(30)

Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.

(29)

This may be viewed as a far-reaching generalisation of Kronecker's second limit formula, to which our result reduces on elliptic schemes.

(28)

This improves on results of Noguchi and Martin-Deschamps in characteristic $0$. In positive characteristic, our result can be used to give the first examples of varieties, which are not embeddable in abelian varieties and satisfy an analog of the Bombieri-Lang conjecture.

(27) with V. Maillot, On a canonical class of Green
currents for the unit sections of abelian schemes. [PDF] (42
pages) *Documenta Math*. **20** (2015),
631--668.

Abstract: We show that on
any abelian scheme over a complex quasi-projective smooth variety,
there is a Green current for the zero-section, which is
axiomatically determined up to $\partial$ and $\bar\partial$-exact
differential forms. On an elliptic curve, this current specialises
to a Siegel function.

We prove generalisations of classical properties of Siegel
functions, like distribution relations and reciprocity laws.
Furthermore, as an application of (a refined version of) the
arithmetic Riemann-Roch theorem, we show that the above current,
when restricted to a torsion section, is the realisation in
analytic Deligne cohomology of an element of the (Quillen) $K_1$
group of the base, the corresponding denominator being given by
the denominator of a Bernoulli number. This generalises the second
Kronecker limit formula and the denominator $12$ computed by
Kubert, Lang and Robert in the case of Siegel units.
Finally, we prove an analog in Arakelov theory of a Chern class
formula of Bloch and Beauville, where the canonical current plays
a key role.

Some of the results of this article were announced in: Elements of the group $K_1$
associated to abelian schemes. [PDF] Oberwolfach Reports 5 (2008), no. 3,
2013--2014.

** (26) **(comme
éditeur) Conference
on Arakelov Geometry and K-theory. Annales
de la faculté des sciences de Toulouse Sér.
6 **23** (2014), no. 3, p.
i-vi : Numéro
spécial (Festschrift) à
l'occasion de la conférence en l'honneur du soixantième
anniversaire de Christophe Soulé, 21-23 mai 2012, Institut
de Mathématiques de Toulouse.

We show that in this situation the group $A(K^{\rm perf})$ is finitely generated (thus generalizing a special case of the Lang-Néron theorem).

Here $K^{\rm perf}=K^{p^{-\infty}}$ is the maximal purely inseparable extension of $K$. This result implies in particular that in the situation described above, the "full" Mordell-Lang conjecture and a conjecture of Esnault and Langer are verified. The proof relies on the theory of semistability (of vector bundles) in positive characteristic and on the existence of the compactification of the universal abelian scheme constructed by Faltings-Chai.

(23) On the Manin-Mumford and Mordell-Lang conjectures in positive characteristic [PDF]

Abstract: We prove that in positive characteristic, the Manin-Mumford conjecture implies the Mordell-Lang conjecture, in the situation where the ambient variety is an abelian variety defined over the function field of a smooth curve over a finite field and the relevant group is a finitely generated group.

In particular, in the setting of the last sentence, we provide a proof of the Mordell-Lang conjecture, which does not depend on tools coming from model theory. After Hrushovski's proof of the Mordell-Lang conjecture in positive characteristic, it had been an open problem for some time to find an algebraic proof of this conjecture. Note that the possibility to prove the implication "Manin-Mumford implies Mordell-Lang" is specific to positive characteristic. In characteristic $0$, it seems unlikely that such an implication could be established, because the Manin-Mumford conjecture is much easier to prove than the Mordell-Lang conjecture.

(22) Infinitely $p$-divisible points on abelian varieties defined over function fields of characteristic $p>0$ [PDF]

Abstract: In this article we answer some questions raised by F. Benoist, E. Bouscaren and A. Pillay. We prove that infinitely $p$-divisible points on abelian varieties defined over function fields of transcendence degree one over a finite field are necessarily torsion points. This can be viewed as an analog in positive characteristic of Manin's "theorem of the kernel". We also prove that when the endomorphism ring of the abelian variety is $\bf Z$ then there are no infinitely $p$-divisible points of order a power of $p$. This statement (see the following unpublished note for an improvement) can be plugged into the main theorem of the article

(21) with R. Pink and an appendix by B. Köck, On the Adams-Riemann-Roch theorem in positive characteristic. [PDF] Math. Z. 270 (2012), no. 3-4, 1067--1076.

Abstract: Let $p>0$ be a prime number. We give a short Frobenius-theoretic proof of the Adams-Riemann-Roch theorem for the $p$-th Adams operation, when the involved schemes live in characteristic $p$ and the morphism is smooth. This result implies the Grothendieck-Riemann-Roch theorem for smooth morphisms in positive characteristic and the Hirzebruch-Riemann-Roch theorem in any characteristic. We also answer a question of B. Köck.

(20) with V. Maillot, On the birational invariance of the BCOV torsion of Calabi-Yau threefolds. [PDF] Comm. Math. Phys. 311 (2012), no. 2, 301--316.

Abstract: Fang, Lu and Yoshikawa conjectured a few years ago that a certain string-theoretic invariant (originally introduced by the physicists M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa) of Calabi-Yau threefolds is a birational invariant. This conjecture can be viewed as a "secondary" analog (in dimension three) of the birational invariance of Hodge numbers of Calabi-Yau varieties established by Batyrev and Kontsevich. Using the arithmetic Riemann-Roch theorem, we prove a weak form of this conjecture.

(19) A note on the ramification of torsion points lying on curves of genus at least two. [PDF] J. Théor. Nombres Bordeaux 22 (2010), no. 2, 475--481.

Abstract: Let $C$ be a curve of genus $g\geqslant 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that ${\rm char}(K)=0$ and that the characteristic of the residue field is not $2$. Suppose that the Jacobian ${\rm Jac}(C)$ has semi-stable reduction over $R$. Embed $C$ in ${\rm Jac}(C)$ using a $K$-rational point.

We show that the coordinates of the torsion points lying on $C$ lie in the unique tamely ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion points on ${\rm Jac}(C)$.

(18) [revised 16/12/09; 06/04/10] avec V. Maillot, Une conjecture sur la torsion des classes de Chern des fibrés de Gauss-Manin. [PDF] Publ. Res. Inst. Math. Sci. 46 (2010), no. 4, 789--828.

Résumé: Pour tout $t\in{\bf N}$ nous définissons un certain entier positif $N_t$ et nous conjecturons: si $H$ est un fibré de Gauss-Manin d'une fibration semi-stable alors la $t$-ème classe de Chern de $H$ est annulée par $N_t$. Nous démontrons diverses conséquences de cette conjecture.

(17) avec V. Maillot, Formes automorphes et théorèmes de Riemann-Roch arithmétiques. [PDF] Astérisque 328 (2009), 237--253 (Festschrift en l'honneur du soixantième anniversaire de J.-M. Bismut).

Résumé: Nous construisons trois familles de formes automorphes au moyen du théorème de Riemann-Roch arithmétique et de la formule de Lefschetz arithmétique. Deux de ces familles ont déjà été construites par Yoshikawa et notre construction met en lumière leur origine arithmétique. Nous prouvons une forme faible d'une conjecture de Yoshikawa sur le corps de définition des coefficients de Fourier de certaines formes automorphes.

(16) with V. Maillot, On the determinant bundles of abelian schemes.[PDF] Compositio Math. 144 (2008), 495--502.

Abstract: Let $\pi:A\to S$ be an abelian scheme and let $L$ be a symmetric, rigidified, relatively ample line bundle on $A$. Suppose that $S$ is quasi-projective over an affine noetherian scheme. We show that there is an isomorphism

$$

\det(\pi_*L)^{\otimes 24}\simeq\big(\pi_*\omega_{A/S}^{\vee}\big)^{\otimes 12d}

$$

of line bundles on $S$, where $d$ is the rank of the (locally free) sheaf $\pi_*L$. We also show that the numbers $24$ and $12d$ are sharp in the following sense: if $N>1$ is a common divisor of $12$ and $24$, then there exist data as above such that

$$

\det(\pi_*L)^{\otimes (24/N)}\not\simeq\big(\pi_*\omega_{A}^{\vee}\big)^{\otimes (12d/N)}.

$$

This answers a question raised by Chai and Faltings in their book on degeneration of abelian varieties (see p. 27, Remark 5.2).

(15) with H. Gillet and C. Soulé, An arithmetic Riemann-Roch theorem in higher degrees.[PDF] Ann. Inst. Fourier 58 (2008), no. 6, 2169--2189.

Abstract: We prove an analogue in Arakelov geometry of the Grothendieck-Riemann-Roch theorem for local complete intersection morphisms. A different approach to this theorem was described by Faltings in his book

(14) An afterthought on the generalized Mordell-Lang conjecture.[PDF] in Model theory with Applications to Algebra and Analysis Vol. 1, 63--71 (Eds. Zoe Chatzidakis, Dugald Macpherson, Anand Pillay, Alex Wilkie) London Math Soc. Lecture Note Series Nr 249, Cambridge Univ Press 2008.

Abstract: The generalized Mordell-Lang conjecture (GML) is the statement that the irreducible components of the Zariski closure of a subset of a group of finite rank inside a semi-abelian variety are translates of closed algebraic subgroups. McQuillan gave a proof of this statement. We revisit his proof, indicating some simplifications.

This text contains a complete elementary proof of the fact that (GML) for groups of torsion points (= generalized Manin-Mumford conjecture), together with (GML) for finitely generated groups imply the full generalized Mordell-Lang conjecture.

(13) with V. Maillot, On the order of certain characteristic classes of the Hodge bundle of semi-abelian schemes.[PDF] Number fields and function fields---two parallel worlds, 287--310, Progr. Math., 239, Birkhäuser Boston, Boston, MA, 2005.

Abstract: We give a new proof of the fact that the even terms (of a multiple of) the Chern character of the Hodge bundles of semi-abelian schemes are torsion classes in Chow theory and we give explicit bounds for almost all the prime powers appearing in their order. These bounds appear in the numerators of modified Bernoulli numbers. We also obtain similar results in an equivariant situation.

(12) A note on the Manin-Mumford conjecture.[PDF] Number fields and function fields---two parallel worlds, 311--318, Progr. Math., 239, Birkhäuser Boston, Boston, MA, 2005.

Abstract: In the article [PR1] (On Hrushovski's proof of the Manin-Mumford conjecture, referenced below), R. Pink and the author gave a short proof of the Manin-Mumford conjecture, which was inspired by an earlier model-theoretic proof by Hrushovski. The proof given in [PR1] uses a difficult unpublished ramification-theoretic result of Serre. It is the purpose of this note to show how the proof given in [PR1] can be modified so as to circumvent the reference to Serre's result.

The result is a short and completely elementary proof of the Manin-Mumford conjecture, which could be described in a first introductory course on algebraic geometry. J. Oesterlé and R. Pink contributed several simplifications and shortcuts to this note.

(11) with R. Pink, A conjecture of Beauville and Catanese revisited.[PDF] Math. Ann.

Abstract: A theorem of Green, Lazarsfeld and Simpson (formerly a conjecture of Beauville and Catanese) states that certain naturally defined subvarieties of the Picard variety of a smooth projective complex variety are unions of translates of abelian subvarieties by torsion points. Their proof uses analytic methods. We refine and give a completely new proof of their result. Our proof combines galois-theoretic methods and algebraic geometry in positive characteristic. When the variety has a model over a function field and its Picard variety has no isotrivial factors, we show how to replace the galois-theoretic results we need by results from model theory (mathematical logic). Furthermore, we prove partial analogs of the conjecture of Beauville and Catanese in positive characteristic.

See also the following preprint by H. Esnault and A. Ogus, where they tackle the conjecture 5.1 of the article (11).

(10) with V. Maillot,

About this article, see the following Bourbaki talk by C. Soulé.

(9) with R. Pink, On psi-invariant subvarieties of
semiabelian varieties and the Manin-Mumford conjecture.[PDF]
J. Algebraic. Geom. ** 13 **
(2004), no. 4, 771--798.

(0) The Riemann-Roch theorem for arithmetic curves.[PDF] ETH Diplomarbeit (1993).

(-1) GEOMETRIA GERBERTI.[PDF] Opuscule de Géométrie Incomplet de Gerbert d'Aurillac. Introduction, Traduction, Notes. Prépublication IHES/M/99/72 de l'Institut des Hautes Etudes Scientifiques.