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{
"Title": "Area Siegel\u2013Veech constants of Abelian strata",
"Definition": "The area Siegel\u2013Veech constant $c_{\\mathrm{area}}(g,\\mu,c)$ is the mean total area of maximal cylinders of circumference less than $L$, divided by $\\pi L^2$, for the Masur\u2013Veech probability measure on the area-one component $c$ of the stratum $\\mathcal H(\\mu)$ of holomorphic Abelian differentials CITE{EKZ}.",
"Parameters": {
"g": {
"type": "Z",
"display": "$g$",
"title": "genus",
"constraints": "$g\\geq2$"
},
"mu": {
"type": "Symbolic",
"display": "$\\mu$",
"title": "zero profile",
"constraints": "$\\mu=(m_1,\\ldots,m_n)$ is a non-increasing partition of $2g-2$ into positive integers"
},
"component": {
"type": "Symbolic",
"display": "$c$",
"title": "connected component",
"constraints": "A connected component allowed by the classification in CITE{KZ}, with the labels defined in Comments"
}
},
"Tags": [
"dynamical systems"
],
"Data properties": {
"type": "R",
"digits": "100",
"rigour": "proven",
"complete": "no",
"complete-note": "it holds all 25 components of all 18 positive zero profiles in genera $2\\leq g\\leq4$, including minimal, principal and intermediate strata and every component type in this range; the genus cutoff is a practical choice supported by complete low-genus exact source tables, favoring breadth over extending only the hyperelliptic formulas, and genus one is excluded because it requires a marked regular point",
"rigour details": "The exact rational Lyapunov sums in CITE{EKZ}, Corollary 2 and the exact columns of Appendix A, are substituted into Theorem 1 to obtain a positive rational coefficient of $1/\\pi^2$. Each coefficient is evaluated with certified Arb arithmetic. At 100 requested significant digits, the 64-bit working guard gives at least 393 bits of relative accuracy. An independent 512-bit MPFI evaluation lies inside the exact uncertainty interval asserted by every written decimal, and 160-digit recomputations agree with all 25 enclosures.\n\nAll source sums were checked against the printed exact column. Six hyperelliptic entries also agree with the genus-zero double-cover formula. Eleven entries agree with cylinder-weighted sums of the configuration constants in CITE{EMZ}, using the Vorobets relation stated in CITE{AEZ}. EKZ itself uses EMZ data, so this is a source-consistency check rather than an independent derivation. Three entries additionally agree with the geometric divisor arguments of CITE{CM}; altogether, additional mathematical checks cover 16 distinct entries. The remaining nine rely on the exact published EKZ data and theorem, with transcription and interval checks. The generator, selection rationale, controls and reproducible checks are attached."
},
"Display properties": {
"number-header": "$c_{\\mathrm{area}}(g,\\mu,c)$"
},
"Numbers": {
"2": {
"2": {
"hyp": {
"number": "0.3377372788077925714795982106990921296811959155741551628186967729805770698741181373040309130854061254",
"comment": "The value is $10/(3\\pi^2)$ CITE{EKZ}, Corollary 2."
}
},
"1,1": {
"hyp": {
"number": "0.3799544386587666429145479870364786458913454050209245581710338696031492036083829044670347772210818911",
"comment": "The value is $15/(4\\pi^2)$ CITE{EKZ}, Corollary 2."
}
}
},
"3": {
"4": {
"hyp": "0.4255489712978186400642937454808560833983068536234355051515579339555271080413888530030789504876117180",
"odd": "0.3647562611124159771979660675550195000556915888200875758441925148190232354640475882883533861322386154"
},
"3,1": {
"connected": "0.3989521605917049750602753863883025781859126752719707860795855630833066637888020496903865160821359856"
},
"2,2": {
"hyp": "0.4728321903309096000714374949787289815536742818038172279461754821728078978237653922256432783195685755",
"odd": "0.3715110066885718286275580317690013426493155071315706791005664502786347768615299510344340043939467379"
},
"2,1,1": {
"connected": "0.4137281665395459000625078081063878588594649965783400744529035469012069105957947181974378685296225036"
},
"1,1,1,1": {
"connected": "0.4233778030769114021047820426977904911360705941661730791048663118435091125921980935489816089034912500"
}
},
"4": {
"6": {
"hyp": "0.5210803730177371102828086679357421429367022697429822512059893068843189078057822689833619801889123077",
"even": "0.4342336441814475919023405566131184524472518914524852093383244224035990898381518908194683168240935898",
"odd": "0.3908102797633028327121065009518066072025267023072366884044919801632391808543367017375214851416842308"
},
"5,1": {
"connected": "0.4221715985097407143494977633738651621014948944676939535233709662257213373426476716300386413567576567"
},
"4,2": {
"even": "0.4593226991785978972122535665507652963664264451808510214334276112535848150288006667334820417961523305",
"odd": "0.3985299889931952343459258886249287130238111803775030921260621921170809424514594020187564774407792279"
},
"4,1,1": {
"connected": "0.4307992508138306882209311358471783337778963537655428263190122201536797152185228713193507037728484859"
},
"3,3": {
"hyp": "0.5699316579881499643718219805547179688370181075313868372565508044047238054125743567005521658316228366",
"nonhyp": "0.4179498825246433072060027857401265104804799455230170139881372565634641239692211949137382549431900802"
},
"3,2,1": {
"connected": "0.4327258884724842322082352074582117911540322668293863023614552403813643707762138634207896073906765981"
},
"3,1,1,1": {
"connected": "0.4381732639371260478772609850501326319553418783709049340198213173649220654516028656353707511501186324"
},
"2,2,2": {
"even": "0.4700937799621977684107921040811687750967997203261889428422401029324248405004617316529078925377950124",
"odd": "0.4052847345693510857755178528389105556174350986889861953824361275766924838489417647648370957024873505"
},
"2,2,1,1": {
"connected": "0.4392518234876157203498972912050215388888072833278372966811867381417238036626173865891165405872218978"
},
"2,1,1,1,1": {
"connected": "0.4441245216322472314956716470693061505307726289800140391065862564694588468844653505548006507073090549"
},
"1,1,1,1,1,1": {
"connected": "0.4484873347563691143155805152022095422324899343085183776819895808313034366199745158828394585235582003"
}
}
},
"Comments": {
"counting": "Each cylinder is counted once, without choosing an orientation; its circumference is the length of a primitive closed core geodesic. The quotient in the definition is independent of $L>0$. The zeros of the differential have positive orders $\\mu=(m_1,\\ldots,m_n)$, and there are no poles or added regular marked points CITE{EKZ}.",
"measure": "The HREF{T560}[Masur\u2013Veech measure] is defined by the relative-period lattice $H^1(X,\\{z_i\\};\\mathbb Z\\oplus i\\mathbb Z)$ of covolume one. All zeros are labelled, including equal-order zeros. The area is $A=(i/2)\\int_X\\omega\\wedge\\bar\\omega=1$, and the phase of $\\omega$ is retained. For $D=2g+n-1$ and $\\omega=r\\omega_1$ with $A(\\omega_1)=1$, the radial convention is $d\\nu=r^{2D-1}dr\\,d\\nu_1$. The restriction to each component is divided by that component's volume to obtain its probability measure CITE{EMZ}.",
"components": "The label hyp denotes the hyperelliptic component; for $\\mathcal H(g-1,g-1)$ its involution interchanges the two zeros. The labels even and odd denote the nonhyperelliptic components of even and odd spin parity; they exclude the hyperelliptic component even when it has the same parity. The label nonhyp denotes the nonhyperelliptic component of $\\mathcal H(g-1,g-1)$ for even $g\\geq4$. The label connected denotes the sole component of any other connected stratum. Both genus-two strata use hyp CITE{KZ}.",
"normalization": "The values are the area constants themselves. In CITE{CM}, the symbol $c$ denotes the scaled constant $\\pi^2c_{\\mathrm{area}}/3$. The exact sums in CITE{EKZ}, Appendix A, are given separately from the experimental individual Lyapunov exponents."
},
"Formulas": {
"lyapunov": "Let $L_{g,\\mu,c}=\\sum_{j=1}^g\\lambda_j$ be the sum of the nonnegative Lyapunov exponents of the real Hodge bundle for the Teichm\u00fcller flow $\\operatorname{diag}(e^t,e^{-t})$, with $\\lambda_1=1$ and the component probability measure used in the definition. Then $\\displaystyle L_{g,\\mu,c}=\\frac1{12}\\sum_{i=1}^n\\frac{m_i(m_i+2)}{m_i+1}+\\frac{\\pi^2}{3}c_{\\mathrm{area}}(g,\\mu,c)$ CITE{EKZ}.",
"hyperelliptic-one": "For $g\\geq2$, $\\displaystyle c_{\\mathrm{area}}(g,(2g-2),\\mathrm{hyp})=\\frac{g(2g+1)}{(2g-1)\\pi^2}$ CITE{EKZ}.",
"hyperelliptic-two": "For $g\\geq2$, $\\displaystyle c_{\\mathrm{area}}(g,(g-1,g-1),\\mathrm{hyp})=\\frac{(g+1)(2g+1)}{2g\\pi^2}$ CITE{EKZ}."
},
"Keywords": [
"Siegel-Veech",
"area Siegel-Veech constant",
"translation surfaces",
"Abelian differentials",
"Teichmuller",
"Teichm\u00fcller",
"flat surfaces",
"cylinder counting"
],
"Links": {
"EKZ-text": {
"title": "Eskin, Kontsevich and Zorich: full text",
"url": "https://arxiv.org/pdf/1112.5872"
}
},
"References": {
"EKZ": {
"bib": "A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichm\u00fcller geodesic flow. Publications Math\u00e9matiques de l\u2019IH\u00c9S 120 (2014), 207\u2013333. Section 1.6, Theorem 1, Corollaries 1\u20132 and Appendix A.",
"arxiv": "1112.5872",
"doi": "10.1007/s10240-013-0060-3"
},
"EMZ": {
"bib": "A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: the principal boundary, counting problems, and the Siegel\u2013Veech constants. Publications Math\u00e9matiques de l\u2019IH\u00c9S 97 (2003), 61\u2013179. Sections 2.1\u20132.2 and 5.2, Examples 14.12\u201314.13 and Appendix B.2.",
"arxiv": "math/0202134"
},
"KZ": {
"bib": "M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Inventiones Mathematicae 153 (2003), 631\u2013678. Theorems 1\u20132.",
"arxiv": "math/0201292",
"doi": "10.1007/s00222-003-0303-x"
},
"CM": {
"bib": "D. Chen and M. M\u00f6ller, Non-varying sums of Lyapunov exponents of Abelian differentials in low genus. Geometry & Topology 16 (2012), 2427\u20132479. Proposition 4.5 and Sections 5.1\u20135.2 and 6.1.",
"arxiv": "1104.3932"
},
"AEZ": {
"bib": "J. S. Athreya, A. Eskin and A. Zorich, with an appendix by J. Chaika, Right-angled billiards and volumes of moduli spaces of quadratic differentials on the complex projective line. Annales Scientifiques de l\u2019\u00c9cole Normale Sup\u00e9rieure 49 (2016), 1311\u20131386. Section 4.13 of the preprint states the Vorobets relation between cylinder and area constants.",
"arxiv": "1212.1660"
}
},
"Similar tables": [
{
"table": "HREF{T560}",
"relation": "the Masur\u2013Veech volumes of the same connected components provide the normalization and boundary-volume ratios used in cylinder-counting formulas"
}
],
"Programs": {
"sage": {
"language": "Sage",
"code": "# Download generate.py from the attached files.\nfrom generate import area_constant\narea_constant(4, (4, 2), 'odd', digits=100)"
},
"sage-hyperelliptic": {
"language": "Sage",
"code": "# Hyperelliptic components: zeros=1 for H(2*g-2), zeros=2 for H(g-1,g-1).\nimport numberdb.sage as numberdb\nfrom sage.rings.integer_ring import ZZ\nfrom sage.rings.rational_field import QQ\nfrom sage.rings.real_arb import RealBallField\n\ndef hyperelliptic_area(g, zeros=1, digits=100):\n g = ZZ(g)\n if g < 2 or zeros not in (1, 2):\n raise ValueError('Use genus g >= 2 and one or two zeros.')\n if zeros == 1:\n q = QQ(g*(2*g+1)) / QQ(2*g-1)\n else:\n q = QQ((g+1)*(2*g+1)) / QQ(2*g)\n field = RealBallField(numberdb.bits(digits, losing=64))\n return field(q) / field.pi()**2\n\nhyperelliptic_area(4, zeros=2, digits=100)"
}
}
}