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On the $p$-adic $L$-function of Bianchi modular forms

Luis Santiago Palacios

Universidad de Santiago de Chile, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloaka5b6a909c55759059724abb1f1d8f118').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addya5b6a909c55759059724abb1f1d8f118 = 'ls&#101;p&#97;l&#97;c&#105;&#111;sm1' + '&#64;'; addya5b6a909c55759059724abb1f1d8f118 = addya5b6a909c55759059724abb1f1d8f118 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_texta5b6a909c55759059724abb1f1d8f118 = 'ls&#101;p&#97;l&#97;c&#105;&#111;sm1' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloaka5b6a909c55759059724abb1f1d8f118').innerHTML += '<a ' + path + '\'' + prefix + ':' + addya5b6a909c55759059724abb1f1d8f118 + '\'>'+addy_texta5b6a909c55759059724abb1f1d8f118+'<\/a>';

Fix $p$ a rational prime, in this poster we present two works about the $p$-adic $L$-function for Bianchi modular forms, that is, the case of automorphic forms for $\mathrm{GL}_2$ over an imaginary quadratic field $K$.

(1) In the first work, when $K$ has class number 1, we obtain the functional equation of: (a) the $p$-adic $L$-function constructed in [Wil17], for a small slope $p$-stabilisation of a cuspidal Bianchi modular form and (b) the $p$-adic $L$-function constructed in [BSWWE18], for a critical slope $p$-stabilisation of a Base change cuspidal Bianchi modular form that is $\Sigma$-smooth. To treat case (b) we use $p$-adic families of Bianchi modular forms.

(2) In the second work we construct the $p$-adic $L$-function of a small slope Eisenstein Bianchi modular form introducing the notions of $C$-cuspidality and partial Bianchi modular symbols, both inspired in [BD15], and modifying accordingly the construction in [Wil17]. We also provide an example where our construction applies: let $\psi$ be a Hecke character of $K$ and consider the base change to $K$ of the theta series of $\psi$, this base change is a non-cuspidal Bianchi modular form that we can $p$-stabilise to have small slope, then we obtain by our methods its $p$-adic $L$-function, even more, we can relate such $p$-adic $L$-function with the Katz $p$-adic $L$-function of $\psi$ introduced in [Kat78]. This is a work in progress.

References

[BD15] Joël Bellaïche and Samit Dasgupta, The $p$-adic $L$-functions of evil Eisenstein series, Compositio Mathematica151(2015), no. 6, 999–1040.

[BSWWE18] Daniel Barrera Salazar, Chris Williams, and Carl Wang-Erickson, Families of Bianchi modular symbols: critical base-change $p$-adic $L$-functions and $p$-adic Artin formalism, arXiv preprint arXiv:1808.09750v6 (2018).

[Kat78] Nicholas M Katz, $p$-adic $L$-functions for CM fields, Inventiones mathematicae 49 (1978), no. 3-4, 199–297.

[Wil17] Chris Williams, $p$-adic $L$-functions of Bianchi modular forms, Proceedings of the London Mathematical Society 114 (2017), no. 4, 614–656.