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Is there a reference that contains explicit examples of component crossing of Hida families at height one primes? The paper of Emerton, Pollack, and Weston addresses component crossing obtained through level raising. I am interested in examples caused by other phenomena (e.g. a CM family meeting another CM family coming from different imaginary quadratics or a CM family meeting a non-CM family.) A question was previously asked that suggests such families exist:

Example of a non-smooth irreducible component of the generic fibre of a Hida family?

I am unable to find examples in the literature, but it's possible that I've looked in the wrong places.

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    $\begingroup$ I asked a question some while back about congruences between CM and non-CM modular forms; this should give some examples. $\endgroup$
    – David Loeffler
    Commented Jan 6, 2015 at 13:33
  • $\begingroup$ @DavidLoeffler Sorry for not being more specific. Such a congruence would give a crossing at a maximal idea. I am primarily interested in crossings at height one primes. I have edited the question. $\endgroup$
    – jkramerm
    Commented Jan 6, 2015 at 14:51

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There is an interesting example in this paper by Dimitrov and Ghate, section 7.3.

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You can write down explicit examples of such crossing with Eisenstein series. If one takes the $p$-adic family of Eisenstein series $E^{(p)}_k(\chi_1,\chi_2)$ and the family $E^{(p)}_k(\chi_2,\chi_1)$, then one sees explicitly (just look at $q$-expansions) that these families meet in weight 1 --- i.e. the order of the characters doesn't matter in weight 1.

In the special case when $\chi_1$ is quadratic, $\chi_2$ is trivial and $\chi_1(p)=1$, there should even be some explicit CM family which also specializes in weight 1 to $E^{(p)}_1(\chi_1,1)$ coming from the the quadratic field cut out by $\chi_1$.

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  • $\begingroup$ I agree that the constant term of the $q$-expansion of $E^p_1(\chi_1,\chi_2)$ at the cusp $\infty$ is zero, but how we proof that the evaluation of $E^p_1(\chi_1,\chi_2)$ is trivial at the $\Gamma_0(p)$ orbit of $\infty$ (in order to say that $E^p_1(\chi_1,\chi_2)$ is cupsidal overconvergent)? $\endgroup$
    – Adel BETINA
    Commented Sep 19, 2016 at 20:40

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