6
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ this is a great question, i've often wondered this myself! $\endgroup$
    – xir
    Jan 24, 2020 at 22:47

1 Answer 1

5
$\begingroup$

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.)

The moral here is that the whole story of L-values vanishing because of "sign" phenomena is specific to self-dual settings.

$\endgroup$
4
  • $\begingroup$ Thank you very much for your clear answer. I do have some interest in the case where the character is quadratic. How would I go about computing the epsilon factor as you did? Do you have a reference that could be useful? Thank you. $\endgroup$
    – tbg93dk
    Jan 25, 2020 at 15:38
  • $\begingroup$ 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). $\endgroup$ Jan 25, 2020 at 16:02
  • $\begingroup$ Alright, thank you for your help, I will try this. $\endgroup$
    – tbg93dk
    Jan 25, 2020 at 21:42
  • $\begingroup$ 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. $\endgroup$ Jan 26, 2020 at 8:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.