Yıl: 2007 Cilt: 5 Sayı: 1 Sayfa Aralığı: 3 - 18 Metin Dili: Türkçe İndeks Tarihi: 29-07-2022

Langlands karşılıklılık ilkesi

Öz:
Bir K global cisminin abelyen genişlemeleri ve bu genişlemelerin aritmetik yapıları tamamen taban cisim K ve buna bağlı değişmezler yardımıyla Artin karşılıklılık yasası ile betimlenmektedir. K global cisminin abelyen olmayan Galois genişlemelerini de içerecek şekilde genel bir kuram hipotetik olarak inşa edilecek olursa Langlands'in karşılıklılık ilkesine, daha genel olarak da Langlands'in fonktörsellik ilkesine varılır. Bu derleme çalışmasında sayı cisimleri için Langlands 'in karşılıklılık ilkesinin ne olduğunu kısaca özetlemeye çalışacağız. İlk olarak, K sayı cismi için, ve bu cismin henselsel v yerlerindeki kapanışlarından elde edilen $K_v$ lokal cisimleri için, sınıf cisim kuramlarının ne olduğunu, ve bu kuramların temelini oluşturan $Theta_K$ global ve $theta_v$ lokal Artin karşılıklılık yasalarını, kısaca özetleyeceğiz. Çalışmanın geri kalan kısmında, K sayı cismi için tanımlı olan Artin karşılıklılık yasasının analitik formülasyonunu kullanarak, global sınıf cisim kuramının $G_K$ mutlak Galois gurubunun 1-boyutlu sürekli temsilleri ile K sayı cisminin belli tip Hecke karakterleri arasında "doğal" bir eşleme olduğunu göreceğiz. Burada "doğal" eşleme ile, karşılık gelen objelere bağlı L-fonks iyonlar inin aynı olması anlaşılmaktadır. Sonuç olarak, Pontrjagin ikilik teoreminin abelyen-olmayan genellemesi, Tannaka ikilik teoremini kullanarak, abelyen-olmayan sınıf cisim kuramının inşası için $G_K$ mutlak Galois gurubunun n-boyutlu sürekli temsillerini K sayı cismine bağlı Hecke karakterlerinin belli çeşit genellemesi olan analitik objelef ile parametrize etmemiz gerekmektedir. Langlands, 1967 yılında, Hecke karakterlerini genelleyen otomorf temsiller kuramını ortaya atmıştır. Çalışmanın geri kalan kısımlarında, bu kuram ve abelyen-olmayan sınıf cisim kuramını, yani Langlands karşılıklılık ilkesini özetleyeceğiz.
Anahtar Kelime:

Langlands reciprocity principle

Öz:
Class field theory studies the arithmetic of abelian extensions of a given global or local field K through the algebraic invariants of the base field K via global or local Artin reciprocity law respectively. If K is a local field with finite residue class field $K_K$, the local class field theory over K is then described by a unique "natural" topological and algebraic homomorphism $theta_v$, which is called the local Artin reciprocity law of K. By the naturality of the map $theta_v$, we should understand certain functorial properties of this mapping. If K is a global field, where we always assume that K is a number field in this text, the class field theory over K is then described by a unique "natural" topological and algebraic homomorphism $Theta_K$, called the global Artin reciprocity law of K, which satisfies certain functorial properties and which should be compatible with the local Artin reciprocity laws of $K_v$, the completions of the global field K at places v. Thus, the remaining problem is to extend class field theory to a general theory that includes the arithmetic description of non-abelian extensions of global and local fields in terms of the ground field. This general theory has a conjectural description, called the Langlands reciprocity law, or more generally the functoriality principle of Langlands. The aim of this survey article is to describe the reciprocity principle of Langlands. In order to describe the reciprocity principle of Langlands, we reformulate the Artin reciprocity law $Theta_K$ over the global field K, so that, 1-dimensional representations of the absolute Galois group $G_K$ of the global field K are parametrized by certain Heche characters of K, which depend on the ground field K only. Taking into account the Pontrjagin duality theorem for locally compact abelian groups, 1-dimensional representations of $G_K$ determines the group $G_K^{ab}$. Moreover, the naturality of the parametrization, the arithmetic of $G_K^{ab}$ (that is, the arithmetic of each finite abelian extension over K) which is encoded in the Artin L-function is encoded in the corresponding Hecke L-function defined over K as well. Therefore, in order to describe the absolute Galois group $G_K$ and the arithmetic of each finite Galois extension over K, following the above recipy, and in view of Tannaka duality for non-abelian compact groups, we would like to parametrize all irreducible h-dimensional representations of the absolute Galois group $G_K$ in terms of certain objects that generalize Hecke characters of K in a natural way. Langlands, in 1967 (cf. Langlands 1970) made a fantastic discovery and realized that the objects generalizing Hecke characters are certain type of representations of the adelic groups $GL{n,Bbb{A}_K}$, called the cuspidal automorphic representations of the group $GL{n,Bbb{A}_K}$. The reciprocity principle of Langlands asserts a natural assignement from n-dimensional irreducible representations p of $G_K$ and cuspidal automorphic representations $Lambda_K^{(n)}$(P) of $G(Bbb{A}_K)$. By the naturality of this assignment we should understand the equality of the Artin L-function L(s,p) and the standard L-function $L(s,Lambda_K^{(n)}(p))$ of Godement and Jacquet. There is also the most general form of the reciprocity principle, which involves the motivic Galois group $mathcal M_K$ over K and the Langlands group$mathcal L_K$ over K. However both of these universal groups are still conjectural in nature.
Anahtar Kelime:

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
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APA BEDİKYAN S, İKEDA K (2007). Langlands karşılıklılık ilkesi. , 3 - 18.
Chicago BEDİKYAN Sevan,İKEDA Kazım İlhan Langlands karşılıklılık ilkesi. (2007): 3 - 18.
MLA BEDİKYAN Sevan,İKEDA Kazım İlhan Langlands karşılıklılık ilkesi. , 2007, ss.3 - 18.
AMA BEDİKYAN S,İKEDA K Langlands karşılıklılık ilkesi. . 2007; 3 - 18.
Vancouver BEDİKYAN S,İKEDA K Langlands karşılıklılık ilkesi. . 2007; 3 - 18.
IEEE BEDİKYAN S,İKEDA K "Langlands karşılıklılık ilkesi." , ss.3 - 18, 2007.
ISNAD BEDİKYAN, Sevan - İKEDA, Kazım İlhan. "Langlands karşılıklılık ilkesi". (2007), 3-18.
APA BEDİKYAN S, İKEDA K (2007). Langlands karşılıklılık ilkesi. İTÜ Dergisi Seri C: Fen Bilimleri, 5(1), 3 - 18.
Chicago BEDİKYAN Sevan,İKEDA Kazım İlhan Langlands karşılıklılık ilkesi. İTÜ Dergisi Seri C: Fen Bilimleri 5, no.1 (2007): 3 - 18.
MLA BEDİKYAN Sevan,İKEDA Kazım İlhan Langlands karşılıklılık ilkesi. İTÜ Dergisi Seri C: Fen Bilimleri, vol.5, no.1, 2007, ss.3 - 18.
AMA BEDİKYAN S,İKEDA K Langlands karşılıklılık ilkesi. İTÜ Dergisi Seri C: Fen Bilimleri. 2007; 5(1): 3 - 18.
Vancouver BEDİKYAN S,İKEDA K Langlands karşılıklılık ilkesi. İTÜ Dergisi Seri C: Fen Bilimleri. 2007; 5(1): 3 - 18.
IEEE BEDİKYAN S,İKEDA K "Langlands karşılıklılık ilkesi." İTÜ Dergisi Seri C: Fen Bilimleri, 5, ss.3 - 18, 2007.
ISNAD BEDİKYAN, Sevan - İKEDA, Kazım İlhan. "Langlands karşılıklılık ilkesi". İTÜ Dergisi Seri C: Fen Bilimleri 5/1 (2007), 3-18.