## Return to Answer

2 know--> no; currently known--> known

One way to answer this question is as follows: Ramanujan's conjecture is a special case of a much more general conjecture that any cuspdial automorphic representation of $GL_n$ over a number field is tempered. This is a technical but fundamental notion, which in the special case of the automorphic representation of $GL_2$ attached to the $\Delta$ function, reduces to Ramanujan's original conjecture. In fact, many people working in the theory of automorphic representations refer to this very general conjecture simply as the Ramanujan conjecture.

When applied to other cuspforms on $GL_2$ (namely Maass forms) it includes Selberg's conjecture that on congruence quotients of the upper half-plane, the spectrum of the hyperbolic Laplacian is bounded below by $1/4$.

The appearance of hyperbolic geometry can be understood in the following way: the quotient of $SL_n(\mathbb R)$ by $SO(n)$ is a non-compact symmetric space, which in the particular case of $SL_2$ is the hyperbolic plane. So, while the particular appearance of hyperbolic geometry may be a bit of a red herring, the appearence of highly symmetric geometry is a reflection of the group representation theory that is underlying the picture.

As of the current moment, know no purely representation-theoretic approach to the (general form of) Ramanujan's conjecture is currently known. (Or rather, a proof strategy involving what is called symmetric power functoriality is known, but the requisite results on symmmetric power functoriality seem very much out of reach at the moment). The only cases that are proved at the moment are cases when one can relate the group theoretic picture of automorphic forms to algebraic geometry (first over $\mathbb C$, then over a number field, and then ultimately over finite fields, so that the Weil conjecture apply). This is how Deligne's proof proceeds. This connection between the geometry of symmetric spaces and arithmetic and geometry over finite fields is one of the profound points of investigation of modern number theory, but despite many positive results related to it, it remains essentially mysterious, even to experts.

1

One way to answer this question is as follows: Ramanujan's conjecture is a special case of a much more general conjecture that any cuspdial automorphic representation of $GL_n$ over a number field is tempered. This is a technical but fundamental notion, which in the special case of the automorphic representation of $GL_2$ attached to the $\Delta$ function, reduces to Ramanujan's original conjecture. In fact, many people working in the theory of automorphic representations refer to this very general conjecture simply as the Ramanujan conjecture.

When applied to other cuspforms on $GL_2$ (namely Maass forms) it includes Selberg's conjecture that on congruence quotients of the upper half-plane, the spectrum of the hyperbolic Laplacian is bounded below by $1/4$.

The appearance of hyperbolic geometry can be understood in the following way: the quotient of $SL_n(\mathbb R)$ by $SO(n)$ is a non-compact symmetric space, which in the particular case of $SL_2$ is the hyperbolic plane. So, while the particular appearance of hyperbolic geometry may be a bit of a red herring, the appearence of highly symmetric geometry is a reflection of the group representation theory that is underlying the picture.

As of the current moment, know purely representation-theoretic approach to the (general form of) Ramanujan's conjecture is currently known. (Or rather, a proof strategy involving what is called symmetric power functoriality is known, but the requisite results on symmmetric power functoriality seem very much out of reach at the moment). The only cases that are proved at the moment are cases when one can relate the group theoretic picture of automorphic forms to algebraic geometry (first over $\mathbb C$, then over a number field, and then ultimately over finite fields, so that the Weil conjecture apply). This is how Deligne's proof proceeds. This connection between the geometry of symmetric spaces and arithmetic and geometry over finite fields is one of the profound points of investigation of modern number theory, but despite many positive results related to it, it remains essentially mysterious, even to experts.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 18, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9415336847305298, "perplexity_flag": "head"}