Beilinson conjectures

수학노트
둘러보기로 가기 검색하러 가기

introduction

  • generalizations of
  1. the Lichtenbaum conjectures for K-groups of number rings
  2. the Hodge conjecture
  3. the Tate conjecture about algebraic cycles
  4. the Birch and Swinnerton-Dyer conjecture about elliptic curves
  5. Bloch's conjecture about K2 of elliptic curves
  • the Beĭlinson conjectures describe the leading coefficients of L-series of varieties over number fields up to rational factors in terms of generalized regulators
    • the very general setting being for L-functions \(L(s)\) associated to Chow motives over number fields
  • Bloch-Beilinson conjecture predicts that ranks of Chow groups of homologically trivial cycles should be related to orders of vanishing of L-functions.


motivation

\[1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots \;=\; \frac{\pi}{4}\]

  • the left-hand side is also \(L(1)\) where \(L(s)\) is the Dirichlet L-function for the Gaussian field.
  • This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1, and also contains four roots of unity, so accounting for the factor 1/4.
  • Q : how to replace \(\pi\) in the Leibniz formula by some other "transcendental" number
  • Beilinson : abstract from the regulator of a number field to some "higher regulator" (the Beilinson regulator), a determinant constructed on a real vector space that comes from algebraic K-theory.


review of Dirichlet class number formula

  • 틀:수학노트
  • let \(F\) be a real quadratic field
  • Dirichlet's class number formula implies

\[ \zeta'_F(0)\sim_{\mathbb{Q}^{\times}}\log(|\epsilon|),\, \epsilon \in \mathcal{O}_{F}^{\times},\,\epsilon\neq \pm 1 \label{cls} \]

  • there are four ingredients
    • an arithmetic/geometric objects (\(\mathcal{O}\))
    • an associated finitely generated abelian group of rank one (\(K(\mathcal{O})\))
    • as associated \(L\)-function \(L(\mathcal{O},s)\) with a simple zero at \(s=0\) (the zeta function \(\zeta_{F}(s)\)
    • a non-zero homomorphism \(r:K(\mathcal{O})\to \mathbb{R}\)
  • with this setup \ref{cls} becomes

\[ L'(\mathcal{O},0)\sim_{\mathbb{Q}^{\times}} r(\alpha),\, \alpha \in K(\mathcal{O})\backslash K(\mathcal{O})_{\operatorname{tor}} \]


Deligne's conjecture

  • seminal 1979 paper
def

The critical values of a (motivic) \(L\)-function \(L(s)\) are the integers arguments of \(s\) at which neither \(\gamma(s)\) nor \(\gamma(k-s)\) has a pole, where \(\gamma(s)\) and \(k\) are defined as \(L^{*}(s)=L(s)\gamma(s)\) satisfying \[ L^{*}(s)=\pm L^{*}(k-s) \]

  • for example, for the Riemann zeta function the critical values are the positive even integers and the negative even integers
conjecture (Deligne, 1979)

The value of \(L(s)\) at any critical value is a non-zero algebraic multiple of the determinant of a certain matrix whose entries are periods


Beilinson's conjecture

  • huge generalization of Deligne's conjecture
  • all values of motivic \(L\)-functions in terms of periods on the variety defining the \(L\)-function and of a regulator


Scholl

  • observed that this regulator can itself be expressed in terms of periods
  • this led to a reformuation of Beilinson's conjecture


conjecture (Deligne-Beilinson-Scholl)

Let \(L(s)\) be a motivic \(L\)-function, \(m\) an arbitrary integer, and \(r\) the order of vanishing of \(L(s)\) at \(s=m\). Then \(L^{(r)}(m)\in \hat{\mathcal{P}}\) where \(\hat{\mathcal{P}}\) denotes the extended period ring.


mixed motives

  • Deligne periods of moxed motives : real numbers obtained by integrating certain differential forms over topological cycles


history

  • 1984 Beilinson generalization of regulator as a map from motivic cohomology to Deligne cohomology
  • 1994 Scholl reformuation of Beilinson conjectures in terms of mixed motives



related items

question and answers(Math Overflow)


expositions

articles

  • Lemma, Francesco. “On Higher Regulators of Siegel Threefolds II: The Connection to the Special Value.” arXiv:1409.8391 [math], September 30, 2014. http://arxiv.org/abs/1409.8391.
  • Miyazaki, Hiroyasu. “Special Values of Zeta Functions of Varieties over Finite Fields via Higher Chow Groups.” arXiv:1406.1390 [math], June 5, 2014. http://arxiv.org/abs/1406.1390.
  • Otsubo, Noriyuki. “On Special Values of Jacobi-Sum Hecke L-Functions.” arXiv:1404.7476 [math], April 29, 2014. http://arxiv.org/abs/1404.7476.
  • Brunault, François. 2006. “Version Explicite Du Théorème de Beilinson Pour La Courbe Modulaire.” Comptes Rendus Mathematique 343 (8) (October 15): 505–510. doi:10.1016/j.crma.2006.09.014.
  • Beilinson, A. A. 1987. “Height Pairing between Algebraic Cycles.” In \(K\)-Theory, Arithmetic and Geometry (Moscow, 1984–1986), 1289:1–25. Lecture Notes in Math. Berlin: Springer. http://www.ams.org/mathscinet-getitem?mr=923131.
  • Beilinson, A. A. 1984. “Higher Regulators and Values of \(L\)-Functions.” In Current Problems in Mathematics, Vol. 24, 181–238. Itogi Nauki I Tekhniki. Moscow: Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. http://www.ams.org/mathscinet-getitem?mr=760999. http://dx.doi.org/10.1007/BF02105861
  • Beilinson, A. A. 1980. “Higher Regulators and Values of \(L\)-Functions of Curves.” Akademiya Nauk SSSR. Funktsional\cprime Ny\uı\ Analiz I Ego Prilozheniya 14 (2): 46–47.
  • Deligne, Values of L-Functions and Periods of Integrals
    • translation of P. Deligne’s Valeurs de Fonctions L et periodes d'integrales, in the Proceedings of Symposia in Pure Mathematics 33, (1979), Part 2, 313-346

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'special'}, {'LOWER': 'values'}, {'LOWER': 'of'}, {'LOWER': 'l'}, {'OP': '*'}, {'LEMMA': 'function'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Beilinson_conjectures&oldid=51831"