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Deligne, Goncharov and Levine have constructed a Tannakian category of mixed Tate motives, MTM($\mathcal{O}_{K,S}$), over the ring of integers of a number field $K$ unramified outside a finite set of places $S$.

In particular there is a category MTM($\mathbb{Z}$) of mixed Tate motives unramified over $\mathbb{Z}$. Its objects are extensions of pure Tate motives $\mathbb{Q}(n)$ where $n$ is an integer. They arise from motivic sheaves on the moduli spaces of marked genus 0 curves.

The $\ell$-adic realization of the pure Tate motive $\mathbb{Q}(n)$ is the $Gal (\overline{\mathbb{Q}}/\mathbb{Q})$-module $\mathbb{Q}_\ell$, the Galois action being given by the $n$-th power of the $\ell$-adic cyclotomic character $\chi_\ell$.

In Langlands philosophy every motive should correspond to an automorphic form. For pure Tate motives $\mathbb{Q}(n)$ class field theory shows how: they correspond to powers of idèle class characters.

Which automorphic form/representation does a truly mixed Tate motive correspond to?

For example we know that $M^1_n := Ext^1(\mathbb{Q}(0), \mathbb{Q}(n)) = K_{2n-1}(\mathbb{Z})\otimes \mathbb{Q}$ for $n > 1$ where $K_.$ are the Milnor K-groups. What makes $M^1_n$ automorphic?

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    $\begingroup$ Just to say that your Galois representation $\rho(\zeta_{\ell^m}\to \zeta_{\ell^m}^a)w= a^n w$ corresponding to $\mathbb{Q}(n)$ factorizes through $Gal(\Bbb{Q}(\zeta_{\ell^\infty})/\Bbb{Q})$ where $Frob(p)= (\zeta_{\ell^m}\to \zeta_{\ell^m}^p)$ so the Artin L-function is $\prod_{p\ne \ell} \frac1{\det(1-\rho(Frob(p)) p^{-s})}=\prod_{p\ne \ell}\frac1{1-p^{n-s}}=(1-p^{n-s})\zeta(s-n)$ which comes from the automorphic representation $\Bbb{A_Q^\times / Q^\times\to C^\times}$, $\pi(x)=|x_\infty|^{-n}\prod_p p^{n \ v_p(x_p)}$ $\endgroup$ – reuns Dec 3 '20 at 5:23
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This answer is a slight addition to Joel's and David's.

In the theory of Galois representations, there is a general philosophy that $p$-adic phenomena (say in Hida families or eigenvarieties) reflect corresponding mod $p$ phenomena. So before asking if all extensions of $\mathbb Q_p(n)$ by $\mathbb Q_p$ can be constructed automorphically, one could ask the corresponding question mod $p$. Here is a slightly more general version, for the number field $\mathbb Q$: if $\psi$ and $\chi$ are two mod $p$ Dirichlet characters (of some conductor), thought of as Galois characters, whose product is odd, can any extension of $\chi$ by $\psi$ be realised in the reduction mod $p$ of a lattice in the (irreducible) $p$-adic Galois representation attached to some cusp form.

I think that the answer to this (and more general version of this) question is believed to be yes (folklore), and perhaps is even known to be yes in the particular case I just discussed. Questions of this type are certainly discussed in Skinner--Wiles: I think they will show that if you have a $p$-adic lattice in an irreducible representation whose reduction gives some extension class, then there is a cusp form giving rise to this extension. (They will need to assume ordinary at $p$, but actually recent work of L. Pan will let you replace ordinary by a more general Fontaine--Mazur-type condition.) Part of their technique is to move from one possible extension class to another. (Kisin used to call this ``Skinner--Wiles tunnelling''.)

I don't know if it's known that every extension of $\chi$ by $\psi$ actually can be lifted to a lattice that is ordinary (or, more generally, geometric) at $p$, though. If it were, one would get a positive answer to the mod $p$ version of the problem in some cases.

On a different note, I believe that Harder (and probably many others too) studied extension classes of automorphic motives arising from Eisenstein cohomology classes, and in this way constructed (or proposed construtions for) actual mixed motives, rather than just Galois representations. The first example of this would be looking at cohomology of open modular curves, or of complete modular curves relative to the cusps. One obtains mixed motives this way, but by Manin--Drinfeld, if you work over $\mathbb Q$ you get split extensions --- non-splitness only occurs with torsion coefficients. I think you can get more interesting things if you both delete some cusps, and then work relative to others (so that the weight filtration on cohomology has length three rather than just two), although it's been a long time since I thought about it (Mazur mentioned these ideas to me, in conjunction with Harder's name, when I was a grad student). I don't know what happens on higher dimensional Shimura varieties, but work of Harris and Zucker on mixed Hodge theory of Shimura varieties is probably relevant.

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  • $\begingroup$ Many thanks for your helpful addition with leads to the literature! Could the Harder paper be "G. HARDER, Eisenstein cohomology of arithmetic groups: The case GL-2 {Invent. Math., Vol. 89, 1987, pp. 37-118)" I wonder. The connection with torsion coefficient is very interesting. $\endgroup$ – Nimas Dec 3 '20 at 18:50
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    $\begingroup$ Not sure about the Harder reference --- probably it's not that one exactly. Perhaps a separate question for a literature reference would be sensible. I have a vague memory of a Springer Lecture Notes, but probably some of this was computations and ideas that weren't published. Chris Skinner might know the reference, if you happen to be in Princeton. $\endgroup$ – Matt E Dec 3 '20 at 19:07
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    $\begingroup$ A colleague also pointed out this reference, which is relevant to the lifting-to-a-lattice problem I discussed: LIFTING AND MODULARITY OF REDUCIBLE MOD p GALOIS REPRESENTATIONS, by NAJMUDDIN FAKHRUDDIN, CHANDRASHEKHAR KHARE, AND STEFAN PATRIKIS $\endgroup$ – Matt E Dec 3 '20 at 20:02
  • $\begingroup$ Thanks a lot for the pointers. On Prof. Harder's website there is a folder "Eisenstein" of unpublished manuscripts at least a couple of which Mix-Mot-2015/17.pdf seem to be along the lines that you are referring to: math.uni-bonn.de/people/harder/Manuscripts/Eisenstein $\endgroup$ – Nimas Dec 3 '20 at 20:28
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You seem to be starting with the answer and trying to deduce the question.

By Langlands philosophy every motive should correspond to an automorphic form. [...] What makes $M^1_n$ automorphic?

I think it's a very interesting philosophical question: if (simple) pure motives correspond to (maybe cuspidal, algebraic, etc) automorphic representations, then should non-semisimple mixed motives correspond to? However, the answer is far from obvious. So we don't know yet what the question "is $M^1_n$ is automorphic?" should mean, which makes it very difficult to answer it in any meaningful way.

What is true is that one can construct lots of interesting examples of mixed motives (or, at least, of their realisations) using modular curves, or more general Shimura varieties. E.g. a very good way of constructing and studying elements in groups like $Ext^1(Q(0), Q(n))$ is to look at the behaviour of Eisenstein series at cusps of a modular curve. But there isn't yet a general theory of "mixed automorphic representations" that gives a unified explanation of all these constructions.

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David is right: there isn't yet a general theory of "mixed automorphic representations". Let me propose some ideas of what a theory could be.

First, I will work with the $p$-adic realizations of mixed motives, rather than with the the mixed motives themself, for concreteness and because motives are harder to construct that Galois representations.

I don't think that it is reasonable to expect that every $p$-adic non-semisimple Galois representations (with Fontaine-Mazur-like conditions) should come from an automorphic representations. In fact, I don't even know what "come" would mean here : to an algebraic automorphic representation, say for $Gl_n$ or for a reductive group G together with a representation of its $L$-group into $Gl_n$, I know at least conjecturally how to attach a semi-simple Galois representations, not a non-trivial extension.

Rather, I expect non semi-simple Galois representations (with suitable local conditions that makes them "motivic") would all come from $p$-adic deformations of automorphic form (in the sense of eigenvarieties). Rather than to try to formulate this in general (that would be mostly guess-work), let me consider an interesting very particular case.

Let $F$ be a number field. I consider extensions of $\mathbb Q_p$ by $\mathbb Q_p(1)$ in the category of $G_F$-representations, which are unramified outside $p$ and crystalline at $p$. The spaces of such extensions is naturally isomorphic, as is well-known (Kümmer Theory ++), to the space $\cal O_F^\ast \otimes_{\mathbb Z} \mathbb Q_p$, whose dimension is determined by Dirichlet's unit Theorem to be $[F:\,mathbb Q]-1$.

Say $F$ is totally real to fix ideas (and different from $\mathbb Q$) because in this case the eigenvariety of over convergent $p$-adic automorphic forms for $GL_2/F$ has been constructed (Andreatta-Iovita-Pilloni-Stevens). There is a (unique) point $x$ on this eigenvariety corresponding on this the Eisenstein series $E_2$ over $F$ with critical refinements at every place above $p$.

The eigenvariety carries a family of Galois representations (rather a pseudo-representation) which at $x$ is the (semi-simple) representation $\mathbb Q_p \oplus \mathbb Q_p(1)$ but is irreducible at points near $x$ (that's a consequence of the choice of "critical" refinements). If you choose any curve $C$ in the eigenvariety through $x$, smooth at $x$, and restrict your family of Galois representation to $C$, and even to the local ring of $C$ at $x$, then the famous "Ribet lemma" gives you an extension of $\mathbb Q_p$ by $\mathbb Q_p(1)$ in the category of Galois representations of $G_F$, unramified outside $p$ and crystalline at $p$.

We can say that this extension is (the $p$-adic realization) of a mixed motive, automorphically constructed.

Then in this special case, you question can be precisely formulated:

Conjecture: All such extensions of $\mathbb Q_p$ by $\mathbb Q_p(1)$ comes from a deformation of $E_2$ this way.

I don't know if that's true beyond the trivial $[F:\mathbb Q]=2$ case. But I think it is reasonable to expect this. A student of me, Yu Fang, proved it is true when the eigenvariety is smooth at $x$.

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  • $\begingroup$ I guess this answer is a mirror-image of mine, in some sense -- I was thinking of the kind of global cohomology classes that form Euler systems, while you were thinking of the kind that come from Eisenstein congruences. $\endgroup$ – David Loeffler Dec 4 '20 at 7:42
  • $\begingroup$ So this prompts the question: let $A$ be a modular elliptic curve, let $P$ be a Heegner point on $A$, and let $E$ be the extension $0 \to V_p A \to E \to \mathbf{Q}_p \to 0$ that is the image of $P$ under the Kummer map. I think $E$ ought to count as a mixed automorphic motive, because it comes from the cycle class of a morphism of Shimura varieties. Can one see $E$ using eigenvarieties (ideally without having to prove Skinner--Urban in the process)? $\endgroup$ – David Loeffler Dec 4 '20 at 7:46

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