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In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca) is a sort of generalization of a Drinfeld module, consisting roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it.

Drinfeld modules were introduced by Template:Harvs, who used them to prove the Langlands conjectures for GL₂ of a function field in some special cases. He later invented shtukas and used shtukas of rank 2 to prove the remaining cases of the Langlands conjectures for GL₂. Laurent Lafforgue proved the Langlands conjectures for GL_n of a function field by studying the moduli stack of shtukas of rank n.

"Shtuka" is a Russian word штука meaning "a single copy", which comes from the German noun “Stück” meaning “piece, item, or unit". In Russian, the word "shtuka" is also used as a slang describing a thing with known properties, but having no name in a speaker's mind.

Drinfeld modules

The ring of additive polynomials

We let L be a field of characteristic p>0. The ring L{τ} is defined to be the ring of noncommutative (or twisted) polynomials a₀ + a₁τ + a₂τ² + ... over L, with the multiplication given by

τa = a^pτ

for a∈ L. The element τ can be thought of as a Frobenius element: in fact, L is a left module over L{τ}, with elements of L acting as multiplication and τ acting as the Frobenius endomorphism of L. The ring L{τ} can also be thought of as the ring of all (absolutely) additive polynomials

$a_0{x}^1+a_1{x}^p+a_2{x}^{p^2}+\cdots =a_0{\tau }^0+a_1\tau +a_2{\tau }^2+\cdots$

in L[x], where a polynomial f is called additive if f(x + y) = f(x) + f(y) (as elements of L[x,y]). The ring of additive polynomials is generated as an algebra over L by the polynomial τ = x^p. The multiplication in the ring of additive polynomials is given by composition of polynomials, not by multiplication of commutative polynomials, and is not commutative.

Definition of Drinfeld modules

If A is a commutative ring, then a Drinfeld A-module over L is defined to be a ring homomorphism ψ from A to L{τ}, such that the image is not contained in L.

The ring A will usually be a ring of functions on some affine curve over a finite field.

The condition that the image of A is not in L is a non-degeneracy condition, put in to eliminate trivial cases.

As L{τ} can be thought of as endomorphisms of the additive group of L, a Drinfeld A-module can be regarded as an action of A on the additive group of L, or in other words as an A-module whose underlying additive group is the additive group of L.

Examples of Drinfeld modules

• Define A to be F_p[T], the usual (commutative!) ring of polynomials over the finite field of order p. In other words A is the coordinate ring of an affine genus 0 curve. Then a Drinfeld module ψ is determined by the image ψ(T) of T, which can be any non-constant element of L{τ}. So Drinfeld modules can be identified with non-constant elements of L{τ}. (In the higher genus case the description of Drinfeld modules is more complicated.)

• The Carlitz module is the Drinfeld module ψ given by ψ(T) = T+τ, where A is F_p[T] and L is a suitable complete algebraically closed field containing A. It was described by L. Carlitz in 1935, many years before the general definition of Drinfeld module. See chapter 3 of Goss's book for more information about the Carlitz module. See also Carlitz exponential.

Shtukas

Suppose that X is a curve over the finite field F_p. A (right) shtuka of rank r over a scheme (or stack) U is given by the following data:

• Locally free sheaves E, E′ of rank r over U×X together with injective morphisms

E → E′ ← (Fr×1)^*E,

whose cokernels are supported on certain graphs of morphisms from U to X (called the zero and pole of the shtuka, and usually denoted by 0 and ∞), and are locally free of rank 1 on their supports. Here (Fr×1)^*E is the pullback of E by the Frobenius endomorphism of U.

A left shtuka is defined in the same way except that the direction of the morphisms is reversed. If the pole and zero of the shtuka are disjoint then left shtukas and right shtukas are essentially the same.

By varying U, we get an algebraic stack Shtuka^r of shtukas of rank r, a "universal" shtuka over Shtuka^r×X and a morphism (∞,0) from Shtuka^r to X×X which is smooth and of relative dimension 2r − 2. The stack Shtuka^r is not of finite type for r > 1.

Drinfeld modules are in some sense special kinds of shtukas. (This is not at all obvious from the definitions.) More precisely, Drinfeld showed how to construct a shtuka from a Drinfeld module. See Drinfeld, V. G. Commutative subrings of certain noncommutative rings. Funkcional. Anal. i Prilovzen. 11 (1977), no. 1, 11–14, 96. for details.

Applications

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The Langlands conjectures for functions fields state (very roughly) that there is a bijection between cuspidal automorphic representations of GL_n and certain representations of a Galois group. Drinfeld used Drinfeld modules to prove some special cases of the Langlands conjectures, and later proved the full Langlands conjectures for GL₂ by generalizing Drinfeld modules to shtukas. The "hard" part of proving these conjectures is to construct Galois representations with certain properties, and Drinfeld constructed the necessary Galois representations by finding them inside the l-adic cohomology of certain moduli spaces of rank 2 shtukas.

Drinfeld suggested that moduli spaces of shtukas of rank r could be used in a similar way to prove the Langlands conjectures for GL_r; the formidable technical problems involved in carrying out this program were solved by Lafforgue after many years of effort.

References

Drinfeld modules

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

Shtukas

• Drinfeld, V. G. Cohomology of compactified moduli varieties of F-sheaves of rank 2. (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 162 (1987), Avtomorfn. Funkts. i Teor. Chisel. III, 107—158, 189; translation in J. Soviet Math. 46 (1989), no. 2, 1789–1821
• Drinfeld, V. G. Moduli varieties of F-sheaves. (Russian) Funktsional. Anal. i Prilozhen. 21 (1987), no. 2, 23—41. English translation: Functional Anal. Appl. 21 (1987), no. 2, 107–122.
• D. Goss, What is a shtuka? Notices of the Amer. Math. Soc. Vol. 50 No. 1 (2003)
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010