Reference request: the dual Coleman family

Question

Recently when I want to understand the construction of triple product p-adic L-function, I am really confused by the notion of dual form. To be precise, assume $f^\circ\in{S_k(N,\chi)}$ is an eigenform of weight k, level N (not necessary new) with nebentype $\chi$ and f be (one of the) p-stabilization of $f^{\circ}$, we can construct several modular forms from f: assume the $q$-expansion of f is $\sum_{n\geq1}a_nq^n$,

$(i)$ The conjugate form $f^*$ defined by $f^*(\tau)=\overline{f(-\bar{\tau})}$, the q-expansion of $f^*$ is $\sum_{n\geq1}\bar{a}_nq^n$;

$(ii)$ The twisted form $f_{\chi^{-1}}$ by $\chi^{-1}$ whose q-expansion is $\sum_{n\geq1}\chi^{-1}(n)a_nq^n$;

$(iii)$ and moreover as in Def.2.4 of the paper a note on p-adic Rankin-Selberg L-functions, the form $f^c$ which is the unique form which has level $Np^r$ for some r and whose Hecke eigenvalue away from N is same as $f_{\chi^{-1}}$.

What is the relation of the three forms? Would anyone provide some reference? Thanks

If F is a Coleman family of tame level N and nebentype $\chi$, could we define the corresponding family $F^*$, $F_{\chi^{-1}}$ or $F^c$ satisfying similar description on $q$-expansion? The paper A p-adic Gross-Zagier formula for diagoanl cycles(page 41) uses $F^*$ (for Hida family), but no details provided there and Lem.3.4 of 1 claims the existence of $F^c$ (at least for F is new at N) but I could not spell out the details. Would anyone please provide me some reference containing more details? Thanks

Answer 1

The definition of $f^c$ that I give in my Canad Math Bulletin paper is only well-defined when $f$ is a p-stabilised newform (i.e. f has level $Np^r$ for some $r \ge 1$ and $p \nmid N$, and either f is new of level $Np^r$, or $r = 1$ and f is the p-stabilisation of a newform of level N). If f is not locally new at the primes dividing N, then I didn't attempt to define $f^c$, and I suspect there is no good way of doing so.

In the "crystalline" case, when f is a p-stabilisation of a level N newform $f^\circ$, then $f^c$ and $f^*$ are both well-defined, and they are both p-stabilised newforms, but they are not generally the same. The relation between the two is the following. There is a level N newform $(f^\circ)^*$, and it has two p-stabilisations at level Np; one of these is $f^*$ and the other one is $f^c$. One computes that $a_p(f^c) = \chi(p)^{-1} a_p(f)$, but $a_p(f^*) = p^{k-1} / a_p(f)$. Hence the observation in my Remark 2.5 that if f is ordinary, then so is $f^c$, but $f^*$ is not ordinary unless k = 1.

(You can see immediately from those two formulae that $a_p(f^*)$ does not interpolate as f varies over specialisations of a Coleman family F, because $p^{k-1}$ is not a p-adically continuous function of k in weight space. I'm not 100% sure what Darmon and Rotger mean by $F^*$ when F is a Hida family, but I suspect they mean what I'm calling $F^c$.)

EDIT. You asked how $F^c$ is constructed. A Coleman family of tame level N has a "tame nebentype" $\chi$, which is a Dirichlet character mod $N$. If you twist $F$ by $\chi^{-1}$, you get another Coleman family whose tame nebentype is $\chi^{-1}$. A priori this family might have tame level larger than $N$, maybe up to $N^2$; but you know that it has many classical specialisations of tame level $N$ (in fact it suffices for it to have a single non-critical-slope specialisation of tame level $N$), so in fact this family has tame level $N$ and you are done.