Zeroes of some degree of two elliptic functions
$begingroup$
Let $tau in {mathbb C}$ with $mathrm{Im} tau > 0$, $a,b in {mathbb Q}$ not both integers (it's not clear to me whether assuming only $a,b in {mathbb R}$ will make a difference to the question or not), $Lambda subset {mathbb C}$ the lattice generated by $1$ and $tau$ and $eta_1$, $eta_2$ the quasi-periods of the Weierstrass $zeta$ function $-int wp$ corresponding to $Lambda$ ( see e.g. https://en.wikipedia.org/wiki/Weierstrass_functions), i.e. the values of the Weierstrass eta function at $1$ and $tau$.
While doing some work in algebraic geometry, I've come across the following degree $2$ elliptic function
begin{equation*}
Upsilon(z) = frac{1}{z(z-a-btau)} + sum_{omega in Lambda backslash {0}} left[ frac{1}{(z-omega)(z-omega-a-btau)} - frac{1}{omega^2} right] - frac{a eta_1 + b eta_2}{a+ b tau}
end{equation*}
Yes, this is quite similar to $wp$.
Since I know nothing about complex analysis, I'd appreaciate any help "identifying" this function -- Does it have a name, can it be expressed in particularly simple way in terms of other functions. Has it appeared anywhere else? It's extremely possible that I'm missing something simple.
What I actually need to know about this function is related to its zeroes. Clearly, it has poles at $0$ and $a+btau$, so we know the sum of the zeroes. Can its zeroes be computed? Is it easier than for $wp$? http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BF01453974/fulltext.pdf (I'm under the impression that $sum_{n geq 0} frac{1}{(an+b)^2}$ is harder than the other $sum_{n geq 0} frac{1}{(an+b)(an+c)}$, so maybe that's not so unlikely.)
P.S. If anyone is curious, the algebraic geometry calculation I was doing was taking place on a geometrically ruled surface over an elliptic curve $E$, specifically, the projectivization of the rank two bundle ${mathcal E}$ which fits in a nonsplit s.e.s. $0 to {mathcal O}_E to {mathcal E} to {mathcal O}_E to 0$.
complex-analysis algebraic-geometry elliptic-functions
edited Dec 24 '18 at 7:40
dmtri
1,7712521
asked Dec 24 '18 at 7:33
azaha89azaha89
264
$endgroup$
add a comment |
$begingroup$
Let $tau in {mathbb C}$ with $mathrm{Im} tau > 0$, $a,b in {mathbb Q}$ not both integers (it's not clear to me whether assuming only $a,b in {mathbb R}$ will make a difference to the question or not), $Lambda subset {mathbb C}$ the lattice generated by $1$ and $tau$ and $eta_1$, $eta_2$ the quasi-periods of the Weierstrass $zeta$ function $-int wp$ corresponding to $Lambda$ ( see e.g. https://en.wikipedia.org/wiki/Weierstrass_functions), i.e. the values of the Weierstrass eta function at $1$ and $tau$.
While doing some work in algebraic geometry, I've come across the following degree $2$ elliptic function
begin{equation*}
Upsilon(z) = frac{1}{z(z-a-btau)} + sum_{omega in Lambda backslash {0}} left[ frac{1}{(z-omega)(z-omega-a-btau)} - frac{1}{omega^2} right] - frac{a eta_1 + b eta_2}{a+ b tau}
end{equation*}
Yes, this is quite similar to $wp$.
Since I know nothing about complex analysis, I'd appreaciate any help "identifying" this function -- Does it have a name, can it be expressed in particularly simple way in terms of other functions. Has it appeared anywhere else? It's extremely possible that I'm missing something simple.
What I actually need to know about this function is related to its zeroes. Clearly, it has poles at $0$ and $a+btau$, so we know the sum of the zeroes. Can its zeroes be computed? Is it easier than for $wp$? http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BF01453974/fulltext.pdf (I'm under the impression that $sum_{n geq 0} frac{1}{(an+b)^2}$ is harder than the other $sum_{n geq 0} frac{1}{(an+b)(an+c)}$, so maybe that's not so unlikely.)
P.S. If anyone is curious, the algebraic geometry calculation I was doing was taking place on a geometrically ruled surface over an elliptic curve $E$, specifically, the projectivization of the rank two bundle ${mathcal E}$ which fits in a nonsplit s.e.s. $0 to {mathcal O}_E to {mathcal E} to {mathcal O}_E to 0$.
complex-analysis algebraic-geometry elliptic-functions
edited Dec 24 '18 at 7:40
dmtri
1,7712521
asked Dec 24 '18 at 7:33
azaha89azaha89
264
$endgroup$
1
$begingroup$
It's probably related to $zeta_Lambda(z)-zeta_Lambda(z-a-btau)$ where $zeta_Lambda$ is the Weierstrass zeta function.
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 8:17
$begingroup$
@Lord Shark the Unknown: yes, that is true! I forgot to say that that's how I got it, so obviously I'd prefer something different.
$endgroup$
– azaha89
Dec 24 '18 at 9:23
1
$begingroup$
Let $c = a+btau$ and $f(z) = sum_{omega in Lambda} frac{1}{(z-omega)(z-omega-c)} - frac{1_{omega ne 0}}{omega^2}$ it converges and it is $Lambda$ periodic and meromorphic with two simples poles at $0,c$ thus $f(z+c/2)$ has two simple poles at $-c/2,c/2$. Then $wp(z)-wp(c/2)$ has two simple zeros at $-c/2,c/2$ and one double pole at $0$ so $f(z+c/2) (wp(z)-wp(c/2))$ has at worst one double pole at $0$ and hence $f(z+c/2) (wp(z)-wp(c/2)) =u, wp(z)+v$ where $u= f(c/2)$
$endgroup$
– reuns
Dec 24 '18 at 11:15
add a comment |
$begingroup$
Let $tau in {mathbb C}$ with $mathrm{Im} tau > 0$, $a,b in {mathbb Q}$ not both integers (it's not clear to me whether assuming only $a,b in {mathbb R}$ will make a difference to the question or not), $Lambda subset {mathbb C}$ the lattice generated by $1$ and $tau$ and $eta_1$, $eta_2$ the quasi-periods of the Weierstrass $zeta$ function $-int wp$ corresponding to $Lambda$ ( see e.g. https://en.wikipedia.org/wiki/Weierstrass_functions), i.e. the values of the Weierstrass eta function at $1$ and $tau$.
While doing some work in algebraic geometry, I've come across the following degree $2$ elliptic function
begin{equation*}
Upsilon(z) = frac{1}{z(z-a-btau)} + sum_{omega in Lambda backslash {0}} left[ frac{1}{(z-omega)(z-omega-a-btau)} - frac{1}{omega^2} right] - frac{a eta_1 + b eta_2}{a+ b tau}
end{equation*}
Yes, this is quite similar to $wp$.
Since I know nothing about complex analysis, I'd appreaciate any help "identifying" this function -- Does it have a name, can it be expressed in particularly simple way in terms of other functions. Has it appeared anywhere else? It's extremely possible that I'm missing something simple.
What I actually need to know about this function is related to its zeroes. Clearly, it has poles at $0$ and $a+btau$, so we know the sum of the zeroes. Can its zeroes be computed? Is it easier than for $wp$? http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BF01453974/fulltext.pdf (I'm under the impression that $sum_{n geq 0} frac{1}{(an+b)^2}$ is harder than the other $sum_{n geq 0} frac{1}{(an+b)(an+c)}$, so maybe that's not so unlikely.)
P.S. If anyone is curious, the algebraic geometry calculation I was doing was taking place on a geometrically ruled surface over an elliptic curve $E$, specifically, the projectivization of the rank two bundle ${mathcal E}$ which fits in a nonsplit s.e.s. $0 to {mathcal O}_E to {mathcal E} to {mathcal O}_E to 0$.
complex-analysis algebraic-geometry elliptic-functions
edited Dec 24 '18 at 7:40
dmtri
1,7712521
asked Dec 24 '18 at 7:33
azaha89azaha89
264
$endgroup$
Let $tau in {mathbb C}$ with $mathrm{Im} tau > 0$, $a,b in {mathbb Q}$ not both integers (it's not clear to me whether assuming only $a,b in {mathbb R}$ will make a difference to the question or not), $Lambda subset {mathbb C}$ the lattice generated by $1$ and $tau$ and $eta_1$, $eta_2$ the quasi-periods of the Weierstrass $zeta$ function $-int wp$ corresponding to $Lambda$ ( see e.g. https://en.wikipedia.org/wiki/Weierstrass_functions), i.e. the values of the Weierstrass eta function at $1$ and $tau$.
While doing some work in algebraic geometry, I've come across the following degree $2$ elliptic function
begin{equation*}
Upsilon(z) = frac{1}{z(z-a-btau)} + sum_{omega in Lambda backslash {0}} left[ frac{1}{(z-omega)(z-omega-a-btau)} - frac{1}{omega^2} right] - frac{a eta_1 + b eta_2}{a+ b tau}
end{equation*}
Yes, this is quite similar to $wp$.
Since I know nothing about complex analysis, I'd appreaciate any help "identifying" this function -- Does it have a name, can it be expressed in particularly simple way in terms of other functions. Has it appeared anywhere else? It's extremely possible that I'm missing something simple.
What I actually need to know about this function is related to its zeroes. Clearly, it has poles at $0$ and $a+btau$, so we know the sum of the zeroes. Can its zeroes be computed? Is it easier than for $wp$? http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BF01453974/fulltext.pdf (I'm under the impression that $sum_{n geq 0} frac{1}{(an+b)^2}$ is harder than the other $sum_{n geq 0} frac{1}{(an+b)(an+c)}$, so maybe that's not so unlikely.)
P.S. If anyone is curious, the algebraic geometry calculation I was doing was taking place on a geometrically ruled surface over an elliptic curve $E$, specifically, the projectivization of the rank two bundle ${mathcal E}$ which fits in a nonsplit s.e.s. $0 to {mathcal O}_E to {mathcal E} to {mathcal O}_E to 0$.
complex-analysis algebraic-geometry elliptic-functions
complex-analysis algebraic-geometry elliptic-functions
edited Dec 24 '18 at 7:40
dmtri
1,7712521
asked Dec 24 '18 at 7:33
azaha89azaha89
264
edited Dec 24 '18 at 7:40
dmtri
1,7712521
asked Dec 24 '18 at 7:33
azaha89azaha89
264
edited Dec 24 '18 at 7:40
dmtri
1,7712521
edited Dec 24 '18 at 7:40
dmtri
1,7712521
edited Dec 24 '18 at 7:40
dmtri
1,7712521
1,7712521
asked Dec 24 '18 at 7:33
azaha89azaha89
264
asked Dec 24 '18 at 7:33
azaha89azaha89
264
asked Dec 24 '18 at 7:33
azaha89azaha89
264
264
1
$begingroup$
It's probably related to $zeta_Lambda(z)-zeta_Lambda(z-a-btau)$ where $zeta_Lambda$ is the Weierstrass zeta function.
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 8:17
$begingroup$
@Lord Shark the Unknown: yes, that is true! I forgot to say that that's how I got it, so obviously I'd prefer something different.
$endgroup$
– azaha89
Dec 24 '18 at 9:23
1
$begingroup$
Let $c = a+btau$ and $f(z) = sum_{omega in Lambda} frac{1}{(z-omega)(z-omega-c)} - frac{1_{omega ne 0}}{omega^2}$ it converges and it is $Lambda$ periodic and meromorphic with two simples poles at $0,c$ thus $f(z+c/2)$ has two simple poles at $-c/2,c/2$. Then $wp(z)-wp(c/2)$ has two simple zeros at $-c/2,c/2$ and one double pole at $0$ so $f(z+c/2) (wp(z)-wp(c/2))$ has at worst one double pole at $0$ and hence $f(z+c/2) (wp(z)-wp(c/2)) =u, wp(z)+v$ where $u= f(c/2)$
$endgroup$
– reuns
Dec 24 '18 at 11:15
add a comment |
1
$begingroup$
It's probably related to $zeta_Lambda(z)-zeta_Lambda(z-a-btau)$ where $zeta_Lambda$ is the Weierstrass zeta function.
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 8:17
$begingroup$
@Lord Shark the Unknown: yes, that is true! I forgot to say that that's how I got it, so obviously I'd prefer something different.
$endgroup$
– azaha89
Dec 24 '18 at 9:23
1
$begingroup$
Let $c = a+btau$ and $f(z) = sum_{omega in Lambda} frac{1}{(z-omega)(z-omega-c)} - frac{1_{omega ne 0}}{omega^2}$ it converges and it is $Lambda$ periodic and meromorphic with two simples poles at $0,c$ thus $f(z+c/2)$ has two simple poles at $-c/2,c/2$. Then $wp(z)-wp(c/2)$ has two simple zeros at $-c/2,c/2$ and one double pole at $0$ so $f(z+c/2) (wp(z)-wp(c/2))$ has at worst one double pole at $0$ and hence $f(z+c/2) (wp(z)-wp(c/2)) =u, wp(z)+v$ where $u= f(c/2)$
$endgroup$
– reuns
Dec 24 '18 at 11:15
1
1
$begingroup$
It's probably related to $zeta_Lambda(z)-zeta_Lambda(z-a-btau)$ where $zeta_Lambda$ is the Weierstrass zeta function.
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 8:17
$begingroup$
It's probably related to $zeta_Lambda(z)-zeta_Lambda(z-a-btau)$ where $zeta_Lambda$ is the Weierstrass zeta function.
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 8:17
$begingroup$
@Lord Shark the Unknown: yes, that is true! I forgot to say that that's how I got it, so obviously I'd prefer something different.
$endgroup$
– azaha89
Dec 24 '18 at 9:23
$begingroup$
@Lord Shark the Unknown: yes, that is true! I forgot to say that that's how I got it, so obviously I'd prefer something different.
$endgroup$
– azaha89
Dec 24 '18 at 9:23
1
1
$begingroup$
Let $c = a+btau$ and $f(z) = sum_{omega in Lambda} frac{1}{(z-omega)(z-omega-c)} - frac{1_{omega ne 0}}{omega^2}$ it converges and it is $Lambda$ periodic and meromorphic with two simples poles at $0,c$ thus $f(z+c/2)$ has two simple poles at $-c/2,c/2$. Then $wp(z)-wp(c/2)$ has two simple zeros at $-c/2,c/2$ and one double pole at $0$ so $f(z+c/2) (wp(z)-wp(c/2))$ has at worst one double pole at $0$ and hence $f(z+c/2) (wp(z)-wp(c/2)) =u, wp(z)+v$ where $u= f(c/2)$
$endgroup$
– reuns
Dec 24 '18 at 11:15
$begingroup$
Let $c = a+btau$ and $f(z) = sum_{omega in Lambda} frac{1}{(z-omega)(z-omega-c)} - frac{1_{omega ne 0}}{omega^2}$ it converges and it is $Lambda$ periodic and meromorphic with two simples poles at $0,c$ thus $f(z+c/2)$ has two simple poles at $-c/2,c/2$. Then $wp(z)-wp(c/2)$ has two simple zeros at $-c/2,c/2$ and one double pole at $0$ so $f(z+c/2) (wp(z)-wp(c/2))$ has at worst one double pole at $0$ and hence $f(z+c/2) (wp(z)-wp(c/2)) =u, wp(z)+v$ where $u= f(c/2)$
$endgroup$
– reuns
Dec 24 '18 at 11:15
add a comment |
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$begingroup$
It's probably related to $zeta_Lambda(z)-zeta_Lambda(z-a-btau)$ where $zeta_Lambda$ is the Weierstrass zeta function.
$endgroup$
– Lord Shark the Unknown
Dec 24 '18 at 8:17
$begingroup$
@Lord Shark the Unknown: yes, that is true! I forgot to say that that's how I got it, so obviously I'd prefer something different.
$endgroup$
– azaha89
Dec 24 '18 at 9:23
1
$begingroup$
Let $c = a+btau$ and $f(z) = sum_{omega in Lambda} frac{1}{(z-omega)(z-omega-c)} - frac{1_{omega ne 0}}{omega^2}$ it converges and it is $Lambda$ periodic and meromorphic with two simples poles at $0,c$ thus $f(z+c/2)$ has two simple poles at $-c/2,c/2$. Then $wp(z)-wp(c/2)$ has two simple zeros at $-c/2,c/2$ and one double pole at $0$ so $f(z+c/2) (wp(z)-wp(c/2))$ has at worst one double pole at $0$ and hence $f(z+c/2) (wp(z)-wp(c/2)) =u, wp(z)+v$ where $u= f(c/2)$
$endgroup$
– reuns
Dec 24 '18 at 11:15