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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-b\tau)} + \sum_{\omega \in \Lambda \backslash \{0\}} \left[ \frac{1}{(z-\omega)(z-\omega-a-b\tau)} - \frac{1}{\omega^2} \right] - \frac{a \eta_1 + b \eta_2}{a+ b \tau} \end{equation*} Yes, this is quite similar to $\wp$.

  1. 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.

  2. What I actually need to know about this function is related to its zeroes. Clearly, it has poles at $0$ and $a+b\tau$, 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. $0 \to {\mathcal O}_E \to {\mathcal E} \to {\mathcal O}_E \to 0$.

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    $\begingroup$ It's probably related to $\zeta_\Lambda(z)-\zeta_\Lambda(z-a-b\tau)$ 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
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    $\begingroup$ Let $c = a+b\tau$ 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

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