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In [[mathematics]], the '''elliptic modular lambda''' function λ(τ) is a highly symmetric holomorphic function on the complex [[upper half-plane]]. It is invariant under the fractional linear action of the [[congruence subgroup|congruence group]] Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the [[modular curve]] ''X''(2). Over any point τ, its value can be described as a [[cross ratio]] of the branch points of a ramified double cover of the projective line by the [[elliptic curve]] <math>\mathbb{C}/\langle 1, \tau \rangle</math>, where the map is defined as the quotient by the [−1] involution. | |||
The q-expansion, where <math>q = e^{\pi i \tau}</math> is the [[Nome (mathematics)|nome]], is given by: | |||
: <math> \lambda(\tau) = 16q - 128q^2 + 704 q^3 - 3072q^4 + 11488q^5 - 38400q^6 + \dots</math>. {{oeis|id=A115977 }} | |||
By symmetrizing the lambda function under the canonical action of the symmetric group ''S''<sub>3</sub> on ''X''(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group <math>SL_2(\mathbb{Z})</math>, and it is in fact Klein's modular [[j-invariant]]. | |||
==Modular properties== | |||
The function <math> \lambda(\tau) </math> is invariant under the group generated by<ref name=C115>Chandrasekharan (1985) p.115</ref> | |||
:<math> \tau \mapsto \tau+2 \ ;\ \tau \mapsto \frac{\tau}{1-2\tau} \ . </math> | |||
The generators of the modular group act by<ref name=C109>Chandrasekharan (1985) p.109</ref> | |||
:<math> \tau \mapsto \tau+1 \ :\ \lambda \mapsto \frac{\lambda}{\lambda-1} \, ;</math> | |||
:<math> \tau \mapsto -1/\tau \ :\ \lambda \mapsto 1 - \lambda \ . </math> | |||
Consequently, the action of the modular group on <math> \lambda(\tau) </math> is that of the [[anharmonic group]], giving the six values of the [[cross-ratio]]:<ref name=C110>Chandrasekharan (1985) p.110</ref> | |||
:<math> \left\lbrace { \lambda, \frac{1}{1-\lambda}, \frac{\lambda-1}{\lambda}, \frac{1}{\lambda}, \frac{\lambda}{\lambda-1}, 1-\lambda } \right\rbrace \ .</math> | |||
== Other appearances == | |||
===Other elliptic functions=== | |||
It is the [[Square (algebra)|square]] of the [[Jacobi modulus]],<ref name=C108>Chandrasekharan (1985) p.108</ref> that is, λ(τ) = ''k''<sup>2</sup>(τ). In terms of [[theta function]]s,<ref name=C108/> | |||
:<math> \lambda(\tau) = \frac{\theta_1^4(0,\tau)}{\theta_3^4(0,\tau)} </math> | |||
where<ref name=C63>Chandrasekharan (1985) p.63</ref> | |||
:<math>\theta_1(0,\tau) = \sum_{n=-\infty}^\infty q^{\left({n+\frac12}\right)^2} \mathrm{ and } \ | |||
\theta_3(0,\tau) = \sum_{n=-\infty}^\infty q^{n^2} </math> | |||
for the [[Nome (mathematics)|nome]] <math>q = e^{\pi i \tau}</math>. | |||
In terms of the half-periods of [[Weierstrass's elliptic functions]], let [ω<sub>1</sub>, ω<sub>2</sub>] be a [[fundamental pair of periods]] with τ = ω<sub>2</sub>/ω<sub>1</sub>. | |||
:<math> e_1 = \wp(\omega_1/2), e_2 = \wp(\omega_2/2), e_3 = \wp((\omega_1+\omega_2)/2) </math> | |||
we have<ref name=C108/> | |||
:<math> \lambda = \frac{e_3-e_2}{e_1-e_2} \, . </math> | |||
Since the three half-period values are distinct, this shows that λ does not take the value 0 or 1.<ref name=C108/> | |||
The relation to the ''j''-invariant is<ref name=C117>Chandrasekharan (1985) p.117</ref><ref>Rankin (1977) pp.226–228</ref> | |||
:<math> j(\tau) = 256 \frac{(1-\lambda+\lambda^2)^3}{\lambda^2 (1-\lambda)^2} \ . </math> | |||
which is the ''j''-invariant of the elliptic curve of [[Legendre form]] <math>y^2=x(x-1)(x-\lambda)</math> | |||
===Little Picard theorem=== | |||
The lambda function is used in the original proof of the [[Little Picard theorem]], that an [[entire function|entire]] non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard in 1879.<ref>Chandrasekharan (1985) p.121</ref> Suppose if possible that ''f'' is entire and does not take the values 0 and 1. Since λ is holomorphic, it has a local holomorphic inverse ω defined away from 0,1,∞. Consider the function ''z'' → ω(''f''(''z'')). By the [[Monodromy theorem]] this is holomorphic and maps the complex plane '''C''' to the upper half plane. From this it is easy to construct a holomorphic function from '''C''' to the unit disc, which by [[Liouville's theorem (complex analysis)|Liouville's theorem]] must be constant.<ref>Chandrasekharan (1985) p.118</ref> | |||
===Moonshine=== | |||
The function <math>\frac{16}{\lambda(2\tau)} - 8</math> is the normalized [[Hauptmodul]] for the group <math>\Gamma_0(4)</math>, and its ''q''-expansion <math>q^{-1} + 20q - 62q^3 + \dots</math> is the graded character of any element in conjugacy class 4C of the [[monster group]] acting on the [[monster vertex algebra]]. | |||
== References == | |||
{{reflist}} | |||
*{{Citation | editor1-last=Abramowitz | editor1-first=Milton | editor1-link=Milton Abramowitz | editor2-last=Stegun | editor2-first=Irene A. | editor2-link=Irene Stegun | title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-61272-0 | year=1972 | zbl=0543.33001 }} | |||
* {{citation | last=Chandrasekharan | first=K. | authorlink=K. S. Chandrasekharan | title=Elliptic Functions | series=Grundlehren der mathematischen Wissenschaften | volume=281 | publisher=[[Springer-Verlag]] | year=1985 | isbn=3-540-15295-4 | zbl=0575.33001 | pages=108–121 }} | |||
*{{citation|first1=John Horton|last1=Conway|author1-link=John Horton Conway|first2=Simon|last2=Norton|author2-link=Simon P. Norton|title=Monstrous moonshine|journal=Bulletin of the London Mathematical Society|volume=11|issue=3|year=1979|pages=308–339|mr=0554399|zbl=0424.20010 |doi=10.1112/blms/11.3.308}} | |||
* {{citation | last=Rankin | first=Robert A. | authorlink=Robert Alexander Rankin | title=Modular Forms and Functions | publisher=[[Cambridge University Press]] | year=1977 | isbn=0-521-21212-X | zbl=0376.10020 }} | |||
*{{dlmf|id=23.15.E6|title=Elliptic Modular Function|first= W. P. |last=Reinhardt|first2=P. L.|last2= Walker}} | |||
{{DEFAULTSORT:Modular Lambda Function}} | |||
[[Category:Modular forms]] | |||
[[Category:Elliptic functions]] |
Revision as of 04:10, 6 January 2013
In mathematics, the elliptic modular lambda function λ(τ) is a highly symmetric holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(2). Over any point τ, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve , where the map is defined as the quotient by the [−1] involution.
The q-expansion, where is the nome, is given by:
By symmetrizing the lambda function under the canonical action of the symmetric group S3 on X(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group , and it is in fact Klein's modular j-invariant.
Modular properties
The function is invariant under the group generated by[1]
The generators of the modular group act by[2]
Consequently, the action of the modular group on is that of the anharmonic group, giving the six values of the cross-ratio:[3]
Other appearances
Other elliptic functions
It is the square of the Jacobi modulus,[4] that is, λ(τ) = k2(τ). In terms of theta functions,[4]
where[5]
for the nome .
In terms of the half-periods of Weierstrass's elliptic functions, let [ω1, ω2] be a fundamental pair of periods with τ = ω2/ω1.
we have[4]
Since the three half-period values are distinct, this shows that λ does not take the value 0 or 1.[4]
The relation to the j-invariant is[6][7]
which is the j-invariant of the elliptic curve of Legendre form
Little Picard theorem
The lambda function is used in the original proof of the Little Picard theorem, that an entire non-constant function on the complex plane cannot omit more than one value. This theorem was proved by Picard in 1879.[8] Suppose if possible that f is entire and does not take the values 0 and 1. Since λ is holomorphic, it has a local holomorphic inverse ω defined away from 0,1,∞. Consider the function z → ω(f(z)). By the Monodromy theorem this is holomorphic and maps the complex plane C to the upper half plane. From this it is easy to construct a holomorphic function from C to the unit disc, which by Liouville's theorem must be constant.[9]
Moonshine
The function is the normalized Hauptmodul for the group , and its q-expansion is the graded character of any element in conjugacy class 4C of the monster group acting on the monster vertex algebra.
References
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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.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
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 - 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.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
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 - 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.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
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 - Template:Dlmf