86.185.237.242: /* Definition */

2015-01-02T18:31:59Z

Definition

en>EmausBot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q13434396

2013-06-12T19:37:37Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q13434396

← Older revision		Revision as of 21:37, 12 June 2013
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	~~Andera~~ is ~~what you can contact her but she~~ by ~~no means really liked that name~~. ~~For many years he~~'~~s been residing~~ in ~~Alaska and he doesn~~'~~t plan on changing it. To climb~~ is ~~something~~ I ~~truly enjoy doing~~. ~~She functions~~ as a ~~travel agent but soon she'll~~ be ~~on her own~~.<br><br>~~Feel free to surf to my blog~~ - ~~tarot readings~~, [~~http~~://~~www~~.~~sirudang~~.~~com~~/~~siroo_Notice~~/~~2110 browse around these guys~~],		In the mathematical theory of [[Riemann surface]]s, the '''first Hurwitz triplet''' is a triple of distinct [[Hurwitz surface]]s with the identical automorphism group of the lowest possible genus, namely 14 (genera 3 and 7 each admit a unique Hurwitz surface, respectively the [[Klein quartic]] and the [[Macbeath surface]]). The explanation for this phenomenon is arithmetic. Namely, in the ring of integers of the appropriate number field, the rational prime 13 splits as a product of three distinct prime ideals. The principal [[congruence subgroup]]s defined by the triplet of primes produce [[Fuchsian group]]s corresponding to the triplet of Riemann surfaces.

			==Arithmetic construction==
			Let <math>K</math> be the real subfield of <math>\mathbb{Q}[\rho]</math> where <math>\rho</math> is a 7th-primitive [[root of unity]].
			The [[ring of integers]] of K is <math>\mathbb{Z}[\eta]</math>, where <math>\eta=2\cos(\tfrac{2\pi}{7})</math>. Let <math>D</math> be the [[quaternion algebra]], or symbol algebra <math>(\eta,\eta)_{K}</math>. Also Let <math>\tau=1+\eta+\eta^2</math> and <math>j'=\tfrac{1}{2}(1+\eta i + \tau j)</math>. Let <math>\mathcal{Q}_{Hur}=\mathbb{Z}[\eta][i,j,j']</math>. Then <math>\mathcal{Q}_{Hur}</math> is a maximal [[Order (ring theory)\|order]] of <math>D</math> (see [[Hurwitz quaternion order]]), described explicitly by [[Noam Elkies]] [1].

			In order to construct the first Hurwitz triplet, consider the prime decomposition of 13 in <math>\mathbb{Z}[\eta]</math>, namely

			:<math>13=\eta (\eta +2)(2\eta-1)(3-2\eta)(\eta+3)</math>,

			where <math>\eta (\eta+2)</math> is invertible. Also consider the prime ideals generated by the non-invertible factors. The principal congruence subgroup defined by such a prime ideal ''I'' is by definition the group

			:<math>\mathcal{Q}^1_{Hur}(I) = \{x \in \mathcal{Q}_{Hur}^1 : x \equiv 1 (</math>mod <math>I\mathcal{Q}_{Hur})\}</math>,

			namely, the group of elements of [[reduced norm]] 1 in <math>\mathcal{Q}_{Hur}</math> equivalent to 1 modulo the ideal <math>I\mathcal{Q}_{\mathrm Hur}</math>. The corresponding Fuchsian group is obtained as the image of the principal congruence subgroup under a representation to P[[SL(2,R)]].

			Each of the three Riemann surfaces in the first Hurwitz triplet can be formed as a [[Fuchsian model]], the quotient of the [[Hyperbolic manifold\|hyperbolic plane]] by one of these three Fuchsian groups.

			==Bound for systolic length and the systolic ratio==
			The [[Gauss–Bonnet theorem]] states that

			:<math>\chi(\Sigma)=\frac{1}{2\pi} \int_{\Sigma} K(u)\,dA,</math>

			where <math>\chi(\Sigma)</math> is the [[Euler characteristic]] of the surface and <math>K(u)</math> is the [[Gaussian curvature]] . In the case <math>g=14</math> we have

			:<math>\chi(\Sigma)=-26</math> and <math>K(u)=-1,</math>

			thus we obtain that the area of these surfaces is

			:<math>52\pi</math>.

			The lower bound on the [[Systolic geometry\|systole]] as specified in [2], namely

			:<math>\frac{4}{3} \log(g(\Sigma)),</math>

			is 3.5187.

			Some specific details about each of the surfaces are presented in the following tables (the number of systolic loops is taken from [3]).The term Systolic Trace refers to the least reduced trace of an element in the corresponding subgroup <math>\mathcal{Q}^1_{Hur}(I)</math>. The systolic ratio is the ratio of the square of the systole to the area.
			{\| class="wikitable"
			\|-
			\| Ideal
			\| <math>3-2\eta\vartriangleleft O_K</math>
			\|-
			\| Systole
			\| 5.9039
			\|-
			\| Systolic Trace
			\| <math>-4\eta^2-8\eta-3</math>
			\|-
			\| Systolic Ratio
			\| 0.2133
			\|-
			\| Number of Systolic Loops
			\| 91
			\|}

			{\| class="wikitable"
			\|-
			\| Ideal
			\| <math>\eta+3 \vartriangleleft O_K</math>
			\|-
			\| Systole
			\| 6.3933
			\|-
			\| Systolic Trace
			\| <math>5\eta^2+11\eta+3</math>
			\|-
			\| Systolic Ratio
			\| 0.2502
			\|-
			\| Number of Systolic Loops
			\| 78
			\|}

			{\| class="wikitable"
			\|-
			\| Ideal
			\| <math>2\eta-1 \vartriangleleft O_K</math>
			\|-
			\| Systole
			\| 6.8879
			\|-
			\| Systolic Trace
			\| <math>-7\eta^2-14\eta-3</math>
			\|-
			\| Systolic Ratio
			\| 0.2904
			\|-
			\| Number of Systolic Loops
			\| 364
			\|}

			==See also==
			*[[(2,3,7) triangle group]]

			==References==
			*{{cite book \|authorlink=Noam Elkies \|last=Elkies \|first=N. \|title=The Klein quartic in number theory. The eightfold way \|pages=51–101 \|series=Math. Sci. Res. Inst. Publ. \|volume=35 \|publisher=Cambridge Univ. Press \|location=Cambridge \|year=1999 }}
			*{{cite journal \|last=Katz \|first=M. \|last2=Schaps \|first2=M. \|last3=Vishne \|first3=U. \|author3-link= Uzi Vishne \|title=Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups \|journal=J. Differential Geom. \|volume=76 \|year=2007 \|issue= \|pages=399–422 \|arxiv=math.DG/0505007 }}
			*{{cite journal \|last=Vogeler \|first=R. \|title=On the geometry of Hurwitz surfaces \|others=Thesis \|publisher=Florida State University \|year=2003 }}

			[[Category:Riemann surfaces]]
			[[Category:Differential geometry of surfaces]]
			[[Category:Systolic geometry]]

129.186.6.83: /* Step two */

2012-05-09T20:28:07Z

Step two

New page

Andera is what you can contact her but she by no means really liked that name. For many years he's been residing in Alaska and he doesn't plan on changing it. To climb is something I truly enjoy doing. She functions as a travel agent but soon she'll be on her own.<br><br>Feel free to surf to my blog - tarot readings, [http://www.sirudang.com/siroo_Notice/2110 browse around these guys],

Variance inflation factor - Revision history

86.185.237.242: /* Definition */

en>EmausBot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q13434396

129.186.6.83: /* Step two */