elliptic cohomology -- references

### Elliptic cohomology

#### General

The concept of [[elliptic cohomology]] originates around:

* {#Landweber88} [[Peter Landweber]], *Elliptic Cohomology and Modular Forms*, in: _Elliptic Curves and Modular Forms in Algebraic Topology_, Lecture Notes in Mathematics **1326** (1988) 55-68 [[doi:10.1007/BFb0078038](https://doi.org/10.1007/BFb0078038)]

* {#LandweberRavenelStong93} [[Peter Landweber]], [[Douglas Ravenel]], [[Robert Stong]], _Periodic cohomology theories defined by elliptic curves_, in [[Haynes Miller]] et. al. (eds.), _The Cech centennial: A conference on homotopy theory_, June 1993, AMS (1995) ([pdf](http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf))

and in the universal guise of [[topological modular forms]] in:

* {#Hopkins02} [[Michael Hopkins]], _Algebraic topology and modular forms_ in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press ([arXiv:math/0212397](http://arxiv.org/abs/math/0212397))

Surveys:

* [[Matthew Greenberg]], _Constructing elliptic cohomology_, McGill University 2002 ([oclc:898194373](https://www.bac-lac.gc.ca/eng/services/theses/Pages/item.aspx?idNumber=898194373), [[GreenbergEllipticCohomology.pdf:file]])

* [[Paul Goerss]], _Topological modular forms (after Hopkins, Miller, and Lurie)_, Séminaire Bourbaki : volume 2008/2009 exposés 997-1011 ([arXiv:0910.5130](https://arxiv.org/abs/0910.5130), [numdam:AST_2010__332__221_0](http://www.numdam.org/item/AST_2010__332__221_0))

* {#Lurie} [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology]]_, in:  _Algebraic Topology_, Abel Symposia Volume 4, 2009, pp 219-277 ([pdf](http://www.math.harvard.edu/~lurie/papers/survey.pdf), [doi:10.1007/978-3-642-01200-6_9](https://doi.org/10.1007/978-3-642-01200-6_9))

* [[Charles Rezk]], _Elliptic cohomology and elliptic curves_, Felix Klein Lectures, Bonn  (2015) $[$[web](http://www.hcm.uni-bonn.de/fkl-rezk), [pdf](https://rezk.web.illinois.edu/felix-klein-lectures-notes.pdf), [[RezkElliptic2015.pdf:file]], video rec: [part 1](https://youtu.be/eFzB_2fkOxY), [2](https://youtu.be/4e9e3AtK7yg), [3](https://youtu.be/ueL5dMH9JoI), [4](https://youtu.be/r_7SsIoU9No)$]$

Textbook accounts:

* {#Thomas02} [[Charles Thomas]], _Elliptic cohomology_, Kluwer Academic, 2002 ([doi:10.1007/b115001](https://link.springer.com/book/10.1007/b115001), [pdf](https://link.springer.com/content/pdf/10.1007%2Fb115001.pdf))

* {#DFHH14} [[Christopher Douglas]], [[John Francis]], [[André Henriques]], [[Michael Hill]] (eds.), _Topological Modular Forms_, Mathematical Surveys and Monographs Volume 201, AMS 2014  ([ISBN:978-1-4704-1884-7](https://bookstore.ams.org/surv-201))

#### Equivariant elliptic cohomology

On [[equivariant elliptic cohomology]] and [[positive energy representations]] of [[loop groups]]:

* [[Eduard Looijenga]], _Root systems and elliptic curves_, Invent. Math. **38** 1 (1976/77) 17-32 [[doi:10.1007/BF01390167](https://doi.org/10.1007/BF01390167)]

* {#Grojnowski94} [[Ian Grojnowski]], _Delocalised equivariant elliptic cohomology_ (1994), in: _Elliptic cohomology_, Volume 342 of London Math. Soc. Lecture Note Ser., pages 114–121. Cambridge Univ. Press, Cambridge, 2007 ([pdf](http://hopf.math.purdue.edu/Grojnowski/deloc.pdf), [doi:10.1017/CBO9780511721489.007]( https://doi.org/10.1017/CBO9780511721489.007))

* {#GinzburgKapranovVasserot95} [[Victor Ginzburg]], [[Mikhail Kapranov]], Eric Vasserot, _Elliptic Algebras and Equivariant Elliptic Cohomology_ ([arXiv:q-alg/9505012](http://arxiv.org/abs/q-alg/9505012))

* {#Ando00} [[Matthew Ando]], _Power operations in elliptic cohomology and representations of loop groups_, Transactions of the American Mathematical Society 352, 2000, pp. 5619-5666. ([jstor:221905](http://www.jstor.org/stable/221905), [pdf](http://www.math.uiuc.edu/~mando/papers/POECLG/poeclg.pdf))

* {#Gepner05} [[David Gepner]], _[[Homotopy topoi and equivariant elliptic cohomology]]_, University of Illinois at Urbana-Champaign, 2005 ([[GepnerElliptic05.pdf:file]])

* {#GepnerMeier20} [[David Gepner]], [[Lennart Meier]], _On equivariant topological modular forms_, ([arXiv:2004.10254](https://arxiv.org/abs/2004.10254))

* [[Michèle Vergne]], *Bouquets revisited and equivariant elliptic cohomology*, International Journal of Mathematics 2021 ([arXiv:2005.00312](https://arxiv.org/abs/2005.00312), [doi:10.1142/S0129167X21400127](https://doi.org/10.1142/S0129167X21400127))

Relation to [[Kac-Weyl characters]] of [[loop group representations]]

* {#Brylinski90} [[Jean-Luc Brylinski]], _Representations of loop groups, Dirac operators on loop space, and modular forms_, Topology, 29(4):461–480, 1990 (doi:10.1016/0040-9383(90)90016-D)

* {#Ganter12} [[Nora Ganter]], _The elliptic Weyl character formula_, Compositio Mathematica, Vol 150, Issue 7 (2014), pp 1196-1234  ([arXiv:1206.0528](http://arxiv.org/abs/1206.0528))

The case of [[twisted ad-equivariant Tate K-theory]]:

* {#Ganter07} [[Nora Ganter]], Section 3.1 in: _Stringy power operations in Tate K-theory_ ([arXiv:math/0701565](https://arxiv.org/abs/math/0701565))

* {#Ganter13} [[Nora Ganter]], _Power operations in orbifold Tate K-theory_, Homology Homotopy Appl. Volume 15, Number 1 (2013), 313-342. ([arXiv:1301.2754](https://arxiv.org/abs/1301.2754), [euclid:hha/1383943680](https://projecteuclid.org/euclid.hha/1383943680))

* [[Zhen Huan]], _Quasi-elliptic cohomology_, 2017 ([hdl](http://hdl.handle.net/2142/97268))

* {#Huan18} [[Zhen Huan]], _Quasi-Elliptic Cohomology I_, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 ([arXiv:1805.06305](https://arxiv.org/abs/1805.06305), [doi:10.1016/j.aim.2018.08.007](https://doi.org/10.1016/j.aim.2018.08.007))

* {#Huan18Quasi} [[Zhen Huan]], *Quasi-theories* ([arXiv:1809.06651](https://arxiv.org/abs/1809.06651))

* {#Luecke19}  [[Kiran Luecke]], *Completed K-theory and Equivariant Elliptic Cohomology*, Advances in Mathematics **410** B (2022) 108754 [[arXiv:1904.00085](https://arxiv.org/abs/1904.00085), [doi:10.1016/j.aim.2022.108754](https://doi.org/10.1016/j.aim.2022.108754)]

* [[Thomas Dove]], _Twisted Equivariant Tate K-Theory_ ([arXiv:1912.02374](https://arxiv.org/abs/1912.02374))

See also:

* [[Daniel Berwick-Evans]], [[Arnav Tripathy]], _A geometric model for complex analytic equivariant elliptic cohomology_, ([arXiv:1805.04146](https://arxiv.org/abs/1805.04146))

#### Via derived $E_\infty$-geometry

Formulation of ([[equivariant elliptic cohomology|equivariant]]) [[elliptic cohomology]] in [[derived algebraic geometry]]/[[E-∞ geometry]] ([[derived elliptic curves]]):

* {#GoerssHopkinsA} [[Paul Goerss]], [[Michael Hopkins]], _Moduli spaces of commutative ring spectra_, in _Structured ring spectra_, London
Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, Cambridge, 2004, pp. 151-200. ([pdf](http://www.math.northwestern.edu/~pgoerss/papers/sum.pdf), [doi:10.1017/CBO9780511529955.009](https://doi.org/10.1017/CBO9780511529955.009))

* {#GoerssHopkinsB} [[Paul Goerss]], [[Michael Hopkins]], _Moduli problems for structured ring spectra_ ([pdf](http://www.math.northwestern.edu/~pgoerss/spectra/obstruct.pdf))

  ([[Goerss-Hopkins-Miller theorem]])

* [[Jacob Lurie]], _[[Elliptic Cohomology I]]: Spectral abelian varieties_, 2018. 141pp ([pdf]( http://math.harvard.edu/~lurie/papers/Elliptic-I.pdf))

* [[Jacob Lurie]], _Elliptic Cohomology II: Orientations_, 2018. 288pp ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-II.pdf))

* [[Jacob Lurie]], _Elliptic Cohomology III: Tempered Cohomology_, 2019. 286pp ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-III-Tempered.pdf))

* [[Jacob Lurie]], _Elliptic Cohomology IV: Equivariant elliptic cohomology_, to appear.

### Elliptic genera

#### General

The general concept of [[elliptic genus]] originates with:

* {#Ochanine87} [[Serge Ochanine]], _Sur les genres multiplicatifs definis par des integrales elliptiques_, Topology, Vol. 26, No. 2, 1987 ([pdf](http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Mult-genre-Ochanine.pdf), doi:10.1016/0040-9383(87)90055-3)

Early development:

* {#Zagier86} [[Don Zagier]], _Note on the Landweber-Stong elliptic genus_ 1986 ([pdf](http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0078047/fulltext.pdf), [edoc:744944](http://edoc.mpg.de/744944))

* {#HirzebruchGergerJung92} [[Friedrich Hirzebruch]], Thomas Berger, Rainer Jung, chapter 2 of: *Manifolds and Modular Forms*, Aspects of Mathematics **20**, Viehweg (1992), Springer (1994) [[doi:10.1007/978-3-663-10726-2](https://doi.org/10.1007/978-3-663-10726-2), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/hirzjung.pdf)]

* {#ChudnovskyChudnovsky88} D.V. Chudnovsky, G.V. Chudnovsky, _Elliptic modular functions and elliptic genera_, Topology, Volume 27, Issue 2, 1988, Pages 163–170 (doi:10.1016/0040-9383(88)90035-3)

* {#Hovey91} [[Mark Hovey]], _Spin Bordism and Elliptic Homology_, Math Z 219, 163–170 (1995) ([doi:10.1007/BF02572356](https://doi.org/10.1007/BF02572356))

* {#KreckStolz93} [[Matthias Kreck]], [[Stephan Stolz]], _$HP^2$-bundles and elliptic homology_, Acta Math, 171 (1993) 231-261 ([[KreckStolzElliptic.pdf:file]], [euclid:acta/1485890737](https://projecteuclid.org/euclid.acta/1485890737))

Review:

* {#Landweber88} [[Peter Landweber]], _Elliptic genera: An introductory overview_ In: P. Landweber (eds.) _Elliptic Curves and Modular Forms in Algebraic Topology_, Lecture Notes in Mathematics, vol 1326. Springer  (1988) ([doi:10.1007/BFb0078036](https://doi.org/10.1007/BFb0078036))

* {#KL96} [[Kefeng Liu]], _Modular forms and topology_, Proc. of the AMS Conference on the Monster and Related Topics, Contemporary Math. (1996) ([pdf](https://www.math.ucla.edu/~liu/Research/vam.pdf), [[LiuModularFormsTopology.pdf:file]], [doi:10.1090/conm/193](http://dx.doi.org/10.1090/conm/193))

* {#Ochanine09} [[Serge Ochanine]], _What is... an elliptic genus?_, Notices of the AMS, volume 56, number 6  (2009) ([pdf](http://www.ams.org/notices/200906/rtx090600720p.pdf))

The [[Stolz conjecture]] on the [[Witten genus]]:

* {#Stolz96} [[Stephan Stolz]], _A conjecture concerning positive Ricci curvature and the Witten genus_, Mathematische Annalen Volume 304, Number 1 (1996) ([doi:10.1007/BF01446319](https://doi.org/10.1007/BF01446319))

* [[Anand Dessai]], _Some geometric properties of the Witten genus_, in: [[Christian Ausoni]], [[Kathryn Hess]], [[Jérôme Scherer]] (eds.) _Alpine Perspectives on Algebraic Topology_, Contemporary Mathematics 504 (2009) ([pdf](http://homeweb2.unifr.ch/dessaia/pub/papers/Arolla/DessaiArollaFinalRevised30June09.pdf), [[DessaiEllipticGenus.pdf:file]],
[doi:10.1090/conm/504](http://dx.doi.org/10.1090/conm/504))

The [[Jacobi form]]-property of the [[Witten genus]]:

* {#AndoFrenchGanter08} [[Matthew Ando]], Christopher French, [[Nora Ganter]], _The Jacobi orientation and the two-variable elliptic genus_, Algebraic and Geometric Topology 8 (2008) p. 493-539 ([pdf](http://www.msp.warwick.ac.uk/agt/2008/08-01/agt-2008-08-016s.pdf))

The identification of [[elliptic genera]], via [[fiber integration]]/[[Pontrjagin-Thom collapse]], as [[complex oriented cohomology theory|complex orientations]] of [[elliptic cohomology]] ([[sigma-orientation]]/[[string-orientation of tmf]]/[[spin-orientation of Tate K-theory]]):

* [[Michael Hopkins]], _Topological modular forms, the Witten genus, and the theorem of the cube_, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, 554–565. MR 97i:11043 ([pdf](http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf))

* {#AndoHopkinsStrickland01} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _Elliptic spectra, the Witten genus and the theorem of the cube_, Invent. Math. 146 (2001) 595–687 MR1869850 ([doi:10.1007/s002220100175](https://doi.org/10.1007/s002220100175), [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/musix.pdf))

* {#AndoHopkinsStrickland02} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _The sigma orientation is an H-infinity map_, American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 ([arXiv:math/0204053](http://arxiv.org/abs/math/0204053), [doi:10.1353/ajm.2004.0008](https://doi.org/10.1353/ajm.2004.0008))

* {#AndoHopkinsRezk} [[Matthew Ando]], [[Michael Hopkins]], [[Charles Rezk]], _Multiplicative orientations of KO-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf), [[AndoHopkinsRezkStringOrientation.pdf:file]])

For the [[Ochanine genus]]:

* [[Dylan Wilson]], _Orientations and Topological Modular Forms with Level Structure_ ([arXiv:1507.05116](http://arxiv.org/abs/1507.05116))

#### Equivariant elliptic genera

Genera in [[equivariant elliptic cohomology]] and the [[rigidity theorem for equivariant elliptic genera]]:

The statement, with a [[string theory]]-motivated plausibility argument, is due to [Witten 87](#Witten87).

The first [[proof]] was given in:

* [[Clifford Taubes]], _$S^1$-actions and elliptic genera_, Comm. Math. Phys. Volume 122, Number 3 (1989), 455-526 ([euclid:cmp/1104178471](https://projecteuclid.org/euclid.cmp/1104178471))

* [[Raoul Bott]], [[Clifford Taubes]], _On the Rigidity Theorems of Witten_, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 137-186 ([doi:10.2307/1990915](https://doi.org/10.2307/1990915))

Reviewed in:

* [[Raoul Bott]], _On the Fixed Point Formula and the Rigidity Theorems of Witten_, Lectures at Cargése 1987. In: ’t Hooft G., Jaffe A., Mack G., Mitter P.K., Stora R. (eds) _Nonperturbative Quantum Field Theory_. NATO ASI Series (Series B: Physics), vol 185. Springer (1988) ([doi:10.1007/978-1-4613-0729-7_2](https://doi.org/10.1007/978-1-4613-0729-7_2))

Further proofs and constructions:

* [[Friedrich Hirzebruch]], _Elliptic Genera of Level $N$ for Complex Manifolds_, In: Bleuler K., Werner M. (eds) _Differential Geometrical Methods in Theoretical Physics_ NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 250. Springer (1988) ([doi:10.1007/978-94-015-7809-7_3](https://doi.org/10.1007/978-94-015-7809-7_3))

* I. M. Krichever, _Generalized elliptic genera and Baker-Akhiezer functions_, Mathematical Notes of the Academy of Sciences of the USSR 47, 132–142 (1990) ([doi:10.1007/BF01156822](https://doi.org/10.1007/BF01156822))

* [[Kefeng Liu]], _On modular invariance and rigidity theorems_, J. Differential Geom. Volume 41, Number 2 (1995), 343-396 ([euclid:jdg/1214456221](https://projecteuclid.org/euclid.jdg/1214456221))

* [[Kefeng Liu]], _On elliptic genera and theta-functions_, Topology Volume 35, Issue 3, July 1996, Pages 617-640 (doi:10.1016/0040-9383(95)00042-9)

* [[Anand Dessai]], Rainer Jung, _On the Rigidity Theorem for Elliptic Genera_, Transactions of the American Mathematical Society Vol. 350, No. 10 (Oct., 1998), pp. 4195-4220 (26 pages) ([jstor:117694](https://www.jstor.org/stable/117694))

* [[Ioanid Rosu]], _Equivariant Elliptic Cohomology and Rigidity_, American Journal of Mathematics 123 (2001), 647-677 ([arXiv:math/9912089](https://arxiv.org/abs/math/9912089))

* [[Matthew Ando]], [[John Greenlees]], _Circle-equivariant classifying spaces and the rational equivariant sigma genus_, Math. Z. 269, 1021–1104 (2011) ([arXiv:0705.2687](https://arxiv.org/abs/0705.2687), [doi:10.1007/s00209-010-0773-7](https://doi.org/10.1007/s00209-010-0773-7))

On manifolds with [[SU(2)]]-[[action]]:

* [[Anand Dessai]], _The Witten genus and $S^3$-actions on manifolds_, 1994 ([pdf](https://homeweb.unifr.ch/dessaia/pub/papers/MZpreprint6_Witten_S3.pdf), [[DessaiWittenGenusS3.pdf:file]])

#### Twisted elliptic genera

Discussion of [[elliptic genera]] [[twisted cohomology|twisted]] by a gauge bundle, i.e. for [[string^c structure]]):

* [[Matthew Ando]], [[Maria Basterra]], _The Witten genus and equivariant elliptic cohomology_, Math Z 240, 787–822 (2002) ([arXiv:math/0008192](https://arxiv.org/abs/math/0008192), [doi:10.1007/s002090100399](https://doi.org/10.1007/s002090100399))

* [[Matthew Ando]], _The sigma orientation for analytic circle equivariant elliptic cohomology_, Geom. Topol., 7:91–153, 2003 ([arXiv:math/0201092](http://arxiv.org/abs/math/0201092), [euclid:gt/1513883094](https://projecteuclid.org/euclid.gt/1513883094))

* {#ABG} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], Section 11 of  _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics, [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society, 2010 ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))

* {#ChenHanZhang10} [[Qingtao Chen]], [[Fei Han]], [[Weiping Zhang]], _Generalized Witten Genus and Vanishing Theorems_, Journal of Differential Geometry 88.1 (2011): 1-39. ([arXiv:1003.2325](http://arxiv.org/abs/1003.2325))

* Jianqing Yu, Bo Liu, _On the Witten Rigidity Theorem for $String^c$ Manifolds_, Pacific Journal of Mathematics 266.2 (2013): 477-508. ([arXiv:1206.5955](http://arxiv.org/abs/1206.5955))

* [[Fei Han]], [[Varghese Mathai]], _Projective elliptic genera and elliptic pseudodifferential genera_, Adv. Math. 358 (2019) 106860  ([arXiv:1903.07035](https://arxiv.org/abs/1903.07035))

* Haibao Duan, [[Fei Han]], Ruizhi Huang, _$String^c$ Structures and Modular Invariants_, Trans. AMS 2020 ([arXiv:1905.02093](https://arxiv.org/abs/1905.02093))

[[!include elliptic genera as partition functions -- references]]


Elliptic Cohomology I

#Contents#
* table of contents
{:toc}

This entry is about the text

* [[Jacob Lurie]], _Elliptic Cohomology I: Spectral Abelian Varieties_, ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-I.pdf))

It is described as the first in a series of papers whose ultimate aim is to carry out the details of the program outlined in [[A Survey of Elliptic Cohomology]].

This first paper provides a discussion of abelian varieties in the setting of [[spectral algebraic geometry]]. It sets out to answer the question:

>What does it mean to give an elliptic curve (or, more generally, an
abelian variety) over a cohomology theory $A$? To what extent do such objects behave like their counterparts in classical algebraic geometry?

It is followed up by the further parts

* [[Jacob Lurie]], _Elliptic Cohomology II: Orientations_, 2018. 288pp ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-II.pdf))

* [[Jacob Lurie]], _Elliptic Cohomology III: Tempered Cohomology_, 2019. 286pp ([pdf](http://www.math.harvard.edu/~lurie/papers/Elliptic-III-Tempered.pdf))

* [[Jacob Lurie]], _Elliptic Cohomology IV: Equivariant elliptic cohomology_, to appear.

### Contents

#### 1 Abelian Varieties in Spectral Algebraic Geometry
* 1.1 Varieties over $E_{\infty}$-Rings
* 1.2 Abelian Group Objects of $\infty$-Categories
* 1.3 Commutative Monoid Objects of $\infty$-Categories
* 1.4 Abelian Varieties
* 1.5 Strict Abelian Varieties

#### 2 Moduli of Elliptic Curves
* 2.1 Comparing Abelian Varieties with Strict Abelian Varieties
* 2.2 Deformation Theory of Abelian Varieties
* 2.3 Deformation Theory of Strict Abelian Varieties
* 2.4 The Moduli Stack of Elliptic Curves

#### 3 Cartier Duality
* 3.1 Coalgebra Objects of $\infty$-Categories
* 3.2 Duality between Algebra and Coalgebra Objects
* 3.3 Bialgebra Objects of $\infty$-Categories
* 3.4 The Spectrum of a Bialgebra
* 3.5 The Affine Line
* 3.6 Smash Products of $E_{\infty}$-Spaces
* 3.7 The Cartier Dual of a Functor
* 3.8 Duality for Bialgebras
* 3.9 Duality for Hopf Algebras

#### 4 Biextensions and the Fourier-Mukai Transform
* 4.1 Line Bundles and Invertible Sheaves
* 4.2 Biextensions of Abelian Varieties
* 4.3 Digression: Tannaka Duality
* 4.4 Biextensions: Tannakian Perspective
* 4.5 Categorical Digression
* 4.6 The Convolution Product
* 4.7 The Fourier-Mukai Transform

#### 5 Duality Theory for Abelian Varieties
* 5.1 Perfect Biextensions
* 5.2 Dualizing Sheaves
* 5.3 Multiplicative Line Bundles
* 5.4 The Functor $\mathcal{P}ic^0_X$
* 5.5 Representability of the Functor $\mathcal{P}ic^m_X$
* 5.6 Existence of the Dual Abelian Variety

#### 6 $p$-Divisible Groups
* 6.1 Finite Flat Group Schemes
* 6.2 Epimorphisms and Monomorphisms
* 6.3 Cartier Duality for Finite Flat Group Schemes
* 6.4 $p$-Torsion Objects of $\infty$-Categories
* 6.5 $p$-Divisible Groups
* 6.6 Cartier Duality for $p$-Divisible Groups
* 6.7 The $p$-Divisible Group of a Strict Abelian Variety
* 6.8 Comparison of Duality Theories

#### 7 The Serre-Tate Theorem
* 7.1 Deformation Theory of the Functor $R \mapsto BT_h(R)$
* 7.2 The Case of a Trivial Square-Zero Extension
* 7.3 Proof of the Serre-Tate Theorem
* 7.4 Application: Lifting Abelian Varieties from Classical to Spectral Algebraic Geometry

category: reference

[[!redirects Elliptic Cohomology I: Spectral Abelian Varieties]]


elliptic curve

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Classically in [[complex geometry]], an _elliptic curve_ is a connected [[Riemann surface]] (a connected compact 1-dimensional [[complex manifold]]) of [[genus of a surface|genus]] 1, hence it is a [[torus]] equipped with the structure of a [[complex manifold]], or equivalently with [[conformal structure]].

The curious term "elliptic" is a remnant from the 19th century, a back-formation which refers to [[elliptic functions]] (generalizing circular functions, i.e., the classical trigonometric functions) and their natural domains as Riemann surfaces.

In more modern frameworks and in the generality of [[algebraic geometry]], an elliptic curve over a [[field]] $k$ or indeed over any [[commutative ring]] may be defined as a complete irreducible non-singular [[algebraic curve]] of [[arithmetic genus]]-1 over $k$, or even as a certain type of algebraic [[group scheme]].

Elliptic curves have many remarkable properties, and their deeper arithmetic study is one of the most profound subjects in present-day mathematics.

The [[moduli stack of elliptic curves]] equipped with its canonical map to the [[moduli stack of formal group laws]] plays a central role in [[chromatic homotopy theory]] at [[chromatic level]] 2, where it serves to parameterize [[elliptic cohomology theories]].

Elliptic curves over the complex numbers are also interpreted as those [[worldsheets]] in [[string theory]] whose [[correlators]] are the [[superstring]]'s [[partition function]], which is the [[Witten genus]]. Via the [[string orientation of tmf]] this connects to to the role of elliptic curves in [[elliptic cohomology theory]].

## Definition

### Over a general ring
 {#DefinitionOverGeneralRing}

Elliptic curves over a general [[commutative ring]] $R$ (hence in [[arithmetic geometry]]) are the well-behaved 1-dimensional [[group objects]] parameterized over the [[space]] $Spec(R)$ (the [[prime spectrum of a commutative ring|prime spectrum]] of $R$). (Notice the count of dimension: over the complex numbers a torus is complex 1-dimensional and in this sense one is looking at 1-dimensional group schemes here.) This we discuss below in

* [Conceptual definition](#OverGeneralRingConceptualDefinition)

More concretely there are various explicit and standard [[coordinates|coordinazations]] of elliptic curves as [[affine schemes]], hence as solution spaces to [[polynomial]] [[equations]]. This we discuss below in

* [Coordinatized as solutions to cubic equations](#OverGeneralRingAsSolutionsToEquations)

#### Conceptual definition
 {#OverGeneralRingConceptualDefinition}

\begin{defn} \label{DefinitionEllipticCurve} {#EllipticCurve}

An **elliptic curve** over a  [[commutative ring]] $R$ is a [[group scheme]] (a [[group object]] in the [[category]] of [[schemes]]) over $Spec(R)$ that is a relative 1-dimensional, [[smooth scheme|smooth]], [[proper scheme|proper]] curve over $R$.

\end{defn}

+-- {: .num_remark}
###### Remark

This implies that an elliptic curve has [[arithmetic genus]] $1$ (by a direct argument concerning the [[Chern class]] of the [[tangent bundle]].)

=--

+-- {: .num_defn}
###### Definition

An elliptic curve over a [[field]] of [[positive characteristic]] whose [[formal group law]] has [[height of a formal group|height]] equal to 2 is called a _[[supersingular elliptic curve]]_. Otherwise the height equals 1 and the elliptic curve is called _ordinary_.

=--

#### Coordinatized as solutions to cubic Weierstrass equations
 {#OverGeneralRingAsSolutionsToEquations}

Elliptic curves are examples of solutions to [[Diophantine equations]] of degree 3. We start by giving the equation valued over general rings, which is fairly complicated compared to the special case that it reduces to in the classical case over the [[complex numbers]]. The more elements in the ground ring are invertible, the more the equation may be simplified.

(See [Silverman 09, III.1](#Silverman09) for a textbook account and for instance ([QuickIntro](#QuickIntro)) for a quick survey.)

##### Weierstrass cubic, discriminant and $j$-invariant
 {#WeierstrassCubicDiscriminantAndJInvariant}

+-- {: .num_defn #GeneralWeierstrassCubic}
###### Definition

Let $R$ be a [[commutative ring]], then [[Zariski topology|Zariski locally]] over $Spec(R)$ a [[cubic curve]] is a solution in the corresponding [[projective space]] to an [[equation]] of the form

$$
  y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6
$$

for [[coefficients]] $a_1, a_2, a_3, a_4, a_6$.  This is called the _[[Weierstrass equation]]_.

=--

+-- {: .num_remark}
###### Remark

Much of the literature on elliptic curves considers def. \ref{GeneralWeierstrassCubic} for the case that $R$ is an [[algebraically closed field]], in which case there is no need to pass to a cover. But for the true global discussion necessary for the [[moduli stack of elliptic curves]] one needs the full generality.

=--

The [[non-singular algebraic variety|non-singular]] such solutions are the elliptic curves over $R$. Non-singularity is embodied in coordinates  as follows.

+-- {: .num_defn #DiscriminantAndJInvariant}
###### Definition

Let

$$
  \begin{aligned}
    b_2 & \coloneqq a_1^2 + 4 a_2
    \\
    b_4 &\coloneqq a_1 a_3 + 2a_4
    \\
    b_6 & \coloneqq a_3^2 + 4 a_6
    \\
    b_8 & \coloneqq a_1^2 a_6 - a_1 a_3 a_4 + a_2 a_3^2 + 4 a_2 a_6 - a_4^2
  \end{aligned}
$$

and in terms of these

$$
  \begin{aligned}
     c_4 & \coloneqq b_2^2 - 24 b_4
     \\
     c_6 & \coloneqq - b_2^3 + 36 b_2 b_4 - 216 b_6
     \\
     \Delta &\coloneqq - b_2^2 b_8 - 8 b_4^3 - 27 b_6^2 + 9 b_2 b_4 b_6
  \end{aligned}
  \,.
$$

Here $\Delta$ is called the **[[discriminant]]**.

Finally let

$$
  j \coloneqq c_4^3/\Delta
  \,,
$$

called the **[[j-invariant]]**.

=--

+-- {: .num_remark #RelationBetweenDiscriminantAndTheModularInvariants}
###### Remark

In def. \ref{DiscriminantAndJInvariant} the discriminant satisfies the relation

$$
  1728 \Delta = c_4^3 - c_6^2
  \,.
$$

=--

+-- {: .num_remark}
###### Remark

Over $R = \mathbb{C}$ the [[complex numbers]] the quantities $c_4$ and $c_6$ in def. \ref{DiscriminantAndJInvariant} are proportional to the [[modular forms]] called the [[Eisenstein series]] (see there)  $G_4$ and $G_6$.

=--

##### Elliptic curves, Nodal curves, Cuspidal curves
 {#EllipticCurvesNodalCurvesCuspidalCurves}

The following is a definition if one takes the coordinate-description as fundamental. If one takes the more abstract characterization of def. \ref{EllipticCurve} as fundamental then the following is a proposition.

+-- {: .num_defn}
###### Definition/Proposition

A solution to the Weierstrass cubic, def. \ref{GeneralWeierstrassCubic}, with modular invariants $c_4$, $c_6$ and discriminant $\Delta$ according to def. \ref{DiscriminantAndJInvariant} is

1. an _elliptic curve_ iff $\Delta \neq 0$;

1. a _[[nodal curve]]_ or _[[cubic curve]] with nodal [[singular point of an algebraic variety|singularity]]_  iff $\Delta = 0$ and $c_4 \neq 0$;

1. a _[[cuspidal curve]]_ or _[[cubic curve]] with [[cusp]] [[singular point of an algebraic variety|singularity]]_  iff $\Delta = 0$ and $c_4 = 0$ (which by remark \ref{RelationBetweenDiscriminantAndTheModularInvariants} is equivalent to $c_4 = 0$ and $c_6 = 0$)

=--

+-- {: .num_remark}
###### Remark

Adding the nodal curve to the [[moduli stack of elliptic curves]] yields its [[compactification]], and the [[formal neighbourhood]] of the nodal curve in that compactification is known as the _[[Tate curve]]_.

=--

##### Localization at 2 and 3

+-- {: .num_prop #WeierstrassEquationLocalAt2And3}
###### Proposition

If $2$ is invertible in $R$ (is a [[unit]] ), and hence generally over the [[localization of a ring|localization]] $R[\frac{1}{2}]$ of $R$ at 2, the general [[Weierstrass equation]], def. \ref{GeneralWeierstrassCubic}, is equivalent, to the equation

$$
  y^2  = 4 x^3  + b_2 x^2 + 2 b_4 x + b_6
$$

with the [[coefficients]] identified as in def. \ref{DiscriminantAndJInvariant}.

If moreover $3$ is also invertible in $R$, hence generally over $R[\frac{1}{2}, \frac{1}{3}]$ then this equation is equivalent to just

$$
  y^2 = x^3 - 27 c_4 x - 54 c_6
 \,.
$$

=--

### Over the complex numbers
 {#OverTheComplexNumbers}

#### In terms of algebraic geometry

+-- {: .num_remark}
###### Remark

If the [[ring]] $R= \mathbb{C}$ is the [[complex numbers]], then complex tori are indeed the solutions to the [[Weierstrass equation]] as in prop. \ref{WeierstrassEquationLocalAt2And3},  parameterized by a [[torus]] $z \in \mathbb{C}/\Lambda$ (as discussed in the section [in terms of complex geometry](#InTermsOfCpomplexGeometry)) via the [[Weierstrass elliptic function]] $\wp$ as
$(x = \wp(z), y = \wp'(z), )$ in the form

$$
  \wp'(z)^2 + 4 \wp(z)^3 - g_2 p(z) - g_3
  \,.
$$

=--

See e.g. ([Hain 08, section 5](#Hain08)) on how complex elliptic curves are expressed in this algebraic geometric fashion.

#### In terms of complex geometry
 {#InTermsOfCpomplexGeometry}

+-- {: .num_prop #CharacterizationOverC}
###### Proposition

An [[elliptic curve]], def. \ref{EllipticCurve}, over the [[complex numbers]] $\mathbb{C}$ is equivalently

* a [[Riemann surface]] $X$ of [[genus]] 1 with a base point $P \in X$ (e.g.[Hain 08, def. 1.1](#Hain08))

* a quotient $\mathbb{C}/\Lambda$ where $\Lambda$ is a [[lattice]] in $\mathbb{C}$;

* a compact complex [[Lie group]] of dimension 1.

* a [[smooth scheme|smooth]] [[algebraic curve]] of degree 3 in $\mathcal{P}$.

=--

+-- {: .num_remark}
###### Remark

From the second definition it follows that to study the
[[moduli space of elliptic curves]] it suffices to study the [[moduli space]] of [[lattices]] in $\mathbb{C}$.

=--

+-- {: .num_defn}
###### Definition

A **[[framed elliptic curve]]** is an elliptic curve $(X,P)$ in the sense of the first item in prop. \ref{CharacterizationOverC}, together with an [[ordering|ordered]] [[basis]] $(a,b)$ of $H_1(X, \mathbb{Z})$ with $(a \cdot b) = 1 $

For $n$ a natural number, a **[[level n-structure on an elliptic curve]]** over the complex numbers is similar data but with [[coefficients]] only in the [[cyclic group]] $\mathbb{Z}/n\mathbb{Z}$.

A **framed lattice** in $\mathbb{C}$ is a lattice $\Lambda$ together with an ordered basis $(\lambda_1, \lambda_2)$ of $\Lambda$ such that $Im(\lambda_2/\lambda_1) \gt 0$.

=--

Hence a framed elliptic curve is the quotient of the complex plane by a lattice _together_ with the information on how this quotient was obtained. This is useful for describing the [[moduli stack of elliptic curves]] over the complex numbers.

### Over the rational numbers
 {#OverTheRationalNumbers}

Over the [[rational numbers]]: [Sagemath: Elliptic curves over the rational numbers](http://www.sagemath.org/doc/reference/plane_curves/sage/schemes/elliptic_curves/ell_rational_field.html)

### Over the $p$-adic integers and $p$-adic numbers
 {#OverpAdics}

Over the [[p-adic integers]], see ([Conrad 07](#Conrad07)).

Over the [[p-adic numbers]], see ([Winter 11](#Winter11)).

## Level structures

In the case over the [[complex numbers]] an elliptic curve $\Sigma$ is equivalently the [[quotient]] of the [[complex plane]] by a framed [[lattice]]. If here one remembers the structure given by that framed lattice, this means equivalently to remember an ordered basis

$$
  (a_1, a_2)\in H_1(\Sigma, \mathbb{Z})
$$

of the [[ordinary homology]] group of $\Sigma$ with [[coefficients]] in the [[integers]].

If here one replaces the integers by a [[cyclic group]] $\mathbb{Z}/n\mathbb{Z}$ then one obtains what is called a
_[[level structure on an elliptic curve|level-n structure on an elliptic curve]]_. Level-$n$ structures on elliptic curves may also be defined over general rings.

These structures are useful in that the [[moduli stack]] of [[elliptic curves with level-n structure]] (a _[[modular curve]]_ in the case over the complex numbers) provides a finite [[covering]] of the full [[moduli stack of elliptic curves]].

## Group structure

A vital reason for the importance of elliptic curves in number theory, arithmetic geometry, and elsewhere is their group structure. This group structure is present by fiat in Definition \ref{DefinitionEllipticCurve}, and can be constructed given a Weierstrass equation.

The [[torsion element|torsion elements]] of this group structure when the elliptic curve is defined over a [[number field]] have good Galois-theoretic properties. See [[torsion points of an elliptic curve]] for more.

## Constructions

### Formal group law

Given an [[elliptic curve]] over $R$, $E \to Spec R$, we get a [[formal group]] $\hat E$ by completing $E$ along its identity [[section]] $\sigma_0$

$$
  E \to Spec(R) \stackrel{\sigma_0}{\to} E
  \,,
$$

we get a [[ringed space]] $(\hat E, \hat O_{E,0})$

+-- {: .num_example}
###### Example

If $R$ is a [[field]] $k$, then the [[structure sheaf]] $\hat O_{E,0} \simeq k[ [z] ]$

then

$$
  \hat O_{E \times E, (0,0)} \simeq
  \hat O_{E,0}
  \hat \otimes_k
  \hat O_{E,0}
  \simeq
  k[[x,y]]
$$

=--

+-- {: .num_example}
###### Example

**(Jacobi quartics)**

$$
  y^2 = 1- 2 \delta x^2 + \epsilon x^4
$$

defines $E$ over $R = \mathbb{Z}[Y_Z,\epsilon, \delta]$.

The corresponding [[formal group law]] is **Euler's formal group law**

\[
  \label{EulerFormalGroupLaw}
  f(x,y)
    \;=\;
  \frac{
x{\sqrt{1- 2 \delta y^2 + \epsilon y^4}} + y{\sqrt{1- 2 \delta x^2 + \epsilon x^4}}
  }
  {1- \epsilon x^2 y^2}
\]

if $\Delta \coloneqq \epsilon(\delta^2 - \epsilon)^2 \neq 0$ then this is a non-trivial elliptic curve.

=--

If $\Delta = 0$ then $f(x,y) \simeq G_m, G_a$ ([[additive group|additive]] or [[multiplicative group|multiplicative]] formal group law corresponding to [[ordinary cohomology]] and [[topological K-theory]] [[KU]], respectively).

### Elliptic cohomology

Elliptic curves, via their [[formal group law]]s, give the name to [[elliptic cohomology]] theories.

See also

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

## Related concepts

* [[abelian variety]]

* [[Tate curve]]

* [[Szpiro's conjecture]]

* [[cubic curve]]

* [[moduli stack of elliptic curves]]

  * [[modular form]], [[Jacobi form]]

  * [[topological modular form]]

* [[modularity theorem]]

* [[elliptic fibration]]

* [[elliptic cohomology]], [[tmf]]

## References

Classical accounts of the general case include

* [[Pierre Deligne]], [[Michael Rapoport]], _Les Schémas de Modules de Courbes Elliptiques_ In: Deligne P., Kuijk W. (eds) _Modular Functions of One Variable II_, Lecture Notes in Mathematics, vol 349. Springer (1973) ([doi:10.1007/978-3-540-37855-6_4](https://doi.org/10.1007/978-3-540-37855-6_4))

* [[Pierre Deligne]], _Courbes Elliptiques: Formulaire (d'apres J. Tate)_ ([web](http://modular.math.washington.edu/Tables/antwerp/deligne/))

* [[Nicholas Katz]], [[Barry Mazur]], _Arithmetic moduli of elliptic curves, Annals of Mathematics Studies_, vol. 108, Princeton University Press, 1985 ([jstor:j.ctt1b9s05p](https://www.jstor.org/stable/j.ctt1b9s05p))

Introductory lecture notes for elliptic curves over the complex numbers include

* {#Hain08} Richard Hain, _Lectures on Moduli Spaces of Elliptic Curves_ ([arXiv:0812.1803](http://arxiv.org/abs/0812.1803))

* {#Neeman07} [[Amnon Neeman]], section 1.2 of _Algebraic and analytic geometry_, London Math. Soc. Lec. Note Series __345__, 2007 ([publisher](http://www.cambridge.org/us/academic/subjects/mathematics/geometry-and-topology/algebraic-and-analytic-geometry))

and for the general case:

* {#QuickIntro} _A quick introduction to elliptic curves_ ([pdf](http://people.reed.edu/~jerry/311/ecintro.pdf))

* R. Sujatha, _Elliptic Curves & Number Theory_ ([[SujahtaElliptic.pdf:file]])

* {#Sutherland21} [[Andrew Sutherland]], _Elliptic curves_, 2021 ([web](https://ocw.mit.edu/courses/18-783-elliptic-curves-spring-2021/))

See also

* {#AndoHopkinsStrickland01} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], appendix B of: _Elliptic spectra, the Witten genus, and the theorem of the cube_, Inventiones Mathematicae, 146:595–687, 2001,  ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/musix.pdf), [doi:10.1007/s002220100175](https://doi.org/10.1007/s002220100175))

  (review of general elliptic curves in the context of [[elliptic genera]], [[elliptic spectra]] and the [[string orientation of tmf]])

Discussion of the [[formal group law]] of elliptic curves:

* Antonia W. Bluher, _Formal groups, elliptic curves, and some theorems of Couveignes_, in: J.P. Buhler (eds.) _Algorithmic Number Theory_ ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer 1998 ([arXiv:math/9708215](https://arxiv.org/abs/math/9708215), [doi:10.1007/BFb0054887]( https://doi.org/10.1007/BFb0054887))

* {#Friedl17} Stefan Friedl, _An elementary proof of the group law for elliptic curves_ ([arXiv:1710.00214](https://arxiv.org/abs/1710.00214))

For more along these lines see also at _[[arithmetic geometry]]_.

In the context of [[elliptic fibrations]]:

* [[Rick Miranda]], _The basic theory of elliptic surfaces_, lecture notes 1988 ([pdf](http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf))

A general textbook account is

* {#Silverman09} [[Joseph Silverman]], _The arithmetic of elliptic curves_, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094 (2010i:11005)

Discussion over the [[rational numbers]] includes

* Sagemath _[Elliptic curves over the rational numbers](http://www.sagemath.org/doc/reference/plane_curves/sage/schemes/elliptic_curves/ell_rational_field.html)_

Discussion of elliptic curves over the [[p-adic numbers]] includes

* {#Conrad07} [[Brian Conrad]], _Arithmetic moduli of generalized elliptic curves_, J. Inst. Math. Jussieu 6 (2007), no. 2, 209-278. ([pdf](http://math.stanford.edu/~conrad/papers/kmpaper.pdf))

* {#Winter11} Rosa Winter, _Elliptic curves over $\mathbb{Q}_p$_, 2011 ([pdf](http://www.math.leidenuniv.nl/scripties/BachWinter.pdf))

See also

* _[[A Survey of Elliptic Cohomology - elliptic curves]]_

[[!redirects elliptic curves]]


elliptic differential operator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[differential operator|differential]] or [[pseudodifferential operator]] is **elliptic** if its [[principal symbol]] is invertible.

## Related concepts

* [[elliptic chain complex]]

* [[generalized elliptic operator]]

* [[index theory]], [[KK-theory]]

* [[hyperbolic differential operator]]

## References

* [eom](http://eom.springer.de/e/e035500.htm)

[[!redirects elliptic operator]]

[[!redirects elliptic operators]]
[[!redirects elliptic differential operators]]

[[!redirects elliptic differential equation]]
[[!redirects elliptic differential equations]]


elliptic fibration

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Elliptic cohomology
+-- {: .hide}
[[!include elliptic cohomology -- contents]]
=--
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _elliptic fibration_ is a [[bundle]] of [[elliptic curves]], possibly including some [[singularity|singular]] [[fibers]].

An _elliptic surface_ is an elliptic fibration over an [[algebraic curve]].

## Properties

### Classification by local systems with modular group coefficients

Write $SL_2(\mathbb{Z})$ for the [[special linear group]] in dimension 2 with [[integer]] [[coefficients]] and write $SL_2(\mathbb{Z}) \to PSL_2(\mathbb{Z})$ for the [[projection]] to the corresponding [[projective linear group]]. Regarding this as the [[Möbius group]] it comes with its natural [[action]] on the [[upper half plane]] $\mathfrak{h}$. The [[homotopy quotient]]  $\mathcal{M}_{ell}(\mathbb{C}) = \mathfrak{h}//SL_2(\mathbb{Z})$ is the [[moduli stack of elliptic curves]] over the complex numbers.

Accordingly, to any $SL_2(\mathbb{Z})$-[[principal bundle]] $P \to B$  (necessarily [[flat connection|flat]] since $SL_2(\mathbb{Z})$ is a [[discrete group]], hence a "[[local system]]") is [[associated bundle|associated]] a $\mathfrak{h}$-[[fiber bundle]] such that a [[section]] of it defines a non-[[singularity|singular]] elliptic fibration.

One may turn this around: Given an elliptic fibration $E \to B$, then away from the points $S\subset B$ over which the fiber is singular, it is given by an $SL_2(\mathbb{Z})$-local system together with a section of the associated upper-half plane bundle on $B-S$.

With due technical care, this data uniquely characterizes the elliptic fibration (e.g. [Miranda 88, prop. VI.3.3](#Miranda88)).

### Classification of the singular fibers
 {#ClassificationOfSingularFibers}

The remaining [[singularity|singular]] fibers follow an [[ADE classification]] ([Kodaira 64](#Kodaira64), [Néron 64](#Neron64), [Kodaira 66](#Kodaira66))

## Examples

### Elliptic fibration of K3-surfaces

See at _[[elliptic fibration of a K3-surface]]_.

## Related entries

* [[F-theory]]

  * [[F/M-theory on elliptically fibered Calabi-Yau 4-folds]]

* [[K3]]

## References

Specifically for [[complex surfaces]]:

* {#Miranda88} [[Rick Miranda]], _The basic theory of elliptic surfaces_, lecture notes 1988 ([pdf](http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf))

* {#FriedmanMorgan94} [[Robert Friedman]], [[John Morgan]], _Smooth Four-Manifolds and Complex Surfaces_, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics (1994) ([doi:10.1007/978-3-662-03028-8](https://www.springer.com/gp/book/9783540570585))

* [[Fedor Bogomolov]], [[Yuri Tschinkel]], *Monodromy of elliptic surfaces* [[[MonodromyOfEllipticSurfaces.pdf:file]], [arXiv:math/0002168](https://arxiv.org/abs/math/0002168)]

* Takahiko Yoshida, _Locally standard torus fibrations_ [pdf](http://www.isc.meiji.ac.jp/~takahiko/papers/sympo.pdf)

See also:

* Wikipedia, _[Elliptic surface](http://en.wikipedia.org/wiki/Elliptic_surface)_

Specifically for [[elliptically fibered K3-surfaces]]:

* Viacheslav Nikulin, _Elliptic fibrations on K3 surfaces_ ([arXiv:1010.3904](http://arxiv.org/abs/1010.3904))

The [[ADE classification]] of the possible singular fibers is due to

* {#Kodaira64} [[Kunihiko Kodaira]],  (1964). "On the structure of compact complex analytic surfaces. I". Am. J. Math. 86: 751–798. doi:10.2307/2373157. Zbl 0137.17501.

* {#Kodaira66} [[Kunihiko Kodaira]],  (1966). "On the structure of compact complex analytic surfaces. II". Am. J. Math. 88: 682–721. doi:10.2307/2373150. Zbl 0193.37701.

and

* {#Neron64} [[André Néron]],  (1964). "Modeles minimaux des variétés abeliennes sur les corps locaux et globaux". Publications Mathématiques de l'IHÉS (in French) 21: 5–128.

[[!redirects elliptic fibrations]]

[[!redirects elliptic surface]]
[[!redirects elliptic surfaces]]

[[!redirects Kodeira classification]]
[[!redirects Kodeira classifications]]

[[!redirects Kodeira's classification]]
[[!redirects Kodeira's classifications]]


elliptic function

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Trigonometry
+-- {: .hide}
[[!include trigonometry -- contents]]
=--
=--
=--

## Examples

* [[Weierstrass elliptic function]]

## References

* Wikipedia, _[Elliptic function](https://en.wikipedia.org/wiki/Elliptic_function)_

[[!redirects elliptic functions]]


elliptic genera as partition functions -- references

### Elliptic genera as super $p$-brane partition functions
  {#EllipticGeneraAsBranePartitionFunctionsReferences}

The interpretation of [[elliptic genera]] (especially the [[Witten genus]]) as the [[partition function]] of a [[2d superconformal field theory]] (or [[Landau-Ginzburg model]]) --  and especially of the [[heterotic string]] ("H-string") or [[type II superstring]] [[worldsheet]] theory has precursors in

* [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomalies and modular invariance in string theory*, Physics Letters B 177 (3-4), 317-323, 1986 (doi:10.1016/0370-2693(86)90760-4)

* [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomalies, characters and strings*, Nuclear Physics B Volume 287, 1987, Pages 317-361 (doi:10.1016/0550-3213(87)90108-8)

* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]],  [[Nicholas P. Warner]], *Anomaly cancelling terms from the elliptic genus*, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (doi:10.1016/0550-3213(88)90468-3)

and then strictly originates with:

* {#Witten87} [[Edward Witten]], _Elliptic genera and quantum field theory_, Comm. Math. Phys. Volume 109, Number 4 (1987), 525-536.  ([euclid:cmp/1104117076](http://projecteuclid.org/euclid.cmp/1104117076))

* [[Edward Witten]], _On the Landau-Ginzburg Description of $N=2$ Minimal Models_, Int. J. Mod. Phys.A9:4783-4800,1994 ([arXiv:hep-th/9304026](http://arxiv.org/abs/hep-th/9304026))

* [[Toshiya Kawai]], [[Yasuhiko Yamada]], Sung-Kil Yang, _Elliptic Genera and $N=2$ Superconformal Field Theory_, Nucl. Phys. B414:191-212, 1994 ([arXiv:hep-th/9306096](http://arxiv.org/abs/hep-th/9306096), doi:10.1016/0550-3213(94)90428-6)

* Sujay K. Ashok, Jan Troost, _A Twisted Non-compact Elliptic Genus_, JHEP 1103:067, 2011 ([arXiv:1101.1059](http://arxiv.org/abs/1101.1059))

* [[Matthew Ando]], [[Eric Sharpe]], _Elliptic genera of Landau-Ginzburg models over nontrivial spaces_, Adv. Theor. Math. Phys. 16 (2012) 1087-1144 ([arXiv:0905.1285](https://arxiv.org/abs/0905.1285))

Review in:

* {#Cheng13} [[Miranda Cheng]], _(Mock) Modular Forms in String Theory and Moonshine_, lecture notes 2016 ([[ChengModularForms16.pdf:file]])

* [[Katrin Wendland]], Section 2.4 in: _Snapshots of Conformal Field Theory_, in: _Mathematical Aspects of Quantum Field Theories_ Mathematical Physics Studies. Springer 2015  ([arXiv:1404.3108](http://de.arxiv.org/abs/1404.3108), [doi:10.1007/978-3-319-09949-1_4](https://doi.org/10.1007/978-3-319-09949-1_4))

#### Formulations

##### Via super vertex operator algebra

Formulation via [[super vertex operator algebras]]:

* Hirotaka Tamanoi, _Elliptic Genera and Vertex Operator Super-Algebras_, Springer 1999 ([doi:10.1007/BFb0092541](https://link.springer.com/book/10.1007/BFb0092541))

* {#DLM02} Chongying Dong, [[Kefeng Liu]], Xiaonan Ma, _Elliptic genus and vertex operator algebras_, Algebr. Geom. Topol. 1 (2001) 743-762 ([arXiv:math/0201135](http://arxiv.org/abs/math/0201135), [doi:10.2140/agt.2001.1.743](http://dx.doi.org/10.2140/agt.2001.1.743))

and for  the [[topological twist|topologically twisted]] [[2d (2,0)-superconformal QFT]] (the [[heterotic string]] with enhanced supersymmetry) via [[sheaves of vertex operator algebras]] in

* {#Cheung10} [[Pokman Cheung]], _The Witten genus and vertex algebras_ ([arXiv:0811.1418](http://arxiv.org/abs/0811.1418))

based on [[chiral differential operators]]:

* [[Vassily Gorbounov]], [[Fyodor Malikov]], [[Vadim Schechtman]], _Gerbes of chiral differential operators_, Math. Res. Lett. 7(1), 55–66 (2000) ([arXiv:math/9906117](http://arxiv.org/abs/math/9906117), [arXiv:math/0003170](http://arxiv.org/abs/math/0003170), [arXiv:math/0005201](http://arxiv.org/abs/math/0005201))

In relation to error-correcting codes:

* Kohki Kawabata, Shinichiro Yahagi, *Elliptic genera from classical error-correcting codes* $[$[arXiv:2308.12592](https://arxiv.org/abs/2308.12592)$]$

##### Via Dirac-Ramond operators on free loop space

Tentative interpretation as [[indices]] of [[Dirac-Ramond operators]] as would-be [[Dirac operators on smooth loop space]]:

* {#Witten87b} [[Edward Witten]], *The Index Of The Dirac Operator In Loop Space*, in: *Elliptic Curves and Modular Forms in Algebraic Topology*, Lecture Notes in Mathematics **1326**, Springer (1988) 161-181 [[doi:10.1007/BFb0078045](https://doi.org/10.1007/BFb0078045), [spire](http://inspirehep.net/record/245523)]

  originating from:

  [[Edward Witten]], p. 92-94 in: *Global anomalies in string theory*, in: W. Bardeen and A. White (eds.) *[Symposium on Anomalies, Geometry, Topology](https://inspirehep.net/conferences/965785?ui-citation-summary=true)*, World Scientific (1985) 61-99 [[[WittenGlobalAnomaliesInStringTheory.pdf:file]], [spire:214913](https://inspirehep.net/literature/214913)]

* {#AlvarezKillingbackManganoWindey87} [[Orlando Alvarez]], T. P. Killingback, Michelangelo Mangano, [[Paul Windey]], _The Dirac-Ramond operator in string theory and loop space index theorems_, Nuclear Phys. B Proc. Suppl., 1A:189–215, 1987, in: _Nonperturbative methods in field theory_, 1987 (doi"10.1016/0920-5632(87)90110-1)

* [[Orlando Alvarez]], T. P. Killingback, Michelangelo Mangano, [[Paul Windey]], _String theory and loop space index theorems_, Comm. Math. Phys., 111(1):1–10, 1987 ([euclid:cmp/1104159462](https://projecteuclid.org/euclid.cmp/1104159462))

* {#Landweber99} [[Gregory Landweber]], _Dirac operators on loop space_, PhD thesis (Harvard 1999) ([pdf](http://math.bard.edu/greg/LoopDirac.pdf))

* [[Orlando Alvarez]], [[Paul Windey]], _Analytic index for a family of Dirac-Ramond operators_, Proc. Natl. Acad. Sci. USA, 107(11):4845–4850, 2010 ([arXiv:0904.4748](https://arxiv.org/abs/0904.4748))

##### Via conformal nets

Tentative formulation via [[conformal nets]]:

* [[Chris Douglas]], [[André Henriques]], _Topological modular forms and conformal nets_, in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_, Proceedings of Symposia in Pure Mathematics, AMS (2011) ([arXiv:1103.4187](https://arxiv.org/abs/1103.4187), [doi:10.1090/pspum/083](https://doi.org/10.1090/pspum/083))

#### Conjectural interpretation in tmf-cohomology
 {#InterpretationInTMFCohomology}

The resulting suggestion that, roughly, deformation-classes ([[concordance]] classes) of [[2d SCFTs]] with [[target space]] $X$ are the [[Whitehead-generalized cohomology|generalized cohomology]] of $X$ with [[coefficients]] in the [[spectrum]] of [[topological modular forms]] ([[tmf]]):

* {#StolzTeichner11} [[Stephan Stolz]], [[Peter Teichner]], _Supersymmetric field theories and generalized cohomology_, in [[Hisham Sati]], [[Urs Schreiber]] (eds.), _[Mathematical foundations of Quantum field theory and String theory](http://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory#ContributionStolzTeichner)_, Proceedings of Symposia in Pure Mathematics, Volume 83, AMS 2011 ([arXiv:1108.0189](https://arxiv.org/abs/1108.0189))

and the more explicit suggestion that, under this identification, the [[Chern-Dold character]] from [[tmf]] to [[modular forms]], sends a [[2d SCFT]] to its [[partition function]]/[[elliptic genus]]/supersymmetric index:

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], Section 5 of: *Holomorphic SCFTs with small index*, Canadian Journal of Mathematics ([arXiv:1811.00589](https://arxiv.org/abs/1811.00589), [doi:10.4153/S0008414X2100002X](https://doi.org/10.4153/S0008414X2100002X))

This perspective is also picked up in [Gukov, Pei, Putrov & Vafa 18](#GukovPeiPutrovVafa18).

Discussion of the [[2d SCFTs]] (namely [[supersymmetry|supersymmetric]] [[SU(2)]]-[[WZW-models]]) conjecturally corresponding, under this conjectural identification, to
the elements of
$\mathbb{Z}/24$
[$\simeq$](tmf#BoardmanHomomorphismInTmfIs6Connected)
$
  tmf^{-3}(\ast)
    =
  \pi_3(tmf)
$
[$\simeq$](tmf#BoardmanHomomorphismInTmfIs6Connected)
$\pi_3(\mathbb{S})$
(the [[third stable homotopy group of spheres]]):

* {#GaiottoJohnsonFreydWitten19} [[Davide Gaiotto]], [[Theo Johnson-Freyd]], [[Edward Witten]], p. 17 of: *A Note On Some Minimally Supersymmetric Models In Two Dimensions*, ([arXiv:1902.10249](https://arxiv.org/abs/1902.10249)) in S. Novikov et al. *Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry*, Proc. Symposia Pure Math., 103(2), 2021 ([ISBN: 978-1-4704-5592-7](https://bookstore.ams.org/pspum-103-2))

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Mock modularity and a secondary elliptic genus* ([arXiv:1904.05788](https://arxiv.org/abs/1904.05788))

* [[Theo Johnson-Freyd]], *Topological Mathieu Moonshine* ([arXiv:2006.02922](https://arxiv.org/abs/2006.02922))

Discussion properly via [[(2,1)-dimensional Euclidean field theory]]:

* [[Daniel Berwick-Evans]], *How do field theories detect the torsion in topological modular forms?* $[$[arXiv:2303.09138](https://arxiv.org/abs/2303.09138)$]$

* [[Daniel Berwick-Evans]], *How do field theories detect the torsion in topological modular forms?*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) $[$[web](Center+for+Quantum+and+Topological+Systems#BerwickEvansMar23), video:[YT](https://www.youtube.com/watch?v=fw7yxFsDmjs)$]$

See also

* Ying-Hsuan Lin, [[Du Pei]], *Holomorphic CFTs and topological modular forms* [[arXiv:2112.10724](https://arxiv.org/abs/2112.10724)]

* Jan Albert, Justin Kaidi, Ying-Hsuan Lin, *Topological modularity of Supermoonshine* $[$[arXiv:2210.14923](https://arxiv.org/abs/2210.14923)$]$

* [[Yuji Tachikawa]], [[Mayuko Yamashita]], [[Kazuya Yonekura]], *Remarks on mod-2 elliptic genus* $[$[arXiv:2302.07548](https://arxiv.org/abs/2302.07548)$]$

#### Occurrences in string theory

##### H-string elliptic genus
{#ReferencesEllipticGeneraAsPartitionFunctionsOfHStrings}

Further on the elliptic genus of the [[heterotic string]] being the [[Witten genus]]:

The interpretation of [[equivariant elliptic genera]] as [[partition functions]] of [[parametrized WZW models]] in [[heterotic string theory]]:

* {#DistlerSharpe07} [[Jacques Distler]], [[Eric Sharpe]], section 8.5 of _Heterotic compactifications with principal bundles for general groups and general levels_, Adv. Theor. Math. Phys. 14:335-398, 2010 ([arXiv:hep-th/0701244](http://arxiv.org/abs/hep-th/0701244))

* {#Ando07} [[Matthew Ando]], _Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe_, [talk 2007](http://www.math.ucsb.edu/~drm/GTPseminar/2007-fall.php) ([lecture notes pdf](http://www.math.ucsb.edu/~drm/GTPseminar/notes/20071026-ando/20071026-malmendier.pdf))

Proposals on physics aspects of lifting the [[Witten genus]] to [[topological modular forms]]:

* [[Yuji Tachikawa]], *Topological modular forms and the absence of a heterotic global anomaly*, Progress of Theoretical and Experimental Physics, **2022** 4 (2022) 04A107 $[$[arXiv:2103.12211](https://arxiv.org/abs/2103.12211), [doi:10.1093/ptep/ptab060](https://doi.org/10.1093/ptep/ptab060)$]$

* [[Yuji Tachikawa]], [[Mayuko Yamashita]], *Topological modular forms and the absence of all heterotic global anomalies*, Comm. Math. Phys. **402** (2023) 1585-1620 $[$[arXiv:2108.13542](https://arxiv.org/abs/2108.13542), [doi:10.1007/s00220-023-04761-2](https://doi.org/10.1007/s00220-023-04761-2)$]$

* [[Yuji Tachikawa]], [[Mayuko Yamashita]], *Anderson self-duality of topological modular forms, its differential-geometric manifestations, and vertex operator algebras* $[$[arXiv:2305.06196](https://arxiv.org/abs/2305.06196)$]$

##### M5-brane elliptic genus
{#ReferencesEllipticGeneraAsPartitionFunctionsOfM5Branes}

On the [[M5-brane elliptic genus]]:

A [[2d SCFT]] argued to describe the  [[KK-compactification]] of the [[M5-brane]] on a [[4-manifold]] (specifically: a [[complex surface]]) originates with

* [[Juan Maldacena]], [[Andrew Strominger]], [[Edward Witten]], _Black Hole Entropy in M-Theory_, JHEP 9712:002, 1997 ([arXiv:hep-th/9711053](https://arxiv.org/abs/hep-th/9711053))

Discussion of the resulting [[elliptic genus]] ([[2d SCFT]] [[partition function]]) originates with:

* [[Davide Gaiotto]], [[Andrew Strominger]], [[Xi Yin]], _The M5-Brane Elliptic Genus: Modularity and BPS States_, JHEP 0708:070, 2007 ([hep-th/0607010](https://arxiv.org/abs/hep-th/0607010))

* [[Davide Gaiotto]], [[Xi Yin]], _Examples of M5-Brane Elliptic Genera_, JHEP 0711:004, 2007 ([arXiv:hep-th/0702012](https://arxiv.org/abs/hep-th/0702012))

Further discussion in:

* Murad Alim, [[Babak Haghighat]], Michael Hecht, [[Albrecht Klemm]], Marco Rauch, Thomas Wotschke, *Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes*, Comm. Math. Phys. **339** (2015) 773–814 $[$[arXiv:1012.1608](https://arxiv.org/abs/1012.1608), [doi:10.1007/s00220-015-2436-3](https://doi.org/10.1007/s00220-015-2436-3)$]$

* {#GukovPeiPutrovVafa18} [[Sergei Gukov]], [[Du Pei]], [[Pavel Putrov]], [[Cumrun Vafa]], *4-manifolds and topological modular forms*,  J. High Energ. Phys. **2021** 84 (2021) $[$[arXiv:1811.07884](https://arxiv.org/abs/1811.07884), doi:10.1007/JHEP05(2021)084, [spire:1704312](https://inspirehep.net/literature/1704312)$]$

##### M-string elliptic genus
{#ReferencesEllipticGeneraAsPartitionFunctionsOfMStrings}

On the [[elliptic genus]] of [[M-strings]] inside [[M5-branes]]:

* [[Stefan Hohenegger]], [[Amer Iqbal]],  _M-strings, Elliptic Genera and $\mathcal{N}=4$ String Amplitudes_, Fortschritte der PhysikVolume 62, Issue 3 ([arXiv:1310.1325](http://arxiv.org/abs/1310.1325))

* [[Stefan Hohenegger]], [[Amer Iqbal]], Soo-Jong Rey, _M String, Monopole String and Modular Forms_, Phys. Rev. D 92, 066005 (2015) ([arXiv:1503.06983](https://arxiv.org/abs/1503.06983))

* M. Nouman Muteeb, _Domain walls and M2-branes partition functions: M-theory and ABJM Theory_ ([arXiv:2010.04233](https://arxiv.org/abs/2010.04233))

* Kimyeong Lee, Kaiwen Sun, Xin Wang, *Twisted Elliptic Genera* [[arXiv:2212.07341](https://arxiv.org/abs/2212.07341)]

##### E-string elliptic genus
{#ReferencesEllipticGeneraAsPartitionFunctionsOfEStrings}

On the [[elliptic genus]] of [[E-strings]] as [[wrapped brane|wrapped]] [[M5-branes]]:

* J. A. Minahan, D. Nemeschansky, [[Cumrun Vafa]], N. P. Warner,
_E-Strings and $N=4$ Topological Yang-Mills Theories_, Nucl. Phys. B527 (1998) 581-623 ([arXiv:hep-th/9802168](https://arxiv.org/abs/hep-th/9802168))

* Wenhe Cai, Min-xin Huang, Kaiwen Sun, _On the Elliptic Genus of Three E-strings and Heterotic Strings_, J. High Energ. Phys. 2015, 79 (2015).
([arXiv:1411.2801](https://arxiv.org/abs/1411.2801), doi:10.1007/JHEP01(2015)079)

On the [[elliptic genus]] of [[E-strings]] as [[M2-branes]] [[brane intersection|ending on]] [[M5-branes]]:

* {#KKLPV14} Joonho Kim, Seok Kim, Kimyeong Lee, [[Jaemo Park]], [[Cumrun Vafa]], _Elliptic Genus of E-strings_, JHEP 1709 (2017) 098 ([arXiv:1411.2324](https://arxiv.org/abs/1411.2324))


elliptic genus

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#### String theory
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{:toc}

## Idea

An _elliptic genus_  is a [[genus]] in [[elliptic cohomology]] ([Landweber-Ravenel-Stong 93](#LandweberRavenelStong93)). In analogy to how there is a "universal elliptic cohomology", namely [[tmf]], there is a universal elliptic genus -- the [[Witten genus]]. This arises as the [[large volume limit]] of the [[partition function]] of the [[superstring]] whose [[target space]] is the given [[manifold]].

## Definition
 {#Definition}

The original definition of elliptic genus is due to ([Ochanine 87](#Ochanine87)) (see the review ([Ochanine 09](#Ochanine09))) and says that an _[[genus]]_ of [[orientation|oriented]] manifolds is called an elliptic genus if it vanishes on manifolds which are [[projective spaces]] of the form $\mathbb{C}P(\xi)$ for $\xi$ an even-dimensional [[complex vector bundle]] over an [[orientation|oriented]] [[closed manifold]].

The terminomology _elliptic_ for this was motivated by the central theorem of ([Ochanine 87](#Ochanine87)) which says that every genus $\phi$ satisfying this condition has a [logarithm](genus#LogarithmAndCharacteristicSeries) $log_\phi$ of the form

$$
  log_\phi(u) = \int_{0^u} (1- 2 \delta t^2+  \epsilon t^4)^{-1/2}
$$

for some constants $\delta, \epsilon$. Hence for non-degenerate choices of parameters ($\delta^2 \neq \epsilon$ and $\epsilon \neq 0$) in the square root this is the expansion at 0 of an elliptic function.

So the logarithm here is an [[elliptic integral]] and that was the original reason for the term "elliptic genus".

## Examples

### Degenerate case: Signature genus

The degenerate case with parameters $\delta = \epsilon = 1$ (as [above](#Definition)) is the [[signature genus]].

### Degenerate case: $\hat A$-genus

The degenerate case with parameters $\delta = - \frac{1}{8}$ and $\epsilon = 0$ (as [above](#Definition)) is the [[A-hat genus]].

### Universal case: Witten genus

Given an elliptic genus with non-degenerate parameters $\delta, \epsilon \in \mathbb{C}$ (as [above](#Definition), see also at _[[j-invariant]]_), the [[Jacobi quartic]] [[Riemann surface]] which is given by the [[equation]]

$$
  Y^2 = X^4 - 2 \delta X^2 + \epsilon
$$

is naturally parameterized by the [[upper half plane]]. Under this identification obe may think of $\epsilon$ and $\delta$ as functions of moduli of [[elliptic curves]] and concretely as [[modular forms]] for the [[subgroup]] $\Gamma_0(2)$ of that of Möbius transformations.

Viewed this way the collection of all elliptic genera provides a single [[genus]] with [[coefficients]] in this ring $MF_\bullet(\Gamma_0(2))$ of [[modular forms]]

$$
  w \colon \Omega^{SO}_\bullet \longrightarrow MF_\bullet(\Gamma_0(2))
$$

(such that postcomposition with evaluation on any elliptic curve parameterized by the given value of $\delta$ and $\epsilon$ produces the corrponding elliptic genus).

This "universal" elliptic genus is the _[[Witten genus]]_.

## Properties

### Integrality on Spin-manifolds

On [[manifolds]] with [[spin structure]] the elliptic genus
takes values in integral series $\mathbb{Z}[ [q] ]$.

([Chudnovsky-Chudnovsky 88](#ChudnovskyChudnovsky88), [Landweber 88, section 5](#Landweber88) [Kreck-Stolz 93](#KreckStolz93), [Hovey 91](#Hovey91))

### Relation to partition functions of superstring

The [[partition function]] of a [[type II superstring]] as a function depending on the [[modulus]] of the [[worldsheet]] [[elliptic curve]] yields an elliptic genus ([Witten 87](#Witten87)). (The analog for the [[heterotic string]] is hence called the _[[Witten genus]]_ with values in the "universal elliptic cohomology" theory, [[tmf]]).

For equivariant/gauged string [[sigma-models]] the elliptic genus should take values in [[equivariant elliptic cohomology]], see at _[gauged WZW mode -- Partition function in elliptic cohomology](gauged+WZW+model#PartitionFunctionInEllipticCohomology)_.

### Rigidity theorem

See _[[rigidity theorem for elliptic genera]]_.

## Related concepts

* [[elliptic cohomology]]

* [[spin orientation of Ochanine elliptic cohomology]]

* [[Witten genus]], [[string orientation of tmf]]

* [[equivariant elliptic genus]]

[[!include genera and partition functions - table]]

## References

[[!include elliptic cohomology -- references]]

[[!redirects elliptic genera]]

[[!redirects Ochanine genus]]
[[!redirects Ochanine elliptic genus]]


elliptic geometry

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## Idea

[[non-Euclidean geometry]] for [[constant function|constant]] [[positive number|positive]] [[sectional curvature]].

## Related concepts

| [[synthetic geometry]]  |
|-------------------------|
| [[Euclidean geometry]]  |
| [[hyperbolic geometry]] |
| [[elliptic geometry]]   |

* [[trigonometry]]

* [[spherical space form]]

## References

* Wikipedia, _[Elliptic geometry](http://en.wikipedia.org/wiki/Elliptic_geometry)_


elliptic Selberg integral

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## Related concepts

* [[Selberg integral]]

## References

* J.F. van Diejen, V.P. Spiridonov, _An elliptic Macdonald-Morris conjecture
and multiple modular hypergeometric sums_, Math. Res. Letters __7__ (2000) 729--746; _Elliptic Selberg integrals_, Internat. Math. Res. Notices __20__, (2001), 1083--1110; _Modular hypergeometric residue sums of elliptic Selberg integrals_, Lett. Math. Phys. __58__ (2001), 223--238.
* [[G. Felder]], L. Stevens, [[A. Varchenko]], _Elliptic Selberg integrals and conformal blocks_, Math. Res. Lett. __10__ (2003), no. 5-6, 671–684, [MR2004i:32024](http://www.ams.org/mathscinet-getitem?mr=2024724), [pdf](https://arxiv.org/pdf/math/0210040.pdf)
* V. P. Spiridonov, _Elliptic beta integrals and solvable models of statistical mechanics_, [arxiv/1011.3798](http://arxiv.org/abs/1011.3798)


elliptic spectrum

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## Idea

An _elliptic spectrum_ is a [[spectrum]] which [[Brown representability theorem|represents]] an [[elliptic cohomology theory]].

## Definition

For $E$ a [[ring spectrum]], write $E^\bullet(\ast)$ for its coefficient ring and generally $E^\bullet(X)$ for its [[generalized cohomology theory|generalized]] [[cohomology ring]] over any $X$.

+-- {: .num_defn #EllipticSpectrum}
###### Definition

An _elliptic spectrum_ is a [[triple]] consisting of

1. an [[elliptic curve]] $A$ over the coefficient ring $E^\bullet(\ast)$;

1. an [[even cohomology theory|even]] [[periodic ring spectrum]] $E$;

1. an [[equivalence]]

   $$
     Spec E^\bullet(BU(1)) \stackrel{\simeq}{\longrightarrow} Pic_A^0
   $$

   between the [[spectrum of a commutative ring|algebraic spectrum]] of the $E$-[[cohomology ring]] over the [[classifying space]] for [[complex line bundles]] (see at _[[complex oriented cohomology theory]]_) and the [[formal Picard group]] $Pic_A^0$ of $A$.

=--

This is due to ([Ando-Hopkins-Strickland01, def. 1.2](AndoHopkinsStrickland01)). See for instance also ([Gepner 05, def. 15](#Gepner05)).

+-- {: .num_remark #RoleOfFormalPicardScheme}
###### Remark
**(role of the [[formal Picard scheme]])**

Originally (and still in many or even most references), def. \ref{EllipticSpectrum} is stated with the [[formal Picard group]] $Pic_A^0$ replaced by the [[formal completion]] $\hat A$ of $A$ at its [[neutral element]].

These two versions of the definition in itself are equivalent, since [[elliptic curves]] are self-[[dual abelian varieties]] equipped with a canonical [[isomorphism]] $\hat A \simeq Pic^0_A$exhibited by the [[Poincaré line bundle]].

But for the development of the theory, notably for application to [[equivariant elliptic cohomology]], for the relation of [[elliptic cohomology]] to [[loop group representations]] etc., it is crucial to understand that $E^\bullet(B U(1))$ is the space of [[sections]] of a [[line bundle]] over a (formal) [[moduli space]] of [[line bundles]] on, in turn, the elliptic curve, instead of directly on the elliptic curve itself.

Indeed, generally for $G$ a [[compact Lie group]], we have that $E^\bullet(B G)$ is the [[space of sections]] of the [[WZW model]]-[[line bundle]] for [[conformal blocks]] (the [[prequantum line bundle]] of the $G$-[[Chern-Simons theory]]) on the (formal) [[moduli space of flat connections]] on $G$-[[principal bundles]] over the elliptic curve. This is the central statement at _[[equivariant elliptic cohomology]]_. As the appearance of the [[WZW model]] here shows, this is also crucial for understanding the role of elliptic spectra in [[quantum field theory]]/[[string theory]], see at _[equivariant elliptic cohomology -- Interpretation in Quantum field theory/String theory](equivariant%20elliptic%20cohomology#InterpretationInQuantumFieldTheory)_ for more on this.

Moreover, understanding $Spec E^\bullet(BU(1))$ as being about [[moduli space of (higher) line bundles|moduli of line bundles]] on the elliptic curve is crucial for understanding the generalization of the concept of elliptic spectra, for instance to [[K3-spectra]]. This is indicated in the following table

[[!include moduli of higher lines -- table]]

=--

## Related concepts

* [[elliptic cohomology]], [[elliptic genus]], [[tmf]]

* [[Eilenberg-MacLane spectrum]]

* [[K3-spectrum]]

## References

The concept of elliptic spectrum was introduced in

* {#AndoHopkinsStrickland01} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _Elliptic spectra, the Witten genus, and the theorem of the cube_, Inventiones Mathematicae, 146:595–687, 2001,  ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/musix.pdf), [doi:10.1007/s002220100175](https://doi.org/10.1007/s002220100175))

A brief review is for instance in  of

* {#Gepner05} [[David Gepner]], section 1 of _[[Homotopy topoi and equivariant elliptic cohomology]]_, 2005.

Survey includes

* [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology]]_

* [[Charles Rezk]], _Elliptic cohomology and elliptic curves_, Felix Klein Lectures, Bonn  2015 ([web](http://www.hcm.uni-bonn.de/fkl-rezk/))

[[!redirects elliptic spectra]]


elliptically fibered K3-surface

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## Idea

Complex algebraic [[K3-surfaces]] admit the [[mathematical structure|structure]] of [[elliptic fibrations]] over the [[Riemann sphere]] $\mathbb{C}P^1$. Equipped with such, they are also called _elliptic K3-surfaces_.

## Properties

### Singular points
 {#SingularPoints}

Counted with multiplicity, these elliptic fibrations have 24 singular points (where the [[elliptic curve]]-fiber degenerates, to a [[nodal curve]] or [[cuspidal curve]]).

(review in [Huybrechts 16, Section 2-2.4](#Huybrechts16), [Propp 18, p. 4](K3-spectrum#Propp18)).

This number 24 of singularities-with-multiplicities is identified with the [[Euler number]] of the [[topological space]] underlying the (complex analytic) [[K3-surface]] (e.g. [Shimada 00, p. 432 (10 of 24)](#Shimada00), [Schütt-Shioda 09, Section 6.7, Theorem 6.10](#SchuttShioda09), [Huybrechts 16, Chapter 11, Remark 1.12](#Huybrechts16), review includes [Marquart 02, Section A.3.3, p. 59](#Marquart02)).

### Classification
 {#Classification}

Up to [[isomorphism]], there are a [[finite number]] of possible such elliptic fibrations.

([Nikulin 10](#Nikulin10),  [Lecacheux 19](#Lecacheux19))

## In F-theory

Elliptically fibered [[Calabi-Yau manifolds]] play a central role in [[F-theory]]:

In passing from [[M-theory]] to [[type IIA string theory]], the locus of any [[Kaluza-Klein monopole]] in 11d becomes the locus of [[D6-branes]] in 10d. The locus of the [[Kaluza-Klein monopole]] in turn (as discussed there) is the locus where the $S^1_A$-circle fibration degenerates. Hence in [[F-theory]] this is the locus where the fiber of the $S^1_A \times S^1_B$-[[elliptic fibration]] degenerates to the [[nodal curve]]. Since the [[T-duality|T-dual]] of [[D6-branes]] are [[D7-branes]], it follows that [[D7-branes]] in F-theory "are" the singular locus of the elliptic fibration.

Now an [[elliptic fibration|elliptically fibered]] complex [[K3-surface]]

$$
  \array{
    T &\longrightarrow& K3
    \\
    && \downarrow
    \\
    && \mathbb{C}\mathbb{P}^1
  }
$$

may be parameterized via the [[Weierstrass elliptic function]] as the solution locus of the equation

$$
  y^2 = x^3 + f(z) x + g(z)
$$

for $x,y,z \in \mathbb{C}\mathbb{P}^1$, with $f$ a [[polynomial]] of degree 8 and $g$ of degree twelve. The [[j-invariant]] of the complex [[elliptic curve]] which this parameterizes for given $z$ is

$$
  j(\tau(z)) = \frac{4 (24 f)^3}{27 g^2 + 4 f^3}
  \,.
$$

The [[poles]] $j\to \infty$ of the [[j-invariant]] correspond to the [[nodal curve]], and hence it is at these poles that the [[D7-branes]] are located.

Since the order of the poles is 24 (the polynomial degree of the [[discriminant]] $\Delta = 27 g^2 + 4 f^3$) there are necessarily _24 D7-branes_ ([Sen 96, page 5](F-theory#Sen96), [Sen 97b](F-theory#Sen97b), see also [Morrison 04, sections 8 and 17](F-theory#Morrison04), [Denef 08, around (3.41)](F-theory#Denef08), [Douglas-Park-Schnell 14](duality+between+M%2FF-theory+and+heterotic+string+theory#DouglasParkSchnell14)).

Under [[T-duality]] this translates to 24 [[D6-branes]] in [[type IIA string theory]] on [[K3]] ([Vafa 96, Footnote 2 on p. 6](#Vafa96)).

Notice that the _net charge_ of these 24 D7-branes is supposed to vanish, due to [[S-duality]] effects (e.g. [Denef 08, below (3.41)](F-theory#Denef08)).

## Related concepts

* [[F/M-theory on elliptically fibered Calabi-Yau 2-folds|F-theory on CY2]]

## References

Basics:

* {#FriedmanMorgan94} [[Robert Friedman]], [[John Morgan]], _Smooth Four-Manifolds and Complex Surfaces_, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics (1994) ([doi:10.1007/978-3-662-03028-8](https://www.springer.com/gp/book/9783540570585))

* {#Shimada00} [[Ichiro Shimada]], _On elliptic K3 surfaces_, Michigan Math. J. Volume 47, Issue 3 (2000), 423-446 ([euclid:mmj/1030132587](https://projecteuclid.org/euclid.mmj/1030132587))

* {#SchuttShioda09} Matthias Schütt, Tetsuji Shioda, Section 12 of: _Elliptic surfaces_, Adv. Stud. Pure Math. Algebraic Geometry in East Asia — Seoul 2008, J. H. Keum, S. Kondō, K. Konno and K. Oguiso, eds. (Tokyo: Mathematical Society of Japan, 2010), 51 - 160 ([arXiv:0907.0298](https://arxiv.org/abs/0907.0298), [euclid:aspm/1543085637](https://projecteuclid.org/euclid.aspm/1543085637))

* {#Huybrechts16} [[Daniel Huybrechts]], Chapter 11 of: _Lectures on K3-surfaces_,  Cambridge University Press  2016 ([pdf](http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf), [[HuybrechtsLecturesOnK3.pdf:file]], [doi:10.1017/CBO9781316594193](https://doi.org/10.1017/CBO9781316594193))

Further review:

* [[Tristan Hübsch]], Section 6.2 of: _Calabi-Yau Manifolds -- A Bestiary for Physicists_, World Scientific 1992 ([doi:10.1142/1410](https://doi.org/10.1142/1410))

* {#Marquart02} [[Monika Marquart]], Section A.3 of: _Applications of Dualities in String Theory_, 2002 ([sundoc:02/02H150](https://sundoc.bibliothek.uni-halle.de/diss-online/02/02H150), [pdf](https://sundoc.bibliothek.uni-halle.de/diss-online/02/02H150/prom.pdf))

Classification:

* {#Nikulin10} Viacheslav Nikulin, _Elliptic fibrations on K3 surfaces_, Proceedings of the Edinburgh Mathematical Society, Volume 57 Issue 1 ([arXiv:1010.3904](http://arxiv.org/abs/1010.3904), [doi:10.1017/S0013091513000953](https://doi.org/10.1017/S0013091513000953))

* {#Lecacheux19} O. Lecacheux, _Weierstrass Equations for the Elliptic Fibrations of a K3 Surface_ In: Balakrishnan J., Folsom A., Lalín M., Manes M. (eds.) _Research Directions in Number Theory Association for Women in Mathematics Series, vol 19. Springer (2019) ([doi:10.1007/978-3-030-19478-9_4](https://doi.org/10.1007/978-3-030-19478-9_4))

* Marie Bertin, _Elliptic Fibrations on K3 surfaces_, 2013 ([pdf](https://webusers.imj-prg.fr/~marie-jose.bertin/WINEurope13.pdf))

Discussion in the context of [[K3-spectra]]:

* {#Propp18} Oron Y. Propp, _Constructing explicit K3-spectra_ ([arXiv:1810.08953](https://arxiv.org/abs/1810.08953))

[[!redirects elliptically fibered K3-surfaces]]

[[!redirects elliptic fibration of a K3-surface]]
[[!redirects elliptic fibrations of a K3-surface]]

[[!redirects elliptically fibered complex K3-surface]]
[[!redirects elliptically fibered complex K3-surfaces]]

[[!redirects elliptically fibered K3]]
[[!redirects elliptically fibered K3s]]


elliptization conjecture

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## References

* Wikipedia, _[Thurston elliptization conjecture](http://en.wikipedia.org/wiki/Elliptization_conjecture)_

[[!redirects Thurston elliptization conjecture]]


Ellis Cooper

* [mathematics genealogy page](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=31280)

## writings

* [[Saunders Mac Lane]], _[[Lectures on category theory]], Bowdoin Summer School 1969, Notes taken by Ellis Cooper

category: people