# Glossary

Ind-category

A quote from somewhere: "For C abelian, can think of Ind(C) as the category of left exact additive Ab-valued contravariant functors on C"

nLab ind-object

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Independence of l

arXiv:1007.4004 On a conjecture of Deligne from arXiv Front: math.NT by Vladimir Drinfeld Let X be a smooth variety over $Fp$. Let E be a number field. For each nonarchimedean place $\lambda$ of E prime to p consider the set of isomorphism classes of irreducible lisse $\bar{E}{\lambda}$-sheaves on X with determinant of finite order such that for every closed point x in X the characteristic polynomial of the Frobenius $F_x$ has coefficents in E. We prove that this set does not depend on $\lambda$.

The idea is to use a method developed by G.Wiesend to reduce the problem to the case where X is a curve. This case was treated by L. Lafforgue.

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Index theorems

arXiv:1210.5249 Noncommutative calculus and operads fra arXiv Front: math.KT av Boris Tsygan This is a survey of current and recent works on deformation quantization and index theorems.

arXiv:1206.1435 Three lectures on Algebraic Microlocal Analysis from arXiv Front: math.AG by Pierre Schapira These three lectures present some fundamental and classical aspects of microlocal analysis. Starting with the Sato's microlocalization functor and the microsupport of sheaves, we then construct a microlocal analogue of the Hochschild homology for sheaves and apply it to recover index theorems for D-modules and elliptic pairs. In the third lecture, we construct the ind-sheaves of temperate and Whitney holomorphic functions and give some applications to the study of irregular holonomic D-modules.

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Infinite loop space

Infinite loop spaces. Various things by May. See the file "Addenda" for a brief guide to these.

http://mathoverflow.net/questions/17569/a-model-structure-on-symmetric-monoidal-categories

A brief but nice introduction is in Weibel's obituary on Thomason. He explains what is meant by infinite loop space, connective such things, and group completion.

Thomason papers:

• First quadrant spectral sequences... (there are two papers with this title, not sure if I have access to any of them).
• Uniqueness of delooping machines
• The uniqueness of infinite loop space machines
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Infinity-algebra

In homotopy theory, there are objects called infinity-algebras. The most common are A-infinity and E-infinity algebras, but there are also others like B-infinity, C-infinity, and G-infinity algebras.

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Infinity-category
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Integrable system

http://nlab.mathforge.org/nlab/show/integrable+system

Integrable systems folder: Intro to classical int systems. Looks very good.

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Integral model

Must find good ref for integral models in AG

Maybe Scholl: Integral elements in K-th etc, in Banff vol.

Schneider survey on the Beilinson conj, says that a proper flat model always exists, and the the image of motivic cohomology of such a model is conjecturally independent of the choice of model. Also, the independence buy not the existence is known for regular proper models.

http://mathoverflow.net/questions/98825/existence-of-proper-integral-models

Check maybe Liu later chapters.

For regular models over DVRs, maybe something is in Liu, Gabber, Lorenzini

http://mathoverflow.net/questions/117179/global-minimal-model-over-a-non-affine-base

http://mathoverflow.net/questions/22998/is-there-a-deep-relationship-between-models-and-etale-cohomology-if-so-why-a

Bosch et al: Neron models, in Elliptic curve folder.

arXiv:0908.1831 Integral Models of Extremal Rational Elliptic Surfaces from arXiv Front: math.AG by Tyler J. Jarvis, William E. Lang, Jeremy R. Ricks Miranda and Persson classified all extremal rational elliptic surfaces in characteristic zero. We show that each surface in Miranda and Persson's classification has an integral model with good reduction everywhere (except for those of type X_{11}(j), which is an exceptional case), and that every extremal rational elliptic surface over an algebraically closed field of characteristic p > 0 can be obtained by reducing one of these integral models mod p.

arXiv:0909.0969 Purity results for $p$-divisible groups and abelian schemes over regular bases of mixed characteristic from arXiv Front: math.AG by Adrian Vasiu, Thomas Zink Let $p$ be a prime. Let $(R,\ideal{m})$ be a regular local ring of mixed characteristic $(0,p)$ and absolute index of ramification $e$. We provide general criteria of when each abelian scheme over $\Spec R\setminus{\ideal{m}}$ extends to an abelian scheme over $\Spec R$. We show that such extensions always exist if $e\le p-1$, exist in most cases if $p\le e\le 2p-3$, and do not exist in general if $e\ge 2p-2$. The case $e\le p-1$ implies the uniqueness of integral canonical models of Shimura varieties over a discrete valuation ring $O$ of mixed characteristic $(0,p)$ and index of ramification at most $p-1$. This leads to large classes of examples of Néron models over $O$. If $p>2$ and index $p-1$, the examples are new.

arXiv:0707.1668 Good Reductions of Shimura Varieties of Hodge Type in Arbitrary Unramified Mixed Characteristic, Part I fra arXiv Front: math.NT av Adrian Vasiu We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we solve a conjecture of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic $(0,p)$ of integral canonical models of projective Shimura varieties of Hodge type; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.

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Integral representation

arXiv:0908.1611 Integral representation for L-functions for GSp(4) x GL(2), II from arXiv Front: math.NT by Ameya Pitale, Ralf Schmidt Based on Furusawa's theory, we present an integral representation for the L-function L(s,\pi \times \tau), where \pi is a cuspidal automorphic representation on GSp(4) related to a holomorphic Siegel modular form, and where \tau is an arbitrary cuspidal automorphic representation on GL(2). As an application, a special value result for this L-function in the spirit of Deligne's conjecture is proved.

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Internal category
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Internal hom
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Intersection theory

The standard reference is Fulton, and perhaps SGA6.

Copy of course notes from Lausanne.

Some really good things are found in the K-theory handbook, notably Gillet: K-theory and Intersection theory, and Geisser.

http://mathoverflow.net/questions/52665/survey-article-on-intersection-theory/

http://mathoverflow.net/questions/26815/deformation-to-the-normal-cone

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Isaksen
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Italian school

Italian school: article in abel vol, in GENERAL folder I think

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Iterated integral

Felix Oprea Tanre in Homotopy theory folder: Rational htpy, formality. Note: Compact Kahler manifolds are fomal. Last chapter: Brief mention of Gelfand-Fuchs cohomology and iterated integrals. Also brief list of refs on MHSs on homotopy groups on pp366

Book by Harris Interated integrals and cycles on algebraic manifolds. In folder AG/Various

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Iwasawa [mathematician]

Collected papers, 2 volumes

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Iwasawa theory MIGRATED TO MS

[CDATA[Iwasawa theory is a branch of number theory with important applications to class groups of number fields and to conjectures on special values of zeta functions. Here are some starting points for learning about Iwasawa theory.

Book references:

• Lang: Cyclotomic fields I and II Google Books
• Washington: Introduction to cyclotomic fields Google Books
• Neukirch, Schmidt, Wingberg: Cohomology of number fields Google Books
• Coates and Sujatha: Cyclotomic fields and zeta values
• Noncommutative Iwasawa Main Conjecture over Totally Real Fields SpringerLink

Surveys and introductions online:

Manfred Kolster: K-theory and arithmetic (Very nice basic introduction to zeta values and Iwasawa theory)

Surveys of Sujatha:

Surveys of Venjakob:

A survey of Greenberg. Other surveys, and a book draft, on Greenberg's webpage

Matthias Flach surveys:

A survey by Mitchell from the Handbook of K-theory. (See also an interesting blog post of Eric Peterson here, for some possible connections with chromatic homotopy theory)

For noncommutative Iwasawa theory, here are some key papers:

Finally, a list of all papers on MathSciNet labelled with subject code 11R23, (Iwasawa theory) and the latest papers on arXiv

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Iwasawa theory of elliptic curves

Soulé: p-adic K-theory of elliptic curves MR0885785, in Duke. See also the review (Kato) for comparisons with other related settings.

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Iwasawa theory of modular forms

Number Theory 3: Iwasawa Theory and Modular Forms, Nobushige Kurokawa, Masato Kurihara, Translations of Mathematical Monographs 242, AMS 2012

See all articles of Lei and coauthors, on arxiv for example.

[arXiv:0912.1263] Wach modules and Iwasawa theory for modular forms from arXiv Front: math.NT by Antonio Lei, David Loeffler, Sarah Livia Zerbes For a crystalline p-adic representation of the absolute Galois group of Qp, we define a family of Coleman maps (linear maps from the Iwasawa cohomology of the representation to the Iwasawa algebra), using the theory of Wach modules. Let f = sum(an q^n) be a normalized new modular eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f, we define two Coleman maps with values in the Iwasawa algebra of Zp^* (after extending scalars to some extension of Qp). Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case ap=0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture.

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