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Arakelov theory & Automorphic forms June 6-9, Berlin A conference at the occasion of J¨ urg Kramer’s 60th birthday Organizers: Gavril Farkas Ulf K¨ uhn Anna von Pippich Gisbert W¨ ustholz
Arakelov theory &Automorphic forms
June 6-9, Berlin
A conference at the occasion ofJurg Kramer’s 60th birthday
Organizers: Gavril FarkasUlf KuhnAnna von PippichGisbert Wustholz
Graphic on the frontpage by Natascha [email protected]
Contents
1 Program 3
2 Abstracts 6
3 Practical information 14
3.1 Lecture hall 14
3.2 Restaurants 15
3.3 Conference dinner 15
3.4 Contact information 15
4 Participants 16
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Acknowledgements
This conference is organized within the International DFG Re-search Training Group Moduli and Automorphic Forms.
We gratefully acknowledge financial support by the GermanResearch Foundation (DFG), the Humboldt-Universitat zu Ber-lin, the Mathematics Department of Humboldt-Universitat zuBerlin, the Berlin Mathematical School (BMS), and the Ein-stein Center for Mathematics Berlin (ECMath).
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1 Program
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Monday, June 6
9:30 Registration
10:30 Coffee break
11:00 Kudla Modularity of generating series for special di-visors on arithmetic ball quotients I
12:00 Ullmo Flows on abelian varieties and Shimura vari-eties
13:00 Lunch break
15:00 Bruinier Modularity of generating series for special di-visors on arithmetic ball quotients II
16:00 Coffee break
16:30 Muller-Stach Arakelov inequalities for special subvarietiesin Mumford–Tate varieties
Tuesday, June 7
9:30 Jorgenson Dedekind sums associated to higher orderEisenstein series
10:30 Coffee break
11:00 Zhang Congruent number problem and BSD conjec-ture
12:00 Imamoglu Another look at the Kronecker limit formulas
13:00 Lunch break
15:00 Michel Moments of L-functions and exponential sums
16:00 Coffee break
16:30 Soule Asymptotic semi-stability of lattices of sec-tions
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Wednesday, June 8
9:30 Freixas Generalizations of the arithmetic Riemann–Roch formula
10:30 Coffee break
11:00 Faltings Arakelov theory on degenerating curves
12:00 Gillet The fiber of a cycle class map
13:00 Lunch break
15:00 Burgos The singularities of the invariant metric of thePoincare bundle
16:00 Coffee break
16:30 BostSpecial lecture
Theta series, euclidean lattices, and Arakelovgeometry
Thursday, June 9
9:30 Edixhoven Gauss composition on primitive integralpoints on spheres, partly following Gunawan
10:30 Coffee break
11:00 Gubler On the pointwise convergence of semipositivemodel metrics
12:00 Viazovska The sphere packing problem in dimensions 8and 24
13:00 Lunch break
15:00 van der Geer Modular forms of low genus
16:00 Coffee break
16:30 Salvati Manni On the 2-torsion points of the theta divisor
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2 Abstracts
Jean-Benoıt BostTheta series, euclidean lattices, and Arakelov geometryUniversite Paris-Sud, Orsay, France
A euclidean lattice is the data (E, ‖.‖) of some Z-module E isomor-phic to Zr, r ∈ N, and of some Euclidean norm ‖.‖ on the associatedR-vector space ER ' Rr. To any Euclidean lattice (E, ‖.‖), one mayattach its theta series
θ(t) :=∑v∈E
e−πt‖v‖2
and the non-negative real number
h0θ(E, ‖.‖) := log θ(1) = log
∑v∈E
e−π‖v‖2.
This talk will explain why this invariant h0θ(E, ‖.‖) is a “natural”
one, from diverse points of view, including large deviations andthe thermodynamic formalism, and the classical analogy betweennumber fields and function fields and its modern developments inArakelov geometry.
I will also discuss some extensions of the invariant h0θ to certain
infinite dimensional generalizations of euclidean lattices, and willpresent some applications to transcendence theory and to alge-braization theorems in Diophantine geometry.
Jan Hendrik BruinierModularity of generating series for special divisors on arithmeticball quotients IITechnische Universitat Darmstadt, Germany
The second lecture (for the first lecture, see Kudla) will sketch theproof of the main theorem, which uses the Borcherds modularitycriterion. The essential point is to determine the divisor of the
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unitary group Borcherds forms on the integral model. This dependson an analysis of the first Fourier–Jacobi coefficient of such a form.Time permitting, the second part of the lecture will give a briefdescription of how the main theorem can be used to establish newcases of the Colmez conjecture.
Jose Ignacio Burgos GilThe singularities of the invariant metric of the Poincare bundleInstituto de Ciencias Matematicas, Madrid, Spain
With J. Kramer and U. Kuhn we have studied the singularities ofthe line bundle of Jacobi forms on the universal elliptic curve. Thenatural generalization of this work is to study the singularities ofthe invariant metric on the Poincare bundle on a family of abelianvarieties and their duals.The singularities of the invariant metric are an avatar of the heightjump phenomenon discovered by R. Hain. In a joint work withD. Holmes and R. de Jong, we prove a particular case of a conjectureby Hain on the positivity of the height jump.Moreover, in joint work with O. Amini, S. Bloch and J. Fresan,we show that the asymptotic behaviour of the height pairing ofzero cycles of degree zero on a family of smooth curves, when oneapproaches a singular stable curve, is governed by the Symanzikpolynomials of the dual graph of the singular curve. This fact isrelated to the physical intuition that the asymptotic behaviour ofstring theory when the length parameter goes to zero is governedby quantum field theory of particles.In this talk I will review the joint work with Jurg as well as thegeneralizations.
Bas EdixhovenGauss composition on primitive integral points on spheres, partlyfollowing GunawanUniversiteit Leiden, Netherlands
Gauss has given formulas for the number of primitive integral pointson the 2-sphere of radius squared n. These formulas are in terms of
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class numbers of imaginary quadratic orders of discriminants closelyrelated to n. It makes one wonder if this is explained by a free andtransitive action of the Picard group of this order on the set of suchprimitive integral points up to global symmetry SO3(Z). This isindeed the case, and the action can be made explicit. The tool thatis used is group schemes over Z, which is more direct than Galoiscohomology plus adeles, and surprisingly elementary. In fact, Bhar-gava and Gross ask for such an approach in their article “Arithmeticinvariant theory”.
Reference: https://openaccess.leidenuniv.nl/handle/1887/38431
Gerd FaltingsArakelov theory on degenerating curvesMax-Planck-Institut fur Mathematik, Bonn, Germany
For a semistable family of curves we investigate the asymptoticsof the Arakelov metric. As an application we get results on theasymptotic of the delta-function.
Gerard Freixas i MontpletGeneralizations of the arithmetic Riemann–Roch formulaCentre National de la Recherche Scientifique, Paris, France
In this talk I review recent extensions of the arithmetic Riemann–Roch theorem, related to the program of Burgos–Kramer–Kuhnabout extending arithmetic intersection theory, in order to deal withtoroidal compactifications of Shimura varieties. Precisely, I will firststate a Riemann–Roch isometry for the trivial sheaf on a cusp com-pactification of a hyperbolic orbicurve. As a consequence I will givethe special value at one of the derivative of the Selberg zeta functionof PSL2(Z), that turns out to “contain” the Faltings heights of theCM elliptic curves corresponding to the elliptic fixed points. Thisfirst part is joint work with Anna von Pippich. Then I will reporton joint work with Dennis Eriksson and Siddarth Sankaran, whoseaim is to explore what can be done in higher dimensions. We focus
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on Hilbert modular surfaces. We give a possible definition of theholomorphic analytic torsion of the trivial sheaf, and by using theJacquet–Langlands correspondence we prove it fits in a weak formof a conjectural arithmetic Riemann–Roch formula. This formulashould contain a boundary contribution given by derivatives at 0 ofShimizu L-functions, in analogy with Hirzebruch’s computations inthe geometric case.
Gerard van der GeerModular forms of low genusUniversiteit van Amsterdam, Netherlands
We give a survey of recent work on Siegel modular forms for genus2 and 3. This is based on joint work with Bergstroem and Faberand with Clery.
Henri GilletThe fiber of a cycle class mapUniversity of Illinois at Chicago, USA
I shall discuss how one might use Milnor K-theory and Nash func-tions to attempt to understand the fiber of (a version of) the cycleclass map.
Walter GublerOn the pointwise convergence of semipositive model metricsUniversitat Regensburg, Germany
At a non-archimedean place, we will show that pointwise conver-gence of semipositive model metrics to a model metric yields thatthe limit metric is semipositive. Using multiplier ideals, this wasshown by Boucksom, Favre and Jonsson under the assumption thatthe residue characteristic is zero. We will prove the claim in generalusing a different strategy. This is joint work with Florent Martin.
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Ozlem ImamogluAnother look at the Kronecker limit formulasEidgenossische Technische Hochschule Zurich, Switzerland
Classical Kronecker limit formulas lead to some beautiful relationsbetween quadratic fields and L-functions. In this talk I will usethem to motivate the study of a new geometric invariant associatedto an ideal class of a real quadratic field. This is joint work withW. Duke and A. Toth.
Jay JorgensonDedekind sums associated to higher order Eisenstein seriesThe City College of New York, USA
Higher order Eisenstein series are, roughly speaking, defined as aseries similar to non-holomorphic Eisenstein series with the inclu-sion of factors involving periods of certain holomorphic forms. Inthis talk we describe results yielding generalizations of Kronecker’slimit formula as well as Dedekind sums associated to the Kroneckerlimit function. Specific evaluations are obtained for certain arith-metic groups. The work is joint with Cormac O’Sullivan and LejlaSmajlovic.
Stephen KudlaModularity of generating series for special divisors on arithmeticball quotients IUniversity of Toronto, Canada
In this pair of talks we will report on joint work with B. Howard, M.Rapoport and T. Yang on the modularity of the generating seriesfor special divisors on integral models of certain ball quotients.
The first lecture (for the second lecture, see Bruinier) will introducethe basic objects: regular integral models of Shimura varieties as-sociated to U(n− 1, 1) and their compactifications, special divisorsand their extensions to the compactifications. Then the construc-tion of Green functions for these divisors by means of Borcherdslifts of harmonic weak Maass forms will be described. Finally, the
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generating series will be defined and the main theorem concerningits modularity will be stated.
Philippe MichelMoments of L-functions and exponential sumsEcole Polytechnique Federale de Lausanne, Switzerland
In this talk, we explain a solution to the long-standing problem ofevaluating asymptotically with a power saving error term, the sec-ond moment of the central value of L-functions of a modular formtwisted by Dirichlet characters of prime modulus. For Eisensteinseries, the problem amounts to evaluating the fourth moment ofcentral values of Dirichlet L-functions and was solved by Young afew years ago. We will discuss the more recent and difficult case ofcusp forms whose proof is a combination of various ingredients in-cluding the analytic theory of automorphic forms, analytic numbertheory, and l-adic cohomology. These are joint works with Blomer,Fouvry, Kowalski, Milicevic, and Sawin.
Stefan Muller-StachArakelov inequalities for special subvarieties in Mumford–Tate va-rietiesJohannes Gutenberg-Universitat Mainz, Germany
We characterize positive-dimensional special subvarieties in Mum-ford–Tate varieties by certain new Arakelov inequalities. As a con-sequence, we get an effective version of the Andre–Oort conjecturein this more general case.
Riccardo Salvati ManniOn the 2-torsion points of the theta divisorLa Sapienza – Universita di Roma, Italy
I will present some recent results, obtained in collaboration withPirola and Auffarth, on the bound for the number of 2-torsion
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points that lie on a given theta divisor. I will present also severalapproaches to attack a conjectural result.
Christophe SouleAsymptotic semi-stability of lattices of sectionsInstitut des Hautes Etudes Scientifiques Paris, France
This is joint work with T. Chinburg and Q. Guignard. Let L bea hermitian line bundle on an arithmetic surface. We consider thelattice of global sections of a high power n of L, equipped with thesup-norm. We ask whether this lattice becomes semi-stable whenn goes to infinity. We get a positive answer when working withcapacities and, in the Arakelov set-up, we obtain an interestinginequality, too weak to imply semi-stability.
Emmanuel UllmoFlows on abelian varieties and Shimura varietiesInstitut des Hautes Etudes Scientifiques Paris, France
I will discuss several questions and some results about algebraicflows, o-minimal flows and holomorphic flows on abelian varietiesand Shimura varieties.
Maryna ViazovskaThe sphere packing problem in dimensions 8 and 24Humboldt-Universitat zu Berlin, Germany
In this talk we will show that the sphere packing problem in di-mensions 8 and 24 can be solved by a linear programing method.In 2003 N. Elkies and H. Cohn proved that the existence of a realfunction satisfying certain constraints leads to an upper bound forthe sphere packing constant. Using this method they obtained al-most sharp estimates in dimensions 8 and 24. We will show thatfunctions providing exact bounds can be constructed explicitly ascertain integral transforms of modular forms. Therefore, we solvethe sphere packing problem in dimensions 8 and 24.
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Shou-Wu ZhangCongruent number problem and BSD conjecturePrinceton University, USA
A thousand years old problem is to determine when a square freeinteger n is a congruent number, i.e., the areas of right angledtriangles with sides of rational lengths. This problem has somebeautiful connection with the BSD conjecture for elliptic curvesEn : ny2 = x3 − x. In fact by BSD, all n = 5, 6, 7 mod 8 shouldbe congruent numbers, and most of n = 1, 2, 3 mod 8 should not becongruent numbers. Recently, Alex Smith has proved that at least41.9% of n = 1, 2, 3 mod 8 satisfy (refined) BSD in rank 0, and atleast 55.9% of n = 5, 6, 7 mod 8 satisfy (weak) BSD in rank 1. Thisimplies in particular that at least 41.9% of n = 1, 2, 3 mod 8 arenot congruent numbers, and 55.9% of n = 5, 6, 7 mod 8 are congru-ent numbers. I will explain the ingredients used in Smith’s proof:including the classical work of Heath-Brown and Monsky on thedistribution of the F2-rank of the Selmer group of En, the complexformula for the central value and the derivative of L-functions ofWaldspurger and Gross–Zagier and their extension by Yuan–Zhang–Zhang, and their mod 2 version by Tian–Yuan–Zhang.
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3 Practical information
3.1 Lecture hall
The lectures take place in the Festsaal of the Humboldt Graduate School,located in the second floor of Luisenstraße 56.
Map data c○ 2016 GeoBasis-DE/BKG ( c○ 2009), Google
The lecture hall is marked by a red arrow. The metro stations, marked by green arrows,are (starting from the top, proceeding clockwise): Naturkundemuseum (U6, 10 min),Oranienburger Tor (U6, 9 min), Friedrichstrasse (S1, S2, S25, S5, S7, S75, U6, 14 min),Bundestag (U55, 11 min) and Hauptbahnhof (S5, S7, S75, U55, 11 min). The bus stopCharite–Campus Mitte, for the buses 147 and TXL, is only 1 walking minute from thebuilding. Additional information on the transportation service can be found on thewebsite: www.bvg.de. The numbers refer to lunch possibilities.
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3.2 Restaurants
The area of Luisenstraße, specifically east of it, is rich in restaurants, cafes,and other lunch possibilities. Some of the closest to the lecture hall arelisted below. The numbers correspond to the ones on the map.
1. Mensa HU nord (cafeteria), Hannoversche Straße 7.Note that it is not possible to pay with cash. One should be either accompanied by a Berlin
student or buy a card (1,55 e) and charge it.
2. Cafeteria CKK (cafeteria), Hufelandweg 12.
3. Cafe Frau Schneider (cafe), Luisenstraße 13.
4. Thai Tasty (thai), Luisenstraße 14.
5. der Eisladen (ice cream), Luisenstraße 14.
6. Sabzi (oriental), Luisenstraße 15.
7. Thurmann (bakery), Luisenstraße 46.
8. Auf die Hand (fine fast food), Luisenstraße 45.
9. Mahlzeit Luise (bistro), Luisenstraße 39.
10. Trattoria Su e Giu (italian), Schumannstraße 16.
11. Ristorante Porta Nova (italian), Robert Koch Platz 12.
12. Mangiarbene di Giancarlo (italian), Platz vor dem neuen Tor 5.
13. Ristorante Cinque (italian), Reinhardtstraße 27.
14. Dehlers (modern german), Reinhardtstraße 14.
15. Monsieur Toche (modern european), Albrechtstrasse 19.
3.3 Conference dinner
The conference dinner will be on Wednesday, June 8 at 7pm at the restau-rant Sale e Tabacchi, in Rudi-Dutschke-Straße 25 (2 minutes walkingdistance from the U-Bahn station Kochstraße).
3.4 Contact information
For any question regarding the conference feel free to contact our secretary:
Marion ThommaPhone: +49 (0)30 2093 5815E-mail: [email protected]
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4 Participants
Agostini, [email protected]
Humboldt-Universitat zu BerlinGermany
Agricola, [email protected]
Phillips-Universitat MarburgGermany
Universite Paris 6France
Altmann, [email protected]
Freie Universitat BerlinGermany
Aryasomayajula, [email protected]
University of HyderabadIndia
Bach, [email protected]
Technische Universitat BraunschweigGermany
Bakker, [email protected]
Humboldt-Universitat zu BerlinGermany
Banaszak, [email protected]
Uniwersytet w PoznaniuPoland
Bost, Jean-Benoı[email protected]
Universite Paris-Sud, OrsayFrance
Botero, Ana Marı[email protected]
Humboldt-Universitat zu BerlinGermany
Bruinier, [email protected]
Technische Universitat DarmstadtGermany
Bruns, [email protected]
Humboldt-Universitat zu BerlinGermany
Buck, [email protected]
Technische Universitat DarmstadtGermany
Burgos Gil, Jose [email protected]
Instituto de Ciencias Matematicas, MadridSpain
Chipot, Renechipot [email protected]
Universitat Basel, Princeton UniversitySwitzerland, USA
Csige, [email protected]
Humboldt-Universitat zu BerlinGermany
D’Addezio, [email protected]
Freie Universitat BerlinGermany
De Gaetano, [email protected]
Humboldt-Universitat zu BerlinGermany
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Edixhoven, [email protected]
Universiteit LeidenNetherlands
Faghihi, [email protected]
Johannes Gutenberg-Universitat MainzGermany
Faltings, [email protected]
Max-Planck-Institut fur Mathematik, BonnGermany
Farkas, [email protected]
Humboldt-Universitat zu BerlinGermany
Freixas i Montplet, [email protected]
Centre National de la Recherche Scientifique, ParisFrance
Friedrich, [email protected]
Humboldt-Universitat zu BerlinGermany
Gajda, [email protected]
Uniwersytet w PoznaniuPoland
van der Geer, [email protected]
Universiteit van AmsterdamNetherlands
Gillet, [email protected]
University of Illinois at ChicagoUSA
Grados, [email protected]
Humboldt-Universitat zu BerlinGermany
Gubler, [email protected]
Universitat RegensburgGermany
Guignard, [email protected]
Institut des Hautes Etudes Scientifiques, ParisFrance
Hao, [email protected]
Freie Universitat BerlinGermany
University of Colorado BoulderUSA
Imamoglu, [email protected]
Eidgenossische Technische Hochschule ZurichSwitzerland
Imkeller, [email protected]
Humboldt-Universitat zu BerlinGermany
de Jong, [email protected]
Universiteit LeidenNetherlands
Jorgenson, [email protected]
The City College of New YorkUSA
Jung, [email protected]
Humboldt-Universitat zu BerlinGermany
Katsigianni, [email protected]
Freie Universitat BerlinGermany
Kleinert, [email protected]
Humboldt-Universitat zu BerlinGermany
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Kohler, [email protected]
Heinrich-Heine-Universitat DusseldorfGermany
Kornhuber, [email protected]
Freie Universitat BerlinGermany
Krason, [email protected]
Uniwersytet SzczecinskiPoland
Kudla, [email protected]
University of TorontoCanada
Kuchler, [email protected]
Humboldt-Universitat zu BerlinGermany
Kuhn, [email protected]
Universitat HamburgGermany
Kuhne, [email protected]
Max-Planck-Institut fur Mathematik, BonnGermany
Kunnemann, [email protected]
Universitat RegensburgGermany
Kurke, [email protected]
Humboldt-Universitat zu BerlinGermany
Lara, [email protected]
Freie Universitat BerlinGermany
Lavanda, [email protected]
Freie Universitat BerlinGermany
Technische Universitat DarmstadtGermany
Li, Yumengyumeng.li [email protected]
Berlin Mathematical SchoolGermany
Lindner, [email protected]
Leibniz Universitat HannoverGermany
van der Lugt, [email protected]
Universiteit LeidenNetherlands
Mandal, [email protected]
Humboldt-Universitat zu BerlinGermany
Martinez, [email protected]
Freie Universitat BerlinGermany
Michel, [email protected]
Ecole Polytechnique Federale de LausanneSwitzerland
Milano, [email protected]
State University of New York at BinghamtonUSA
Mocz, [email protected]
Princeton UniversityUSA
Mohajer, [email protected]
Johannes Gutenberg-Universitat MainzGermany
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Muller-Stach, [email protected]
Johannes Gutenberg-Universitat MainzGermany
Ortega, [email protected]
Humboldt-Universitat zu BerlinGermany
Pandharipande, [email protected]
Eidgenossische Technische Hochschule ZurichSwitzerland
Pineiro, [email protected]
Bronx Community CollegeUSA
von Pippich, [email protected]
Technische Universitat DarmstadtGermany
Proskurkin, [email protected]
Immanuel Kant Baltic Federal UniversityRussia
Reich, [email protected]
Freie Universitat BerlinGermany
Ren, [email protected]
Freie Universitat BerlinGermany
Roczen, [email protected]
Humboldt-Universitat zu BerlinGermany
Salvati Manni, [email protected]
La Sapienza – Universita di RomaItaly
Schwagenscheidt, [email protected]
Technische Universitat DarmstadtGermany
Sertoz, [email protected]
Humboldt-Universitat zu BerlinGermany
Smelov, [email protected]
Immanuel Kant Baltic Federal UniversityRussia
Sodoudi, [email protected]
Berlin Mathematical SchoolGermany
Soule, [email protected]
Institut des Hautes Etudes Scientifiques ParisFrance
Srivastava, Tanya [email protected]
Freie Universitat BerlinGermany
Sullivan, [email protected]
Technische Universitat BerlinGermany
Tewari, Ayush [email protected]
National Institute of Science Education andResearch BhubaneswarIndia
Tischendorf, [email protected]
Humboldt-Universitat zu BerlinGermany
Ullmo, [email protected]
Institut des Hautes Etudes Scientifiques, ParisFrance
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Ungureanu, [email protected]
Humboldt-Universitat zu BerlinGermany
Valloni, [email protected]
Universitat BonnGermany
Viada, [email protected]
Eidgenossische Technische Hochschule ZurichSwitzerland
Viazovska, [email protected]
Humboldt-Universitat zu BerlinGermany
Volz, [email protected]
Technische Universitat DarmstadtGermany
Warmuth, [email protected]
Humboldt-Universitat zu BerlinGermany
Wilms, [email protected]
Johannes Gutenberg-Universitat MainzGermany
Wisniewski, [email protected]
Technische Universitat BerlinGermany
Wright, [email protected]
Wofford CollegeUSA
Wustholz, [email protected]
Eidgenossische Technische Hochschule ZurichSwitzerland
Zhang, [email protected]
Princeton UniversityUSA
Zink, [email protected]
Humboldt-Universitat zu BerlinGermany
Zomervrucht, [email protected]
Universiteit Leiden, Freie Universitat BerlinNetherlands, Germany
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