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MATHEMATISCHES FORSCHUNGSINSTITUT OBERWOLFACH T a g u n g s b e r i c h t Algebraic Number Theory MATHEMATISCHES FORSCHUNGSINSTITUT OBERWOLFACH T a g u n g s b e r i c h t Algebraic Number Theory The meeting has b een organized by Christopher Deninger M unster Peter Schneider M unster and Anthony Scholl Durham The sub ject of the conference was Algebraic Number Theory and Arithmetic Algebraic Ge ometry In particular the following asp ects have b een treated in the talks Iwasawa Theory Galois group actions Mo dular Forms and Galois representations Langlands corresp ondences padic Ho dge Theory rigid analytic metho ds in Arithmetic Geometry classical and padic p olylogarithms Higher dimensional class eld theory Cycles and Chow groups Some sp eakers rep orted on their pro ofs of outstanding conjectures Henniart Laorgue on lo cal resp ectively global Langlands corresp ondences Fontaine on the weakly admiss admissconjecture joint work with Colmez VORTRAGSAUSZUGE Amnon Besser BeerSheva Israel padic regulators for K theory of elds and padic p olylogarithms joint work with Rob de Jeu Goncharov conjectured the existence of complexes for a eld F n F n B B F B F F n n n Q Q Q in degrees n where B is generated by symbols x x mathbbP F f g and an isomorphism n n r r H F n H F Q n Zagier further conjectured that when F is a number eld the M comp osition n1 H F nH F Q n H F R n R M D F C where the eigenspace with resp ect to the action of complex conjugation on the embeddings is given by x P x where P is a version of the complex p olylogarithm de Jeu n Z ag n Z ag n constructed a complex with similar prop erties and a map H H F Q n M M n n M and proved the analogue of Zagiers conjecture In this work we treat the padic analogue when K Q we have a syntomic regulator p H K Q n K Gros Niziol Nekovar Besser We state the following conjecture M Conjecture If F K where F is a number eld then the comp osed r eg H H F Q n H K Q n K M p n M M is given by x L x where L is a version of Colemans padic p olylogarithm n mo dn mo dn We prove this conjecture when jxj jx j and derive consequences for syntomic regu lators of cyclotomic elements in K theory Don Blasius UCLA Motives for Hilb ert Mo dular Forms According to a basic theorem of Shimura there is attached to each holomorphic newform f of weight with rational Hecke eigenvalues an elliptic curve E dened over Q such that f LE s Lf s It is exp ected that this result will generalize to totally real elds F the f result is known if F Q is o dd or f b elongs to the discrete series at some nite prime v We show that if the Ho dge Conjecture is true then the sought elliptic curve always exists More generally if the Hecke eld of f is T there exists under the same hypothesis an ab elian f variety A dened over F with dim A T Q having a natural action of T such that f f f f LA s Lf s Thus Shimuras full result over Q will generalize to F f For metho ds we employ base change and the JacquetLanglands corresp ondence to nd a motive for LSym f s in the cohomology of a Shimura surface A formal argument pro duces over C an ab elian variety B of dimension T Q whose H contains a submotive f isomorphic to S y m f The Ho dge conjecture is invoked to descend B to a number eld L and relate its l adic representations to those attached to f by Taylor From this p oint the conclusion follows easily that the sought A is a factor of R B f LF It may b e hop ed that this result may b e proved by direct construction using for example Shimura varieties attached to unitary groups However the construction ab ove leads to the following general question Is the MumfordTate group of the category of motives dened by Shimura varieties ie o ccuring in their cohomology simplyconnected Christophe Breuil CNRS Universite dOrsay Classication of group schemes and applications padic Riemann conditions ShimuraTaniyamaWeil conjecture Let p b e a prime p k a p erfect eld of char p W W k the Witt vectors K Frac W K a nite totally ramied extension of K O the ring of integers and K a xed uniformizer in O K ie u Let S b e the padic completion of W u Fil S the kernel of the map S O u iN K i p 1 and S S the lifting of Frobenius induced by u u Let j one has Fil S p Fil S S Dene Mo dS to b e the category whose ob jects are an S mo dule M a submo dule Fil M Fil S M (E (u)x) 1 a semilinear application Fil M M st sx s (E (u)) 1 s Fil S x M where E u in the minimal p olynomial of and whose morphisms are the obvious com patible ones Finally let Mo dF I S b e the full sub category of Mo dS such that n i M Sp S jI j iI Fil M spans M over S and dene a strongly divisible mo dule to b e any M in Mo dS st d M S Fil M spans M over S M Fil M has no ptorsion Theorem There is an equivalence of categories b etween M odF I S and the category of n nite at pgroups over O such that k er p is still at over O for any n This equivalence K K preserves short exact sequences Theorem There is an equivalence of categories b etween strongly divisible mo dules and pdivisible groups over O K The pro of uses syntomic techniques introduced by Mazur Fontane Messing in the case where K K Using this on can get Theorem padic Riemann conditions Assume k F Then every crystalline repre p Q for K K with Ho dgeTate weights f g is isomorphic to T G sentation V of Gal p p Z p some pdivisible group GO K This is already known for K K p Fontaine Messing Laaille Corollary Let A b e an ab elian variety over K and k F p i A has go o d reduction over O T A Q is crystalline K p p ii A has semistable reduction over O T A Q is semistable K p p Corollary The p erio d map of Rap op ortZink for pdivisible groups is surjective Corollary i was known Fontaine Coleman Iovita the implication of ii was known b efore to o Fontaine The ob jects killed by p of Mo dF I S classify al l nite at group schemes over O killed by p By developing a decent data formalism on such ob jects K and using it to compute the dimension of tangent spaces of sp ecic deformation problems of p otential BarsottiTate representations joint work with B Conrad F Diamond and R Taylor one can settle the last cases of the ShimuraTaniyamaWeil conjecture following of course Wiles Taylors strategy Thus Theorem Wiles Every ellipic curve over Q is mo dular Kevin Buzzard Imp College London Galois representations and weight mo dular forms My talk was a rep ort on the current status of Taylors program for attacking Artins con jecture for representations Gal Q Q GL C and also contained some remarks on some of the technicalities b ehind one of the results used in the program The talk centered around a theorem of myself and Taylor which broadly sp eaking was a weight analogue of Wiles lifting theorem if is mo dular and lots of technical conditions hold then is mo dular G GL Z etc Q p I explained how this lifting theorem already gave new examples of Artins conjecture and also how mo dulo some technical details would give many more new examples The key problem is that to get these many new examples one has to prove many standard results ab out mo d p mo dular forms in the case when p The main standard result which is still unproved is a companion forms result proved by Gross mo dulo unchecked compatibilities for all primes and by Coleman and Voloch not assuming these compatibilities for p It now seems that if this result were to b e unconditionally established for p then one would Q Q GL C which b e able to prove that many icosahedral Galois representations Gal were o dd would come from weight mo dular forms and would hence have the prop erty that their Lfunctions had analytic continuation and functional equation Henri Darmon McGill University Montreal The padic Birch and SwinnertonDyer Conjecture This talk summarizes the results of dierent works all joint with Massimo Bertolini and Adrian Iovita and Michael Spie Let E Q b e an elliptic curve and LE s its HasseWeil zetafunction Thanks to Breuils lecture we know E is mo dular so there exists H N E C R s s and sLE s y f iy dy where if z dz so that f is a cusp form of weight on n The Birch and SwinnertonDyer conjecture predicts ord LE s rank E Q s But b oth conjectured inequalities are very mysterious If E Q has multiplicative reduction there is a rigid analytic uniformization H p p E C with a padic arithmetic subgroup of SL Q Following an idea of Schneider p p and IovitaSpie we dene L E s by taking a kind of padic Mellin transform of the rigid p analytic mo dular form of weight on attached to E We are then able to give formulae for L E relating these values to padic p erio ds and Heegner p oints p on E extend the denition of L f to forms f of higher even weight k and express p k L f in terms of an Linvariant of Teitelbaum and in terms of certain Heegner p cycles in the Chow groups of KugaSato varieties joint work with Iovita and Spie prove ord L E s rank E Q in fact we supply a more precise lower b ound for s p ord L E s which is conjectually sharp s p Jonathan Dee Cambridge UK Chow Groups for selfpro ducts of CM Elliptic Curves I discuss certain prop erties of the image of the cycle class map for cycles of arbitrary co di mension on the dfold selfpro duct of an elliptic curve over Q with complex multiplication Theorem Let E jQ b e an elliptic curve with complex multiplication by O

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