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An elementary introduction is available in Litt - Local–global compatibility and applications to the arithmetic of modular curves.

Historically, the first proof dealing with the simplest case came out in Deligne’s letter to Piatetski-Shapiro (the completed argument appeared in Brylinski’s appendice). Deligne’s method was generalized to the full $\operatorname{GL}_2$ case by Carayol and later to arbitrary $\operatorname{GL}_n$ by Harris–Taylor in their book.

As for as I know, the state-of-the-art results start from Caraiani‘s thesis Local-global compatibility and the action of monodromy on nearby cycles (see also her second paper, which extends the results to the case $\ell=p$).

An elementary introduction is available in Litt - Local–global compatibility and applications to the arithmetic of modular curves.

Historically, the first proof dealing with the simplest case came out in Deligne’s letter to Piatetski-Shapiro (the completed argument appeared in Brylinski’s appendice). Deligne’s method was generalized to the full $\operatorname{GL}_2$ case by Carayol and later to arbitrary $\operatorname{GL}_n$ by Harris–Taylor in their book.

As far as I know, the state-of-the-art results start from Caraiani‘s thesis Local-global compatibility and the action of monodromy on nearby cycles (see also her second paper, which extends the results to the case $\ell=p$).

An elementary introduction is available in Litt - Local–global compatibility and applications to the arithmetic of modular curves.

Historically, the first proof dealing with the simplest case came out in Deligne’s letter to Piatetski-Shapiro (the completed argument appeared in Brylinski’s paper). Deligne’s method was generalized to the full $\operatorname{GL}_2$ case by Carayol and later to arbitrary $\operatorname{GL}_n$ by Harris–Taylor in their book.

As for as I know, the state-of-the-art results start from Caraiani‘s thesis Local-global compatibility and the action of monodromy on nearby cycles (see also her second paper, which extends the results to the case $\ell=p$).

An elementary introduction is available in Litt - Local–global compatibility and applications to the arithmetic of modular curves.

Historically, the first proof dealing with the simplest case came out in Deligne’s letter to Piatetski-Shapiro (the completed argument appeared in Brylinski’s appendice). Deligne’s method was generalized to the full $\operatorname{GL}_2$ case by Carayol and later to arbitrary $\operatorname{GL}_n$ by Harris–Taylor in their book.

As for as I know, the state-of-the-art results start from Caraiani‘s thesis Local-global compatibility and the action of monodromy on nearby cycles (see also her second paper, which extends the results to the case $\ell=p$).

An elementary introduction is available here.

Historically, the first proof dealing with the simplest case came out in Deligne’s letter to Piatetski-Shapiro(the completed argument appeared in Brylinski‘s paper). Deligne’s method was generalized to the full $\operatorname{GL}_2$ case by Carayol and later to arbitrary $\operatorname{GL}_n$ by Harris-Taylor in their book.

As for as I know, the state-of-the-art results start from Caraiani‘s thesis(see also her second paper, which extends the results to the case $\ell=p$).

An elementary introduction is available in Litt - Local–global compatibility and applications to the arithmetic of modular curves.

Historically, the first proof dealing with the simplest case came out in Deligne’s letter to Piatetski-Shapiro  (the completed argument appeared in Brylinski’s paper). Deligne’s method was generalized to the full $\operatorname{GL}_2$ case by Carayol and later to arbitrary $\operatorname{GL}_n$ by Harris–Taylor in their book.

As for as I know, the state-of-the-art results start from Caraiani‘s thesis Local-global compatibility and the action of monodromy on nearby cycles  (see also her second paper, which extends the results to the case $\ell=p$).