# Functional equation of twisted triple product L-function

Let $$\mathbb{E}=E_1\times E_2\times E_3$$ denote the product of three elliptic curves over $$\mathbb{Q}$$ of prime level $$p$$ and consider the $$p$$-adic Galois representation $$V_p(\mathbb{E})=H^1_{et}(E_{1/\bar{\mathbb{Q}}}, \mathbb{Q}_p)\otimes H^1_{et}(E_{2/\bar{\mathbb{Q}}}, \mathbb{Q}_p)\otimes H^1_{et}(E_{3/\bar{\mathbb{Q}}}, \mathbb{Q}_p).$$ Denote by $$L(\mathbb{E}, s)=L(V_p(\mathbb{E}),s)$$ the associated triple product $$L$$-function. It has a functional equation centered at $$s=2$$ with global sign equal to $$a_p(E_1)a_p(E_2)a_p(E_2)\in \{ \pm 1 \}$$ (cf. Gross-Kudla '92). Here, $$a_p(E_i)$$ denotes the $$p$$-th Fourier coefficient of the weight 2 normalized newform of level $$\Gamma_0(p)$$ associated to $$E_i$$ by modularity.

Let $$\chi$$ be a Dirichlet character modulo $$p$$ and denote by $$L(\mathbb{E}\otimes \chi, s)$$ the $$L$$-function attached to the Galois representation $$V_p(\mathbb{E})\otimes \chi$$. My question is: what is the functional equation of $$L(\mathbb{E}\otimes \chi, s)$$ and what is its global sign?

Thank you in advance for any help.

• this is a great question, i've often wondered this myself!
– xir
Jan 24, 2020 at 22:47

There is a functional equation for $$L(\mathbb{E} \otimes \chi, s)$$, but it relates $$L(\mathbb{E} \otimes \chi, s)$$ to $$L(\mathbb{E} \otimes \bar\chi, 4-s)$$. If $$\chi$$ is not trivial or quadratic, then $$L(\mathbb{E} \otimes \chi, s)$$ and $$L(\mathbb{E} \otimes \bar\chi, s)$$ are different functions, so you cannot use this to deduce anything in particular about vanishing at the central value. The Langlands $$\varepsilon$$-factor is still defined, and it is a complex number of absolute value 1, but it isn't $$\pm 1$$, so you can't reasonably call it a "sign"; at a guess it is probably something like $$\tau(\chi)^4 / p^{2n}$$, where $$p^n$$ is the conductor of $$\chi$$ and $$\tau(\chi)$$ is the Gauss sum.
(You can try to cheat by considering the function $$M(s) = L(E \otimes \chi, s) L(E \otimes \bar\chi, s)$$, which does satisfy a functional equation relating $$M(s)$$ and $$M(4-s)$$, but unfortunately the order of vanishing of $$M(s)$$ at $$s = 2$$ is automatically even anyway, since $$L(E \otimes \bar\chi, s) = \overline{L(E \otimes \chi, \overline{s})}$$, so both factors have the same order of vanishing. So although $$M(s)$$ does have a functional equation, you can't get any nontrivial vanishing information out of it.)
• I would do this by computing the Deligne $\varepsilon$-factor of the associated Weil-Deligne representation at $p$ (which might sound scary, but is actually rather explicit and simple since all the WD reps have a very simple shape). Jan 25, 2020 at 16:02
• If you get stuck, you might like to consult Ramakrishnan's paper "Modularity of the Rankin--Selberg L-series", where he proves the existence of the Rankin--Selberg lifting $GL_2 \times GL_2 \to GL_4$; he does this using the converse theorem, so the core of the proof is a very careful analysis of the analytic continuation and functional equation of the $GL_2 \times GL_2 \times GL_2$ triple product. Jan 26, 2020 at 8:09