(32) [revised 19th July 2017] On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic II.[PDF] Submitted.

Abstract: Let $A$ be an abelian variety over the function field  $K$ of a curve over a finite field. We provide several conditions ensuring that the group $A(K^{\rm perf})$ is finitely generated and that the $p$-primary torsion subgroup of $A(K^{\rm sep})$ is finite. This gives partial answers to questions of Scanlon, Ziegler Ghioca and Moosa, Esnault and Langer, and Poonen and Voloch. We also describe the basics of a theory (used to prove our results) relating the Harder-Narasimhan filtration of vector bundles and the structure of finite flat group schemes over projective curves over finite fields.

(31) with T. Szamuely, Cohomology and torsion cycles over the maximal cyclotomic extension.[PDF] (18 pages)
To appear in Crelle's Journal.

Abstract: A classical theorem by K. Ribet asserts that an abelian variety defined over the maximal cyclotomic extension K of a number field has only finitely many torsion points. We show that this statement can be viewed as a particular case of a much more general one, namely that the absolute Galois group of K acts with finitely many fixed points on the étale cohomology with ${\bf Q}/{\bf Z}$-coefficients of a smooth proper K-variety defined over K. We also present a conjectural generalization of Ribet’s theorem to torsion cycles of higher codimension. We offer supporting evidence for the conjecture in codimension 2, as well as an analogue in positive characteristic.

Strongly semistable sheaves and the Mordell-Lang conjecture over function fields.[PDF] (14 pages) Submitted.

Abstract: We give a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety.
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.

with G. Kings, Higher analytic torsion, polylogarithms and norm compatible elements on abelian schemes.[PDF] (27 pages) To appear in The Jean-Michel Bismut 65th Anniversary Volume (Birkhäuser).

Abstract: We give a simple axiomatic description of the degree $0$ part of the polylogarithm on abelian schemes and show that its realisation in analytic Deligne cohomology can be described in terms of the Bismut-Köhler higher analytic torsion form of the Poincaré bundle.

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

[final version Christmas 2016] with H. Gillet,
Rational points of varieties with ample cotangent bundle over function fields. [PDF] (20 pages) To appear in Mathematische Annalen.

Abstract: Let $K$ be the function field of a smooth curve over an algebraically closed field $k$. Let $X$ be a scheme, which is smooth and projective over $K$. Suppose that the cotangent bundle $\Omega_{X/K}$ is ample. Let $R:={\rm Zar}(X(K)\cap X)$ be the Zariski closure of the set of all $K$-rational points of $X$, endowed with its reduced induced structure. We prove that for each irreducible component ${\mathfrak R}$ of $R$, there is a projective variety ${\mathfrak R}'_0$ over $k$ and a finite and surjective $K^{\rm sep}$-morphism ${\mathfrak R}'_{0,K^{\rm sep}}\to {\mathfrak R}_{K^{\rm sep}}$, which is birational when ${\rm char}(K)=0$.
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.

(25) On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic. [PDF] (12 pages) Commentarii Mathematici Helvetici 90 (2015), 23--32.

Abstract: Let $K$ be the function field of a smooth and proper curve $S$ over an algebraically closed field $k$ of characteristic $p>0$. Let $A$ be an ordinary abelian variety over $K$. Suppose that the Néron model $\cal A$ of $A$ over $S$ has some closed fibre ${\cal A}_s$, which is an abelian variety of $p$-rank $0$.
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.

(24) (comme éditeur, avec B. Halimi et S. Maronne) Introduction  : la théorie de l’homotopie en perspective. Annales de la faculté des sciences de Toulouse Sér. 6 22 (2013), no. 5, p. i-vii : Numéro Spécial à l’occasion du Workshop Homotopie, 20-21 octobre 2011, Institut mathématique de Toulouse.

(23) On the Manin-Mumford and Mordell-Lang conjectures in positive characteristic [PDF] Algebra and Number Theory 7 (2013), no. 8, 2039--2057.

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] Notre Dame Journal of Formal Logic 54 (2013),  no. 3-4, 579--589.

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 The Brauer-Manin obstruction for subvarieties of abelian varieties over function fields, Ann. of Math. (2) 171 (2010), no. 1, 511--532, by B. Poonen and J-F Voloch.

(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. MaillotOn 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 Lectures on the Arithmetic Riemann-Roch Theorem (Princeton Univ. Press, Annals of Mathematics Studies 127). Our proof is based on Bismut's immersion theorem for the higher analytic torsion form.

(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. 330 (2004), no. 2, 293--308.

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, On the periods of motives with complex multiplication and a conjecture of Gross-Deligne.[PDF] Annals of Math. 160 (2004), 727--754.

Abstract: We prove that the existence of an automorphism of finite order on a ${\bf Q}$-variety $X$ implies the existence of algebraic linear relations between the logarithm of certain periods of $X$ and the logarithm of special values of the $\Gamma$-function. This implies that a slight variation of results by Anderson, Colmez and Gross on the periods of CM abelian varieties is valid for a larger class of CM motives. In particular, we prove a weak form of the period conjecture of Gross-Deligne (this should not be confused with the conjecture by Deligne relating periods and values of $L$-functions.). Our proof relies on the arithmetic fixed point formula (equivariant arithmetic Riemann-Roch theorem) proved by K. K\"ohler and the second author and the vanishing of the equivariant analytic torsion for the de Rham complex.
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.

Abstract: Let A be a semiabelian variety over an algebraically closed field of arbitrary characteristic, endowed with a finite morphism psi from A to A. In this paper we give an essentially complete classification of all psi-invariant subvarieties of A.
For example, under some mild assumptions on (A,psi) we prove that every psi-invariant subvariety is a finite union of translates of semiabelian subvarieties. This result is then used to prove the Manin-Mumford conjecture in arbitrary characteristic and in full generality. Previously, it had been known only for the group of torsion points of order prime to the characteristic of K. The proofs involve only algebraic geometry, though scheme theory and some arithmetic arguments cannot be avoided.

(8) with R. Pink, On Hrushovski's proof of the Manin-Mumford conjecture.[PDF] Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 539--546, Higher Ed. Press, Beijing, 2002.

Abstract: The Manin-Mumford conjecture in characteristic zero was first proved by Raynaud. Later, Hrushovski gave a different proof using model theory. His main result from model theory, when applied to abelian varieties, can be rephrased in terms of algebraic geometry. In this paper we prove that intervening result using classical algebraic geometry alone. Altogether, this yields a new proof of the Manin-Mumford conjecture using only classical algebraic geometry.
About this article, see also the following letter of J. Oesterlé to D. Roessler.

(7)  avec V. Maillot, Conjectures sur les dérivées logarithmiques des fonctions $L$ d'Artin aux entiers négatifs.[PDF] Math. Res. Lett. 9 (2002), no. 5-6, 715--724.

Résumé: Nous formulons plusieurs variantes d'une conjecture reliant le degré arithmétique de certains fibrés hermitiens aux valeurs prises aux entiers négatifs par la dérivée logarithmique des fonctions $L$ d'Artin des corps CM.
Cette conjecture peut être vue comme une vaste généralisation de la formule de Chowla et Selberg évaluant les périodes des courbes elliptiques CM en termes de la fonction $\Gamma$. Nous annoncons plusieurs résultats en direction de ces énoncés.

Abstract: We formulate several variants of a conjecture relating the arithmetic degree of certain hermitian fibre bundles with the values of the logarithmic derivative of Artin’s L functions at negative integers.
This conjecture can be viewed as a far-reaching generalisation of the formula of Chowla and Selberg, which compute the periods of CM elliptic curves in terms of the $\Gamma$-function. We announce several results in the direction of these statements.
(6)  with  K. Koehler, A fixed point formula of Lefschetz type in Arakelov geometry II: a residue formula.[PDF] Ann. Inst. Fourier 52 (2002), no. 1, 81--103.

Abstract: This is the second of a series of papers dealing with an Arakelovian analog of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; here recent results of Bismut-Goette on the equivariant analytic torsion play a key role.
(5)  with  K. Koehler, A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof.[PDF] Invent. Math. 145 (2001), no. 2, 333--396; announced [PDF] in C. R. Acad. Sci. 326 (1998), 719--722.

Abstract: We consider arithmetic varieties endowed with an action of the group scheme of n-th roots of unity and we define equivariant arithmetic Grothendieck groups for these varieties. We use the equivariant analytic torsion to define direct image maps in this context and we prove a Riemann-Roch theorem for the natural transformation of equivariant arithmetic Grothendieck groups induced by the restriction to the fixed point scheme; this theorem can be viewed as an analog, in the context of Arakelov geometry, of the regular case of the theorem proved by P. Baum, W. Fulton and G. Quart. We show that it implies an equivariant refinement of the arithmetic Riemann-Roch theorem, in a form conjectured by J.-M. Bismut.
(4) Lambda structure on arithmetic Grothendieck groups.[PDF] Israel J. Math. 122 (2001), 279--304; announced [PDF] in C. R. Acad. Sci. 322 (1996), 251--254.

Abstract: We define a "compactification" of the representation ring of the linear group scheme over ${\rm Spec} {\bf Z}$, in the spirit of Arakelov geometry. We show that it is a $\lambda$-ring which is canonically isomorphic to a localized polynomial ring and that it plays a universal role with respect to natural operations on the $K_{0}$-theory of hermitian bundles defined by Gillet-Soul\'e. As a byproduct, we prove that the natural pre-$\lambda$-ring structure of the $K_{0}$-theory of hermitian bundles is a $\lambda$-ring structure.

(3)  with K. Koehler, A fixed point formula of Lefschetz type in Arakelov geometry IV: the modular height of C.M. abelian varieties.[PDF] J. reine angew. Math. 556 (2003), 127--148.

Abstract: We give a new proof of a slightly weaker form of a theorem of P. Colmez. This theorem gives a formula for the Faltings height of abelian varieties with complex multiplication by a C.M. field whose Galois group over Q is abelian; it reduces to the formula of Chowla and Selberg in the case of elliptic curves. We show that the formula can be deduced from the arithmetic fixed point formula proved in [KR2]. Our proof is intrinsic in the sense that it does not rely on the computation of the periods of any particular abelian variety.

(2) Analytic torsion forms for cubes of vector bundles and Gillet's Riemann-Roch theorem.[PDF] J. Algebraic Geom. 8 (1999), 497--518.

Abstract: We present an analytic proof of Gillet's Riemann-Roch theorem for the Beilinson regulator in the case of compact fibrations, thereby extending to higher $K$-theory the analytic approach to the Grothendieck-Riemann-Roch theorem.
Our proof depends essentially on Burgos-Wang's description of the regulator and on the properties of Bismut-K\"ohler's higher analytic torsion forms. Moreover, our proof shows how to define analogs of these analytic torsion forms for cubes of vector bundles.

(1) An Adams-Riemann-Roch theorem in Arakelov geometry.[PDF] Duke Math. J. 96 (1999), 61--126; announced [PDF] in C. R. Acad. Sci. 322 (1996), 749--752.

Abstract: We prove an analog of the classical Riemann-Roch theorem for Adams operations acting on K-theory, in the context of Arakelov geometry.

(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.