# Zeroes of some degree of two elliptic functions

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. $$0 \to {\mathcal O}_E \to {\mathcal E} \to {\mathcal O}_E \to 0$$.

• It's probably related to $\zeta_\Lambda(z)-\zeta_\Lambda(z-a-b\tau)$ where $\zeta_\Lambda$ is the Weierstrass zeta function. – Lord Shark the Unknown Dec 24 '18 at 8:17
• @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. – azaha89 Dec 24 '18 at 9:23
• 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 $u= f(c/2)$ – reuns Dec 24 '18 at 11:15