Congruence sheaves and congruence differential equations Beyond hypergeometric functions
|
|
- Blaise Paul
- 5 years ago
- Views:
Transcription
1 Congruence sheaves and congruence differential equations Beyond hypergeometric functions Vasily Golyshev Lille, March 6, 204 / 45
2 Plan of talk Report on joint work in progress with Anton Mellit and Duco van Straten monodromy Calabi-Yau Correspondence Idea of Congruence sheaves Example of Outlook 2 / 45
3 Fano: complex projective variety F with positive K. Classification: dim : P dim 2: del Pezzo surfaces dim 3: H 2 (F,Z) = Z: Iskovskikh, 7 families; higher rank: Mori Mukai, 05 families 3 / 45
4 New approach to Classify Fanos by classifying mirror dual objects: Landau Ginzburg models; respective Picard Fuchs differential equations Golyshev, Classification problems and mirror duality Corti, Coates, Galkin, Golyshev, Kasprzyk, Mirror symmetry and Fano manifolds Pioneered for CY by van Straten van Enckevort. 4 / 45
5 A famous hypergeometric series φ(t) := n=0 5n! n! 5tn Satisfies hypergeometric differential equation Pφ(t) = 0 P := D t(d+ 5 )(D+ 2 5 )(D+ 3 5 )(D+ 4 5 ) D := t t Three singular points Σ := {0, /5 5, } P 5 / 45
6 Candelas, de la Ossa, Green, Parkes (99) Picard-Fuchs equation of family of Calabi-Yau threefolds. π : Y P rankh 3 (Y t ) = 4, h 3,0 = h 2, = h,2 = h 0,3 = Defines symplectic local system R 3 π (Z) on P \Σ Contains enumerative information of general quintic X P 4 : numbers of lines, conics, etc on X. 6 / 45
7 Logarithmic solution: q-coordinate: φ (t) := log(t) φ(t)+f (t), f (t) tq[[t]] q := exp(φ (t)/φ(t) = t+770t t Express P in q-coordinate P D 2 5 K(q) D2 K(q) = 5+ d n d d 3 q d q d n = 2875, n 2 = , n 3 = ,... 7 / 45
8 D 4 Nt(D +α )(D+α 2 )(D+α 3 )(D+α 4 ) α +α 4 = = α 2 +α 3 Mirror to complete intersections in weighted projective spaces. α α 2 α α / 45
9 Monodromy G = h 0,h,h GL n (C) 9 / 45
10 Differential equations D3 Differential operator of type D3 can be written as D 3 +t ( )( ) D + 2 a3 (D 2 +D)+b +t 2 (D+) ( ) a 2 (D +) 2 +b 0 +a t 3 (D +2) ( D + 3 2) (D+) +a 0 t 4 (D +3)(D+2)(D+). Th. [, Przyjalkowski] Regularized quantum differential equations of rank Fano threefolds of index d and degree 2d 2 N are exactly (N,d) modular D3 s: given by the expansions of level N weight 2 modular forms with respect to the d th root of the Hauptmodul on X 0 (N) +N. 0 / 45
11 Calabi-Yau Operators Differential equations D3 Abstraction of Picard-Fuchs for families Y P of Calabi-Yau 3-folds/motives defined over Z. Definition: P Z[t, D] is called Calabi-Yau, when: Fuchsian of order four. MUM (maximal unipotent monodromy) at 0. Monodromy Sp 4 (Z). Integrality: φ(x) Z[[t]], n d Z. / 45
12 Beyond hypergeometrics Simplest family of with four singular points D 4 +t(ad 4 +2aD 3 +bd 2 +cd+e)+ft 2 (D+α )(D+α 2 )(D+α 3 ))(D+α 4 ) singularities are at the roots of +at+ft 2 = 0. b, c, e: accesory parameters. 2 / 45
13 Find all such with coming from geometry. or: with monodromy Sp 4 (Z) or: with an integral solution φ Z[[x]] Seems very difficult: we not know position of the the four points: j-invariant we not know the value of the accesory parameters...but we know many examples ( 85 cases). List complete? 3 / 45
14 Example: Coming from a Calabi-Yau 3-fold in Grass(3, 6): D 4 t(65d 4 +30D 3 +05D 2 +40D+6)+4 3 t 2 (D+/4)(D+) 2 (D+3/4) Solution holomorphic at 0: φ(x) = a n t n n=0 a n = k,l,m( n k)( n l)( m k)( m l )( k+l )( n 2 k m) = k,l( n k) 2 ( n l ) 2 ( k+l ) 2 n 4 / 45
15 A step back Special Heun equations: D 2 +t(ad 2 +ad +λ)+bt 2 (D +) 2 singularities are at the roots of +at+bt 2 = 0, λ: accesory parameter. Basically four cases; studied by F. Beukers, A. Beauville, D. Zagier and others. New systematic approach: Congruence sheaves. Works very nicely for this case. 5 / 45
16 Correspondence:...very Number field case: K Q Gal(K/K)-representations Automorphic representations ρ : Gal(K/K) G π ρ L 2 ( L G(A)/ L G(Q)) Example: Elliptic curve E/Q: H (E,Q l ) Hecke Eigenform n=0 a nq n Tr(F p ) a p 6 / 45
17 Correspondence:...very Function field case: K = k(c), C curve over k := F p Gal(K/K)-representations Automorphic representations l-adic sheaves on C Hecke-Eigensheaves on Bun. 7 / 45
18 Correspondence:...very Example: Family of elliptic curves π : E C = P /F p L = R π (Q l ) is an l-adic sheaf: x : Spec(k) C point (in the scheme sense), k = q = p r. x (L) is Gal(k/k)-representation. Frobenius F x Gal(k/k) (inverse to) a a q. F x Aut(x (L)). Frobenius F x Hecke-operator H x 8 / 45
19 The glueing problem Correspondence:...very Families over π : X P := P Z For p prime get X mod p P mod p and thus l-adic sheaves L p. Relation between L p? 9 / 45
20 Provisional definition: Condition on P x (t) = det(f x t) characteristic polynomial of F x. For L of rank 2 and weight : P x (t) = (t α 0 )(t α ) α 0 = mod N, α = p mod N If this happens for almost all p and x C(F p ), we say: L is a N-congruence sheaf. +p Tr(F x ) 0 mod N 20 / 45
21 For L of rank 4 and weight 3: P(t) = (t α 0 )(t α )(t α 2 )(t α 3 ) α 0 = mod N, α 3 = p 3 mod N α = p mod N 2, α 2 = p 2 mod N 2 If this happens for almost allp and x C(F p ), we say: L is a (N,N 2 )-congruence sheaf. Works as glue between various L p 2 / 45
22 A special Hecke-algebra Kontsevich (2005) Notes on motives in finite characterisic Makes correspondence explicit for SL 2 -local systems on P \{four points} with unipotent local monodromies 22 / 45
23 From data Σ = {roots of t 3 +at 2 +bt+c = 0, } get a polynomial in three variables: P(x,y,z) = (b xy yz xz) 2 4(xyz +c)(x+y +z +a) It is the discriminant of the polynomial (t 3 +at 2 +bt+c) (t x)(t y)(t z) 23 / 45
24 Picture of the surface P(x,y,z) = 0 a = b = c = 0 24 / 45
25 Picture of the surface P(x,y,z) = 0 a > 0,b = c = 0 25 / 45
26 Picture of the surface P(x,y,z) = 0 26 / 45
27 Now take a prime number p and a point x P (F p ). Define a (p+) (p+)-matrix H x by (H x ) yz = 2+#{w w 2 = P(x,y,z)}+correction For x,y P (F p ) one has (miracle!) [H x,h y ] = 0 Hecke-algebra: algebra generated by H x,x P (F). 27 / 45
28 Example p = 7, a = 3,b = 4,c = H = / 45
29 The trace function of L is the Hecke-eigenvector Hecke-algebra acts on functions on P (F p ), written as vectors v = (v x ), x P (F p ) (H x v) y = z (H x ) yz v z Drinfelds theorem tells: SL 2 (F p )-automorphic representations correspond to normalised common eigenvectors H x v = v x.v, v x v y = z H xyz v z and hence to l-adic local systems L on P \Σ. L v = (Tr(F x ) x P ) 29 / 45
30 Let s calculate! The trace function of L is the Hecke-eigenvector Strategy Run over p = 3,5,7,,......and all f = t 3 +at 2 +bt+c F p [t]. Determine rational common eigenvectors for Hecke-algebra. Sort them for various p using congruences. 30 / 45
31 The trace function of L is the Hecke-eigenvector Decomposition of the Hecke-algebra Example: p = 7 a,b,c 0,0, ,0,2 6 0,0,3 6 0,, ,, 7 0,, ,3, ,3, 3 4 0,3,2 7 0,3, / 45
32 The trace function of L is the Hecke-eigenvector The 8 non-trivial rational eigenvectors for p = 7. a, b, c Eigenvector v N j 0,0,2 [ ] 3 0 0,0,3 [ ] 3 0 0,,0 [ ] 4 6 0,3,3 [ ] 5 4 0,3,0 [ ] 6 6 0,3,0 [ ] 6 6 0,3,0 [ ] 8 6 0,0, [ ] 9 0 N: the congruence. j: j-invariant F p of the four points. 32 / 45
33 The trace function of L is the Hecke-eigenvector Observations: Running over p = 3,5,7,,... we see: There are very few Q-Hecke-eigenvectors. All of these have a non-trivial congruence. There are no 7, 0,, 2,... congruences. 3, 4, 5, 6, 8-congruences are persistent. 33 / 45
34 The trace function of L is the Hecke-eigenvector Focus on 5-congruence: p a, b, c j Eigenvector 7 0,3,3 4 [ 2,8, 2,3,3, 2, 2,8] 0,2,0 [2, 3, 3,2,2, 3,2, 3,2,2,2,2] 0,2,0 [2,2,2,2, 3,2, 3,2,2, 3, 3,2] 3 0,2,5 4 [4,4,, 6,,,,4,4,4,4, 6, 6,4] 7 0,,2 [3, 2, 2,3, 2, 7, 2, 2, 2,3, 2, 7,3,3,8,3,8, 9 0,2,4 5 [5,5, 5,0,20,5,0, 20, 20,0,0,0,0, 5,0,0,5, 5, 5 34 / 45
35 The trace function of L is the Hecke-eigenvector j-invariant for 5-congruence trace function p j Reconstruct j Q from reductions mod p (Wang algorithm): j = ! Simplest choice: a =,b =,c = / 45
36 Special Heun equation with Riemann symbol where P = f 2 +f +t λ f = t(t 2 +at+b) Singular points: 0, roots of t 2 +at+b,. λ: the accesory parameter. 36 / 45
37 Automorphic property of Multiplication theorem: Let φ(t) = φ λ (t) = +b (λ)t+b 2 (λ)t be solution of differential equation Then one has: φ(x)φ(y) = 2πi Pφ(t) = 0 K(x,y,z)φ(z) dz z where K(x,y,z) = b P(x,y,b/z) 37 / 45
38 Automorphic property of Complete analogy: for C = P (F p ) v x v y = z H xyz v z Geometric for C = P (C): φ(x)φ(y) = K(x,y,z)φ(z) dz 2πi z 38 / 45
39 Automorphic property of Change from point-basis to power-basis Functions on F p P (F p ) δ x, x F p x n, n = 0,...,p v φ p (x) = p n=0 Inverse Vandermonde-Transformation p.... p... (p ) p a n x n a 0 a. a p = v p.. v v 0 39 / 45
40 Automorphic property of p = 7: p = 3: Eigenvector Transform Eigenvector Transform p = 43: / 45
41 Automorphic property of x = 0 mod 7, x = 4 mod 3, x = 8 mod 43 x = 47 mod We get a series φ(t) = +3t+9t 2 +47t 3 +25t with φ(t) φ p (t) mod p 4 / 45
42 Automorphic property of Apéry numbers Generating function, 3, 9, 47,... a n := n ( ) 2 ( ) n n+k k k k=0 φ(t) := satisfies the differential equation a n t n n=0 (D 2 t(d 2 +D +3)+t 2 (D+) 2 )φ(t) = 0 42 / 45
43 Automorphic property of F. Beukers (98) Modular interpretation Operator arises as Picard-Fuchs operator of modular elliptic surface for Γ (5). π : X P with singular four singular fibres of type I,I,I 5,I 5. Number of points in fibre X t (F p ) divisible by 5. +p Tr(F p ) 0 mod 5 43 / 45
44 Automorphic property of So indeed: R π (Q l ) is a 5-congruence sheaf! What did we? Use explicit SL 2 -Hecke-algebra. Found 5-congruence trace-functions. Found from it Apéry sequence. Found arithmetic D2-equation. It works for all Beukers-Zagier-Beauville. Bonus: Get not only differential equation, but also get the Frobenius traces of all fibres! 44 / 45
45 What one might hope to Find explicit Hecke-algebras and multiplication theorems for general Heun-equations. Same for D3s Find explicit Hecke-algebra for Sp 4 -case. Find bi-congruence trace-functions. We claim: is substitute for modularity of D2 and D3! Find corresponding differential of CY-type. 45 / 45
Hodge structures from differential equations
Hodge structures from differential equations Andrew Harder January 4, 2017 These are notes on a talk on the paper Hodge structures from differential equations. The goal is to discuss the method of computation
More informationThe geometry of Landau-Ginzburg models
Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror
More informationPROOF OF THE GAMMA CONJECTURE FOR FANO 3-FOLDS OF PICARD RANK ONE
PROOF OF THE GAMMA CONJECTURE FOR FANO 3-FOLDS OF PICARD RANK ONE VASILY GOLYSHEV AND DON ZAGIER Dedicated to the memory of Andrei Andreevich Bolibrukh Abstract. We verify the first Gamma Conjecture, which
More informationFeynman integrals as Periods
Feynman integrals as Periods Pierre Vanhove Amplitudes 2017, Higgs Center, Edinburgh, UK based on [arxiv:1309.5865], [arxiv:1406.2664], [arxiv:1601.08181] Spencer Bloch, Matt Kerr Pierre Vanhove (IPhT)
More informationPeriods and generating functions in Algebraic Geometry
Periods and generating functions in Algebraic Geometry Hossein Movasati IMPA, Instituto de Matemática Pura e Aplicada, Brazil www.impa.br/ hossein/ Abstract In 1991 Candelas-de la Ossa-Green-Parkes predicted
More informationSOME MONODROMY GROUPS OF FINITE INDEX IN Sp 4 (Z)
J. Aust. Math. Soc. 99 (5) 8 6 doi:.7/s678875 SOME MONODROMY GROUPS OF FINITE INDEX IN Sp (Z) JÖRG HOFMANN and DUCO VAN STRATEN (Received 6 January ; accepted June ; first published online March 5) Communicated
More informationarxiv: v3 [math.ag] 26 Jun 2017
CALABI-YAU MANIFOLDS REALIZING SYMPLECTICALLY RIGID MONODROMY TUPLES arxiv:50.07500v [math.ag] 6 Jun 07 CHARLES F. DORAN, ANDREAS MALMENDIER Abstract. We define an iterative construction that produces
More informationMonodromy of the Dwork family, following Shepherd-Barron X n+1. P 1 λ. ζ i = 1}/ (µ n+1 ) H.
Monodromy of the Dwork family, following Shepherd-Barron 1. The Dwork family. Consider the equation (f λ ) f λ (X 0, X 1,..., X n ) = λ(x n+1 0 + + X n+1 n ) (n + 1)X 0... X n = 0, where λ is a free parameter.
More informationCalabi-Yau Geometry and Mirror Symmetry Conference. Cheol-Hyun Cho (Seoul National Univ.) (based on a joint work with Hansol Hong and Siu-Cheong Lau)
Calabi-Yau Geometry and Mirror Symmetry Conference Cheol-Hyun Cho (Seoul National Univ.) (based on a joint work with Hansol Hong and Siu-Cheong Lau) Mirror Symmetry between two spaces Mirror symmetry explains
More informationArithmetic Mirror Symmetry
Arithmetic Mirror Symmetry Daqing Wan April 15, 2005 Institute of Mathematics, Chinese Academy of Sciences, Beijing, P.R. China Department of Mathematics, University of California, Irvine, CA 92697-3875
More informationFrobenius polynomials for Calabi Yau equations
communications in number theory and physics Volume 2, Number 3, 537 561, 2008 Frobenius polynomials for Calabi Yau equations Kira Samol and Duco van Straten We describe a variation of Dwork s unit-root
More informationCounting problems in Number Theory and Physics
Counting problems in Number Theory and Physics Hossein Movasati IMPA, Instituto de Matemática Pura e Aplicada, Brazil www.impa.br/ hossein/ Encontro conjunto CBPF-IMPA, 2011 A documentary on string theory
More informationCounting curves on a surface
Counting curves on a surface Ragni Piene Centre of Mathematics for Applications and Department of Mathematics, University of Oslo University of Pennsylvania, May 6, 2005 Enumerative geometry Specialization
More informationExtremal Laurent Polynomials
Extremal Laurent Polynomials Alessio Corti London, 2 nd July 2011 1 Introduction The course is an elementary introduction to my experimental work in progress with Tom Coates, Sergei Galkin, Vasily Golyshev
More informationOverview of classical mirror symmetry
Overview of classical mirror symmetry David Cox (notes by Paul Hacking) 9/8/09 () Physics (2) Quintic 3-fold (3) Math String theory is a N = 2 superconformal field theory (SCFT) which models elementary
More informationLecture 4. Frits Beukers. Arithmetic of values of E- and G-function. Lecture 4 E- and G-functions 1 / 27
Lecture 4 Frits Beukers Arithmetic of values of E- and G-function Lecture 4 E- and G-functions 1 / 27 Two theorems Theorem, G.Chudnovsky 1984 The minimal differential equation of a G-function is Fuchsian.
More informationMirror Symmetry: Introduction to the B Model
Mirror Symmetry: Introduction to the B Model Kyler Siegel February 23, 2014 1 Introduction Recall that mirror symmetry predicts the existence of pairs X, ˇX of Calabi-Yau manifolds whose Hodge diamonds
More informationArithmetic Properties of Mirror Map and Quantum Coupling
Arithmetic Properties of Mirror Map and Quantum Coupling arxiv:hep-th/9411234v3 5 Dec 1994 Bong H. Lian and Shing-Tung Yau Department of Mathematics Harvard University Cambridge, MA 02138, USA Abstract:
More informationCOMPUTING ARITHMETIC PICARD-FUCHS EQUATIONS JEROEN SIJSLING
COMPUTING ARITHMETIC PICARD-FUCHS EQUATIONS JEROEN SIJSLING These are the extended notes for a talk given at the Fields Institute on August 24th, 2011, about my thesis work with Frits Beukers at the Universiteit
More informationMirror symmetry, Langlands duality and the Hitchin system I
Mirror symmetry, Langlands duality and the Hitchin system I Tamás Hausel Royal Society URF at University of Oxford http://www.maths.ox.ac.uk/ hausel/talks.html April 200 Simons lecture series Stony Brook
More informationHomological mirror symmetry via families of Lagrangians
Homological mirror symmetry via families of Lagrangians String-Math 2018 Mohammed Abouzaid Columbia University June 17, 2018 Mirror symmetry Three facets of mirror symmetry: 1 Enumerative: GW invariants
More informationRational points on elliptic curves. cycles on modular varieties
Rational points on elliptic curves and cycles on modular varieties Mathematics Colloquium January 2009 TIFR, Mumbai Henri Darmon McGill University http://www.math.mcgill.ca/darmon /slides/slides.html Elliptic
More informationChern numbers and Hilbert Modular Varieties
Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point
More informationHOW TO CLASSIFY FANO VARIETIES?
HOW TO CLASSIFY FANO VARIETIES? OLIVIER DEBARRE Abstract. We review some of the methods used in the classification of Fano varieties and the description of their birational geometry. Mori theory brought
More informationTopics in Geometry: Mirror Symmetry
MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:
More informationON THE COMPUTATION OF THE PICARD GROUP FOR K3 SURFACES
ON THE COMPUTATION OF THE PICARD GROUP FOR K3 SURFACES BY ANDREAS-STEPHAN ELSENHANS (BAYREUTH) AND JÖRG JAHNEL (SIEGEN) 1. Introduction 1.1. In this note, we will present a method to construct examples
More informationIrrationality proofs, moduli spaces, and dinner parties
Irrationality proofs, moduli spaces, and dinner parties Francis Brown, IHÉS-CNRS las, Members Colloquium 17th October 2014 1 / 32 Part I History 2 / 32 Zeta values and Euler s theorem Recall the Riemann
More informationA conjecture of Evans on Kloosterman sums
Evan P. Dummit, Adam W. Goldberg, and Alexander R. Perry 2009 Wisconsin Number Theory REU Project Notation We will make use of the following notation: p odd prime Additive character ψ : F p C, x e 2πix
More informationList of Works. Alessio Corti. (Last modified: March 2016)
List of Works Alessio Corti (Last modified: March 2016) Recent arxiv articles 1. The Sarkisov program for Mori fibred Calabi Yau pairs (with A.-S. Kaloghiros), arxiv:1504.00557, 12 pp., accepted for publication
More informationGENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY. Sampei Usui
GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY Sampei Usui Abstract. This article is a geometric application of polarized logarithmic Hodge theory of Kazuya Kato and Sampei Usui. We prove generic Torelli
More informationOn the Virtual Fundamental Class
On the Virtual Fundamental Class Kai Behrend The University of British Columbia Seoul, August 14, 2014 http://www.math.ubc.ca/~behrend/talks/seoul14.pdf Overview Donaldson-Thomas theory: counting invariants
More informationEnumerative Geometry: from Classical to Modern
: from Classical to Modern February 28, 2008 Summary Classical enumerative geometry: examples Modern tools: Gromov-Witten invariants counts of holomorphic maps Insights from string theory: quantum cohomology:
More informationFANO THREEFOLDS WITH LARGE AUTOMORPHISM GROUPS
FANO THREEFOLDS WITH LARGE AUTOMORPHISM GROUPS CONSTANTIN SHRAMOV Let G be a finite group and k a field. We can consider the notion of G-rationality, G- nonrationality, etc., by considering G-equivariant
More informationCalculation and arithmetic significance of modular forms
Calculation and arithmetic significance of modular forms Gabor Wiese 07/11/2014 An elliptic curve Let us consider the elliptic curve given by the (affine) equation y 2 + y = x 3 x 2 10x 20 We show its
More information15 Elliptic curves and Fermat s last theorem
15 Elliptic curves and Fermat s last theorem Let q > 3 be a prime (and later p will be a prime which has no relation which q). Suppose that there exists a non-trivial integral solution to the Diophantine
More informationPietro Fre' SISSA-Trieste. Paolo Soriani University degli Studi di Milano. From Calabi-Yau manifolds to topological field theories
From Calabi-Yau manifolds to topological field theories Pietro Fre' SISSA-Trieste Paolo Soriani University degli Studi di Milano World Scientific Singapore New Jersey London Hong Kong CONTENTS 1 AN INTRODUCTION
More informationLattice methods for algebraic modular forms on orthogonal groups
Lattice methods for algebraic modular forms on orthogonal groups John Voight Dartmouth College joint work with Matthew Greenberg and Jeffery Hein and Gonzalo Tornaría Computational Challenges in the Theory
More informationDifferential equations concerned with mirror symmetry of toric K3 hypersurfaces with arithmetic properties. Atsuhira Nagano (University of Tokyo)
Differential equations concerned with mirror symmetry of toric K3 hypersurfaces with arithmetic properties Atsuhira Nagano (University of Tokyo) 1 Contents Section 1 : Introduction 1: Hypergeometric differential
More informationMirror symmetry. Mark Gross. July 24, University of Cambridge
University of Cambridge July 24, 2015 : A very brief and biased history. A search for examples of compact Calabi-Yau three-folds by Candelas, Lynker and Schimmrigk (1990) as crepant resolutions of hypersurfaces
More informationIntroduction Curves Surfaces Curves on surfaces. Curves and surfaces. Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway
Curves and surfaces Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway What is algebraic geometry? IMA, April 13, 2007 Outline Introduction Curves Surfaces Curves on surfaces
More informationHypergeometric Functions and Hypergeometric Abelian Varieties
Hypergeometric Functions and Hypergeometric Abelian Varieties Fang-Ting Tu Louisiana State University September 29th, 2016 BIRS Workshop: Modular Forms in String Theory Fang Ting Tu (LSU) Hypergeometric
More informationFeynman Integrals, Regulators and Elliptic Polylogarithms
Feynman Integrals, Regulators and Elliptic Polylogarithms Pierre Vanhove Automorphic Forms, Lie Algebras and String Theory March 5, 2014 based on [arxiv:1309.5865] and work in progress Spencer Bloch, Matt
More information(www.math.uni-bonn.de/people/harder/manuscripts/buch/), files chap2 to chap
The basic objects in the cohomology theory of arithmetic groups Günter Harder This is an exposition of the basic notions and concepts which are needed to build up the cohomology theory of arithmetic groups
More informationA Motivated Introduction to Modular Forms
May 3, 2006 Outline of talk: I. Motivating questions II. Ramanujan s τ function III. Theta Series IV. Congruent Number Problem V. My Research Old Questions... What can you say about the coefficients of
More informationTopics in Geometry: Mirror Symmetry
MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:
More informationINTEGRAL SOLUTIONS OF APÉRY-LIKE RECURRENCE EQUATIONS. Don Zagier. 1. Beukers s recurrence equation. Beukers [4] considers the differential equation
INTEGRAL SOLUTIONS OF APÉRY-LIKE RECURRENCE EQUATIONS Don Zagier Abstract. In [4], Beukers studies the differential equation `(t 3 + at + btf (t + (t λf (t = 0, (* where a, b and λ are rational parameters,
More informationFrom K3 Surfaces to Noncongruence Modular Forms. Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015
From K3 Surfaces to Noncongruence Modular Forms Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015 Winnie Li Pennsylvania State University 1 A K3 surface
More informationElliptic dilogarithms and two-loop Feynman integrals
Elliptic dilogarithms and two-loop Feynman integrals Pierre Vanhove 12th Workshop on Non-Perturbative Quantum Chromodynamics IAP, Paris June, 10 2013 based on work to appear with Spencer Bloch Pierre Vanhove
More informationDifferential Equations and Associators for Periods
Differential Equations and Associators for Periods Stephan Stieberger, MPP München Workshop on Geometry and Physics in memoriam of Ioannis Bakas November 2-25, 26 Schloß Ringberg, Tegernsee based on: St.St.,
More informationMahler measure of K3 surfaces
Mahler measure of K3 surfaces Marie José BERTIN Institut Mathématique de Jussieu Université Paris 6 4 Place Jussieu 75006 PARIS bertin@math.jussieu.fr November 7, 2011 Introduction Introduced by Mahler
More informationMirror symmetry for G 2 manifolds
Mirror symmetry for G 2 manifolds based on [1602.03521] [1701.05202]+[1706.xxxxx] with Michele del Zotto (Stony Brook) 1 Strings, T-duality & Mirror Symmetry 2 Type II String Theories and T-duality Superstring
More informationFrom de Jonquières Counts to Cohomological Field Theories
From de Jonquières Counts to Cohomological Field Theories Mara Ungureanu Women at the Intersection of Mathematics and High Energy Physics 9 March 2017 What is Enumerative Geometry? How many geometric structures
More informationPart A: Frontier Talks. Some Mathematical Problems on the Thin Film Equations
Title and Part A: Frontier Talks Some Mathematical Problems on the Thin Film Equations Kai-Seng Chou The Chinese University of Hong Kong The thin film equation, which is derived from the Navier-Stokes
More informationPoint counting and real multiplication on K3 surfaces
Point counting and real multiplication on K3 surfaces Andreas-Stephan Elsenhans Universität Paderborn September 2016 Joint work with J. Jahnel. A.-S. Elsenhans (Universität Paderborn) K3 surfaces September
More informationLinear systems and Fano varieties: introduction
Linear systems and Fano varieties: introduction Caucher Birkar New advances in Fano manifolds, Cambridge, December 2017 References: [B-1] Anti-pluricanonical systems on Fano varieties. [B-2] Singularities
More informationarxiv: v1 [math.ag] 27 Sep 2017
PICARD-FUCHS OPERATORS FOR OCTIC ARRANGEMENTS I (THE CASE OF ORPHANS) SLAWOMIR CYNK AND DUCO VAN STRATEN arxiv:1709.09752v1 [math.ag] 27 Sep 2017 Abstract. We report on 25 families of projective Calabi-Yau
More informationarxiv: v1 [math.ag] 3 Aug 2017
1 HODGE NUMBERS OF LANDAU-GINZBURG MODELS ANDREW HARDER arxiv:1708.01174v1 [math.ag] 3 Aug 2017 Abstract. WestudytheHodgenumbersf p,q oflandau-ginzburgmodelsasdefinedbykatzarkov, Kontsevich and Pantev.
More informationMirror symmetry, Langlands duality and the Hitchin system
Mirror symmetry, Langlands duality and the Hitchin system Tamás Hausel Royal Society URF at University of Oxford http://www.maths.ox.ac.uk/ hausel/talks.html June 2009 Workshop on mirror symmetry MIT,
More informationThree-dimensional imprimitive representations of PSL 2 (Z) and their associated vector-valued modular forms
Three-dimensional imprimitive representations of PSL 2 (Z) and their associated vector-valued modular forms U-M Automorphic forms workshop, March 2015 1 Definition 2 3 Let Γ = PSL 2 (Z) Write ( 0 1 S =
More informationHodge Theory, Moduli and Representation Theory (Stony Brook, August 14-18, 2017)
Hodge Theory, Moduli and Representation Theory (Stony Brook, August 14-18, 2017) Monday Tuesday Wednesday Thursday Friday 10-10:50 Pearlstein Griffiths Corti Filip Totaro 11:10-12 Zhang Green van Straten
More informationElliptic Curves with 2-torsion contained in the 3-torsion field
Elliptic Curves with 2-torsion contained in the 3-torsion field Laura Paulina Jakobsson Advised by Dr. M. J. Bright Universiteit Leiden Universita degli studi di Padova ALGANT Master s Thesis - 21 June
More informationUdine Examples. Alessio Corti, Imperial College London. London, 23rd Sep Abstract
Udine Examples Alessio Corti, Imperial College London London, rd Sep 014 Abstract This is a list of exercises to go with the EMS School New perspectives on the classification of Fano manifolds in Udine,
More informationHomological Mirror Symmetry and VGIT
Homological Mirror Symmetry and VGIT University of Vienna January 24, 2013 Attributions Based on joint work with M. Ballard (U. Wisconsin) and Ludmil Katzarkov (U. Miami and U. Vienna). Slides available
More informationCusp forms and the Eichler-Shimura relation
Cusp forms and the Eichler-Shimura relation September 9, 2013 In the last lecture we observed that the family of modular curves X 0 (N) has a model over the rationals. In this lecture we use this fact
More informationIV. Birational hyperkähler manifolds
Université de Nice March 28, 2008 Atiyah s example Atiyah s example f : X D family of K3 surfaces, smooth over D ; X smooth, X 0 has one node s. Atiyah s example f : X D family of K3 surfaces, smooth over
More informationThe kappa function. [ a b. c d
The kappa function Masanobu KANEKO Masaaki YOSHIDA Abstract: The kappa function is introduced as the function κ satisfying Jκτ)) = λτ), where J and λ are the elliptic modular functions. A Fourier expansion
More informationDel Pezzo surfaces and non-commutative geometry
Del Pezzo surfaces and non-commutative geometry D. Kaledin (Steklov Math. Inst./Univ. of Tokyo) Joint work with V. Ginzburg (Univ. of Chicago). No definitive results yet, just some observations and questions.
More information0. Introduction 1 0. INTRODUCTION
0. Introduction 1 0. INTRODUCTION In a very rough sketch we explain what algebraic geometry is about and what it can be used for. We stress the many correlations with other fields of research, such as
More informationProjects on elliptic curves and modular forms
Projects on elliptic curves and modular forms Math 480, Spring 2010 In the following are 11 projects for this course. Some of the projects are rather ambitious and may very well be the topic of a master
More informationThe structure of algebraic varieties
The structure of algebraic varieties János Kollár Princeton University ICM, August, 2014, Seoul with the assistance of Jennifer M. Johnson and Sándor J. Kovács (Written comments added for clarity that
More informationQUANTIZATION OF SPECTRAL CURVES FOR MEROMORPHIC HIGGS BUNDLES THROUGH TOPOLOGICAL RECURSION OLIVIA DUMITRESCU AND MOTOHICO MULASE
QUANTIZATION OF SPECTRAL CURVES FOR MEROMORPHIC HIGGS BUNDLES THROUGH TOPOLOGICAL RECURSION OLIVIA DUMITRESCU AND MOTOHICO MULASE Abstract. A geometric quantization using a partial differential equation
More informationThe Galois representation associated to modular forms pt. 2 Erik Visse
The Galois representation associated to modular forms pt. 2 Erik Visse May 26, 2015 These are the notes from the seminar on local Galois representations held in Leiden in the spring of 2015. The website
More informationComputing zeta functions of nondegenerate toric hypersurfaces
Computing zeta functions of nondegenerate toric hypersurfaces Edgar Costa (Dartmouth College) December 4th, 2017 Presented at Boston University Number Theory Seminar Joint work with David Harvey (UNSW)
More informationRefined Donaldson-Thomas theory and Nekrasov s formula
Refined Donaldson-Thomas theory and Nekrasov s formula Balázs Szendrői, University of Oxford Maths of String and Gauge Theory, City University and King s College London 3-5 May 2012 Geometric engineering
More informationOn Flux Quantization in F-Theory
On Flux Quantization in F-Theory Raffaele Savelli MPI - Munich Bad Honnef, March 2011 Based on work with A. Collinucci, arxiv: 1011.6388 Motivations Motivations The recent attempts to find UV-completions
More informationAN ARITHMETIC FORMULA FOR THE PARTITION FUNCTION
AN ARITHMETIC FORMULA FOR THE PARTITION FUNCTION KATHRIN BRINGMANN AND KEN ONO 1 Introduction and Statement of Results A partition of a non-negative integer n is a non-increasing sequence of positive integers
More informationSpecial cubic fourfolds
Special cubic fourfolds 1 Hodge diamonds Let X be a cubic fourfold, h H 2 (X, Z) be the (Poincaré dual to the) hyperplane class. We have h 4 = deg(x) = 3. By the Lefschetz hyperplane theorem, one knows
More informationON THE INTEGRAL HODGE CONJECTURE FOR REAL THREEFOLDS. 1. Motivation
ON THE INTEGRAL HODGE CONJECTURE FOR REAL THREEFOLDS OLIVIER WITTENBERG This is joint work with Olivier Benoist. 1.1. Work of Kollár. 1. Motivation Theorem 1.1 (Kollár). If X is a smooth projective (geometrically)
More informationTwisted integral orbit parametrizations
Institute for Advanced Study 4 Octoer 017 Disclaimer All statements are to e understood as essentially true (i.e., true once the statement is modified slightly). Arithmetic invariant theory This talk is
More informationRicci-flat metrics on complex cones
Ricci-flat metrics on complex cones Steklov Mathematical Institute, Moscow & Max-Planck-Institut für Gravitationsphysik (AEI), Potsdam-Golm, Germany Quarks 2014, Suzdal, 3 June 2014 arxiv:1405.2319 1/20
More informationGeneralized Tian-Todorov theorems
Generalized Tian-Todorov theorems M.Kontsevich 1 The classical Tian-Todorov theorem Recall the classical Tian-Todorov theorem (see [4],[5]) about the smoothness of the moduli spaces of Calabi-Yau manifolds:
More information1 The Classical Theory [in brief]
An Introduction to Modular Symbols This is a preparatory talk for Rob Harron's talk; he will talk about overconvergent modular symbols and families of p-adic modular forms. The goal of this talk is to
More informationRiemann-Hilbert problems from Donaldson-Thomas theory
Riemann-Hilbert problems from Donaldson-Thomas theory Tom Bridgeland University of Sheffield Preprints: 1611.03697 and 1703.02776. 1 / 25 Motivation 2 / 25 Motivation Two types of parameters in string
More informationHOMOLOGICAL GEOMETRY AND MIRROR SYMMETRY
HOMOLOGICAL GEOMETRY AND MIRROR SYMMETRY Alexander B. GIVENTAL Dept. of Math., UC Berkeley Berkeley CA 94720, USA 0. A popular example. A homogeneous degree 5 polynomial equation in 5 variables determines
More informationCongruences, graphs and modular forms
Congruences, graphs and modular forms Samuele Anni (IWR - Universität Heidelberg) joint with Vandita Patel (University of Warwick) (work in progress) FoCM 2017, UB, 12 th July 2017 The theory of congruences
More informationFiberwise stable bundles on elliptic threefolds with relative Picard number one
Géométrie algébrique/algebraic Geometry Fiberwise stable bundles on elliptic threefolds with relative Picard number one Andrei CĂLDĂRARU Mathematics Department, University of Massachusetts, Amherst, MA
More informationHigher Hasse Witt matrices. Masha Vlasenko. June 1, 2016 UAM Poznań
Higher Hasse Witt matrices Masha Vlasenko June 1, 2016 UAM Poznań Motivation: zeta functions and periods X/F q smooth projective variety, q = p a zeta function of X: ( Z(X/F q ; T ) = exp m=1 #X(F q m)
More informationMahler Measure Variations, Eisenstein Series and Instanton Expansions.
arxiv:math.nt/0502193 v1 9 Feb 2005 Mahler Measure Variations, Eisenstein Series and Instanton Expansions. Jan Stienstra Mathematisch Instituut, Universiteit Utrecht, the Netherlands Abstract This paper
More informationHomological mirror symmetry
Homological mirror symmetry HMS (Kontsevich 1994, Hori-Vafa 2000, Kapustin-Li 2002, Katzarkov 2002,... ) relates symplectic and algebraic geometry via their categorical structures. A symplectic manifold
More informationBjorn Poonen. MSRI Introductory Workshop on Rational and Integral Points on Higher-dimensional Varieties. January 17, 2006
University of California at Berkeley MSRI Introductory Workshop on Rational and Integral Points on Higher-dimensional (organized by Jean-Louis Colliot-Thélène, Roger Heath-Brown, János Kollár,, Alice Silverberg,
More informationPseudoholomorphic Curves and Mirror Symmetry
Pseudoholomorphic Curves and Mirror Symmetry Santiago Canez February 14, 2006 Abstract This survey article was written for Prof. Alan Weinstein s Symplectic Geometry (Math 242) course at UC Berkeley in
More informationPeriod Domains. Carlson. June 24, 2010
Period Domains Carlson June 4, 00 Carlson - Period Domains Period domains are parameter spaces for marked Hodge structures. We call Γ\D the period space, which is a parameter space of isomorphism classes
More informationThe Geometry of Landau-Ginzburg models
The Geometry of Landau-Ginzburg models Andrew Harder Department of Mathematics and Statistics University of Alberta A thesis submitted in partial fulfillment of the requirements for the degree of Doctor
More informationThe Grothendieck-Katz Conjecture for certain locally symmetric varieties
The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-
More informationSMOOTH FINITE ABELIAN UNIFORMIZATIONS OF PROJECTIVE SPACES AND CALABI-YAU ORBIFOLDS
SMOOTH FINITE ABELIAN UNIFORMIZATIONS OF PROJECTIVE SPACES AND CALABI-YAU ORBIFOLDS A. MUHAMMED ULUDAĞ Dedicated to Mehmet Çiftçi Abstract. We give a classification of smooth complex manifolds with a finite
More informationFOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2
FOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2 CAMERON FRANC AND GEOFFREY MASON Abstract. We prove the following Theorem. Suppose that F = (f 1, f 2 ) is a 2-dimensional vector-valued
More informationElliptic curves and modularity
Elliptic curves and modularity For background and (most) proofs, we refer to [1]. 1 Weierstrass models Let K be any field. For any a 1, a 2, a 3, a 4, a 6 K consider the plane projective curve C given
More informationAN ARITHMETIC FORMULA FOR THE PARTITION FUNCTION
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 11, November 007, Pages 3507 3514 S 000-9939(07)08883-1 Article electronically published on July 7, 007 AN ARITHMETIC FORMULA FOR THE
More informationFrom alternating sign matrices to Painlevé VI
From alternating sign matrices to Painlevé VI Hjalmar Rosengren Chalmers University of Technology and University of Gothenburg Nagoya, July 31, 2012 Hjalmar Rosengren (Chalmers University) Nagoya, July
More informationLINEAR RELATIONS BETWEEN MODULAR FORM COEFFICIENTS AND NON-ORDINARY PRIMES
LINEAR RELATIONS BETWEEN MODULAR FORM COEFFICIENTS AND NON-ORDINARY PRIMES YOUNGJU CHOIE, WINFRIED KOHNEN, AND KEN ONO Appearing in the Bulletin of the London Mathematical Society Abstract. Here we generalize
More information