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I think that Eichler and Shimura's proof of the Ramanujan--Petersson conjecture for weight two modular forms provides an example. Recall that this conjecture is a purely analytic statement: namely that if $f$ is a weight two cuspform on some congruence subgroup of $SL_2(\mathbb Z)$, which is an eigenform for the Hecke operator $T_p$ ($p$ a prime not dividing the level of the congruence subgroup in question) with eigenvalue $\lambda_p$, then $| \lambda_p | \leq 2 p^{1/2}.$ Unfortunately, no purely analytic proof of this result is known. (Indeed, if one shifts one's focus from holomorphic modular forms to Maass forms, then the corresponding conjecture remains open.)

What Eichler and Shimura realized is that, somewhat miraculously, $\lambda_p$ admits an alternative characterization in terms of counting solutions to certain congruences modulo $p$, and that estimates there due to Hasse and Weil (generalizing earlier estimates of Gauss and others) can be applied to show the desired inequality.

This argument was pushed much further by Deligne, who handled the general case of weight $k$ modular forms (for which the analogous inequality is $| \lambda_p | \leq 2 p^{(k-1)/2}$), using etale cohomology of varieties in characteristic $p$ (which is something of a subtle and more technically refined analogue of the notion of a congruence mod $p$). (Ramanujan's original conjecture was for the unique cuspform of weight 12 and level 1.)

The idea that there are relationships (some known, others conjectural) between automorphic forms and algebraic geometry over finite fields and number fields has now become part of the received wisdom of algebraic number theorists, and lies at the heart of the Langlands program. (And, of course, at the heart of the proof of FLT.) Thus the striking idea of Eichler and Shimura has now become a basic tenet of a whole field of mathematics.

Note: Tim in his question, and in some comments, has said that he wants "first non-trivial instances" rather than difficult arguments that involve a whole range of ideas and techniques. In his comment to Terry Tao's answer regarding Perelman, he notes that long, difficult proofs might well include within them instances of such examples. Thus I am offering this example as perhaps a "first non-trivial instance" of the kind of insights that are involved in proving results like Sato--Tate, FLT, and so on.

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I think that Eichler and Shimura's proof of the Ramanujan--Petersson conjecture for weight two modular forms provides an example. Recall that this conjecture is a purely analytic statement: namely that if $f$ is a weight two cuspform on some congruence subgroup of $SL_2(\mathbb Z)$, which is an eigenform for the Hecke operator $T_p$ ($p$ a prime not dividing the level of the congruence subgroup in question) with eigenvalue $\lambda_p$, then $| \lambda_p | \leq p^{1/2}.$ Unfortunately, no purely analytic proof of this result is known. (Indeed, if one shifts one's focus from holomorphic modular forms to Maass forms, then the corresponding conjecture remains open.)

What Eichler and Shimura realized is that, somewhat miraculously, $\lambda_p$ admits an alternative characterization in terms of counting solutions to certain congruences modulo $p$, and that estimates there due to Hasse and Weil (generalizing earlier estimates of Gauss and others) can be applied to show the desired inequality.

This argument was pushed much further by Deligne, who handled the general case of weight $k$ modular forms (for which the analogous inequality is $| \lambda_p | \leq p^{(k-1)/2}$), using etale cohomology of varieties in characteristic $p$ (which is something of a subtle and more technically refined analogue of the notion of a congruence mod $p$). (Ramanujan's original conjecture was for the unique cuspform of weight 12 and level 1.)

The idea that there are relationships (some known, others conjectural) between automorphic forms and algebraic geometry over finite fields and number fields has now become part of the received wisdom of algebraic number theorists, and lies at the heart of the Langlands program. (And, of course, at the heart of the proof of FLT.) Thus the striking idea of Eichler and Shimura has now become a basic tenet of a whole field of mathematics.

Note: Tim in his question, and in some comments, has said that he wants "first non-trivial instances" rather than difficult arguments that involve a whole range of ideas and techniques. In his comment to Terry Tao's answer regarding Perelman, he notes that long, difficult proofs might well include within them instances of such examples. Thus I am offering this example as perhaps a "first non-trivial instance" of the kind of insights that are involved in proving results like Sato--Tate, FLT, and so on.

I think that Eichler and Shimura's proof of the Ramanujan--Petersson conjecture for weight two modular forms provides an example. Recall that this conjecture is a purely analytic statement: namely that if $f$ is a weight two cuspform on some congruence subgroup of $SL_2(\mathbb Z)$, which is an eigenform for the Hecke operator $T_p$ ($p$ a prime not dividing the level of the congruence subgroup in question) with eigenvalue $\lambda_p$, then $| \lambda_p | \leq p^{1/2}.$ Unfortunately, no purely analytic proof of this result is known. (Indeed, if one shifts one's focus from holomorphic modular forms to Maass forms, then the corresponding conjecture remains open.)
What Eichler and Shimura realized is that, somewhat miraculously, $\lambda_p$ admits an alternative characterization in terms of counting solutions to certain congruences modulo $p$, and that estimates there due to Hasse and Weil (generalizing earlier estimates of Gauss and others) can be applied to show the desired inequality.
This argument was pushed much further by Deligne, who handled the general case of weight $k$ modular forms (for which the analogous inequality is $| \lambda_p | \leq p^{(k-1)/2}$), using etale cohomology of varieties in characteristic $p$ (which is something of a subtle and more technically refined analogue of the notion of a congruence mod $p$). (Ramanujan's original conjecture was for the unique cuspform of weight 12 and level 1.)