector with components in \(k(A)\). The module \(\mathfrak M(A)\) of closed/exact hyperclasses is a canonical module of dimension \(= \dim A\), and is presumably some \(H^1(A, ?)\). A Riemann form can be introduced by using \(\varphi_Y\). The possibility of going on to a theory of \(p\)-adic “period matrices” and “abelian functions” (generalizing H. Morikawa’s results [Nagoya Math. J. 20, 1–27 (1962; Zbl 0115.39001); 21, 231–250 (1962; Zbl 0115.39002)]) is alluded to in a sentence near the end, which is otherwise devoted to tying up these results with previous statements and conjectures of the author, true and false.

Reviewer: A. Mattuck (M. R. 35 #193)

### MSC:

14-XX | Algebraic geometry |

### Citations:

Zbl 0132.41302; Zbl 0163.15301; Zbl 0121.16104; Zbl 0092.27203; Zbl 0128.15603; Zbl 0115.39001; Zbl 0115.39002
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\textit{I. Barsotti}, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 19, 277--330, 481--512 (1966; Zbl 0168.18601)

### References:

[1] | Mc I. Barsotti , Moduli canonici e gruppi analitici commutativi , Ann. Scuola Norm. Sup. , 13 , 1959 , p. 303 . Numdam | MR 113887 | Zbl 0092.27203 · Zbl 0092.27203 |

[2] | 1 I. Barsotti , Abelian varieties over fields of positive characteristic , Rend. Circ. Matem. Palermo , 5 , 1956 , p. 145 . MR 83181 | Zbl 0075.30103 · Zbl 0075.30103 |

[3] | 2 I. Barsotti , Gli endomorfismi delle varietà abeliane su corpi di caratteristica positiva , Ann. Scuola Norm. Sup. , 10 , 1956 , p. 1 . Numdam | MR 82182 | Zbl 0075.30201 · Zbl 0075.30201 |

[4] | 3 I. Barsotti , Repartitions on abelian varieties , Illinois Journ. of Math. , 2 , 1958 , p. 43 . Article | MR 103893 | Zbl 0091.33401 · Zbl 0091.33401 |

[5] | 4 I. Barsotti , Local properties of algebraic correspondences , Trans. Amer. Math. Soc. , 71 , 1951 , p. 349 . MR 44868 | Zbl 0045.24101 · Zbl 0045.24101 |

[6] | 5 A. Weil , Varietes abeliennes et courbes algebriques , Parigi , 1948 . MR 29522 | Zbl 0037.16202 · Zbl 0037.16202 |

[7] | 6 J.-P. Serre , Quelques proprietes des varietes abeliennes en caracteristique p, Amer. Journ. of Math. 80 , 1958 , p. 715 . MR 98100 | Zbl 0099.16201 · Zbl 0099.16201 |

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