Journées Arithmétiquesà Villetaneuse
16-17 mars 2017
Université Paris 13
Institut GaliléeJeudi 16 mars (Amphi A)
Orateurs: G. Böckle, S. Brochard, M. Harris

Vendredi 17 mars (Salle Darwin)
Orateurs: D. Disegni, D. Loeffler, E. Urban, S. Zerbes

Organisateurs: Farrell Brumley, Cédric Pépin, Alberto Vezzani

Programme du 16 mars– Amphi A –• 10h Accueil / Café viennoiseries• 10h30-12h Gebhard Böckle (Heidelberg): Potential automorphy over global function
fields I• 13h-14h30 Michael Harris (Columbia, Paris 7): Potential automorphy over global
function fields II
Abstract: Let G be a split reductive group over a finite field Fq and let K be a global
function field over Fq with adele ring AK . By fundamental work of Vincent Lafforgue
any cuspidal automorphic representation of G(AK ) gives rise to a compatible systemof Galois representation of Gal(K sep /K) valued in the dual group Ĝ of G. Suppose
conversely that we are given an l-adic representation ρ of Gal(K sep /K) into Ĝ. If ρ ispotentially unramified and has Zariski dense image, then in joint work with C. Khareand J. Thorne we show that ρ is potentially automorphic, i.e., there is a finite extension
K 0 of K, such that the restriction of ρ to Gal(K sep /K 0 ) arises from V. Lafforgue’sconstruction over K 0 . The talk aims at explaining some of the ideas that go into theproof of the result and the main steps, and at suggesting some potential applications.
• 15h-16h Sylvain Brochard (Montpellier): De Smit’s conjecture: a new freeness criterion
Abstract: Let A → B be a morphism of Artin local rings with the same embedding
dimension (the embedding dimension of a local ring is the dimension of its tangent spaceover its residue field). Bart de Smit conjectured in 1997 that any A-flat B-module isautomatically B-flat. We will first recall the context in which that conjecture arose, andhow it allows to slightly simplify Wiles’ proof of Fermat’s Last Theorem. We will thenexplain the main lines of its proof. Finally we will discuss some possible continuations.

Programme du 17 mars

– Salle Darwin –• 9h30 Accueil / Café viennoiseries• 10h-11h Eric Urban (Columbia): Euler systems and deformations of Galois representations
Abstract: The purpose of this talk is to present a new and general construction of Eulersystems. In particular, I will discuss a case which can be solved completely and which
gives the construction of an higher rank Euler system together with its relation to p-adic
L-functions.
• 11h-11h20 Pause Café
• 11h20-12h20 Sarah Zerbes (University College London): An Euler system for Siegel
modular forms
Abstract: I will outline the construction of an Euler system for the spin Galois representation attached to a genus two Siegel modular form. This is joint work with DavidLoeffler and Chris Skinner.
• 13h45-14h45 David Loeffler (Warwick): Iwasawa theory for the symmetric square ofa modular form
Abstract: In this talk, I will describe some recent results (joint with Sarah Zerbes) on
the arithmetic of the 3-dimensional symmetric square Galois representation attached toa modular form. We use the Euler system of Beilinson–Flach elements, introduced inour earlier works with Lei and with Kings, to prove an upper bound for the Selmer groupof this Galois representation over the p-adic cyclotomic tower, in terms of a suitablep-adic L-function.
• 15h-16h Daniel Disegni (Orsay): The universal p-adic Gross-Zagier formula
Abstract: Following Howard and Fouquet, one can construct a ’universal Heegner point’:
it is a section of a sheaf of Selmer groups over the ordinary locus in the eigenvariety forGL(2)xU(1) over a totally real field. I will tall about a general formula for its height interms of derivatives of p-adic L-functions. It has applications to the p-adic analogue ofthe Bloch-Kato conjecture and to Iwasawa theory of Hilbert modular forms.

Pour venir au LAGA:• En voiture, à partir de Paris:– Porte de la Chapelle Autoroute A1 [direction Lille]– Sortie numéro 2 (Saint-Denis - Stade de France)– puis direction Villetaneuse Université• En transports en commun depuis Paris– Train ligne H, de la Gare du Nord (quais 30 à 36), jusqu’à la gare d’EpinayVilletaneuse. [En Gare du Nord, suivre l’une des directions Persan-Beaumont,Valmondois, Monsoult-Maffliers ou Pontoise en vérifiant, sur le quai de départ,que le train s’arrête en gare d’Epinay-Villetaneuse] Gare d’Epinay-Villetaneuse,sortie côté Villetaneuse puis bus 156 [direction Université Paris 13] ou bus 361[direction Gare de Pierrefite-Stains RER) jusqu’à l’arrêt Université Paris 13.
– Vous pouvez également emprunter le RER D, la ligne 13 ou encore la ligne H, puis
changer pour le Tramway T8 et descendre au terminus Villetaneuse Université.

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