Publications
(31) [NEW
June 7th 2015] with T. Szamuely, Cohomology and torsion cycles
over the maximal cyclotomic extension.[PDF] (18 pages)
Preprint.
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 \'etale
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.
(30) Strongly semistable sheaves
and the Mordell-Lang conjecture over function fields.[PDF]
(14
pages) Preprint.
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.
(29) with G.
Kings, Higher
analytic torsion, polylogarithms and norm compatible
elements on abelian schemes.[PDF] (27 pages)
Submitted.
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.
(28) [formulation of main theorem corrected
13.04.2015] with H. Gillet,
Rational points of
varieties with ample cotangent bundle over function fields.
[PDF] (20 pages) Submited.
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 $R_0$ of $R$, there is a projective variety
$X_0$ over $k$ and a finite and surjective $K^{\rm
sep}$-morphism $X_{0,K^{\rm sep}}\to R_{0,K^{\rm sep}}$, which is
birational when ${\rm char}(K)=0$.
Using our result, one can give the first examples of varieties,
which are not embeddable in abelian varieties and satisfy a positive
characteristic 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).
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. 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 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.
(MATHSCINET
FEATURED
REVIEW
DECEMBER 2003)
(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.
(MATHSCINET
FEATURED
REVIEW
DECEMBER 2003)
(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.