Bounds on Hecke eigenvalues












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Let $pi$ be an automorphic representation of a certain general linear group $GL(k)$. Write its L-function as
$$L(s,pi) = sum_{n>0} frac{lambda(n)}{n^s}$$



for $Re(s) gg 1$. What is the best known bound (without assuming Ramanujan-Petersson) for $lambda(n)$?






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    Let $pi$ be an automorphic representation of a certain general linear group $GL(k)$. Write its L-function as
    $$L(s,pi) = sum_{n>0} frac{lambda(n)}{n^s}$$



    for $Re(s) gg 1$. What is the best known bound (without assuming Ramanujan-Petersson) for $lambda(n)$?






    number-theory automorphic-forms







    share|cite|improve this question

























      1












      1








      1


      3





      Let $pi$ be an automorphic representation of a certain general linear group $GL(k)$. Write its L-function as
      $$L(s,pi) = sum_{n>0} frac{lambda(n)}{n^s}$$



      for $Re(s) gg 1$. What is the best known bound (without assuming Ramanujan-Petersson) for $lambda(n)$?






      number-theory automorphic-forms







      share|cite|improve this question













      Let $pi$ be an automorphic representation of a certain general linear group $GL(k)$. Write its L-function as
      $$L(s,pi) = sum_{n>0} frac{lambda(n)}{n^s}$$



      for $Re(s) gg 1$. What is the best known bound (without assuming Ramanujan-Petersson) for $lambda(n)$?






      number-theory automorphic-forms




      number-theory automorphic-forms






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 27 at 8:13









      TheStudent

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          Let $pi$ be a cuspidal unitary automorphic representation of $mathrm{GL}_n(mathbb{A}_F)$, where $mathbb{A}_F$ denotes the ring of adèles of an algebraic number field $F$. Then
          [L(s,pi) = sum_{substack{mathfrak{a} subset mathcal{O}_F \ mathfrak{a} neq {0}}} frac{lambda_{pi}(mathfrak{a})}{N_{F/mathbb{Q}}(mathfrak{a})^s}]
          converges absolutely for $Re(s) > 1$. For $mathfrak{a}$ a prime ideal $mathfrak{p}$, we have that
          [lambda_{pi}(mathfrak{p}) = alpha_{pi,1}(mathfrak{p}) + cdots + alpha_{pi,n}(mathfrak{p})]
          for some complex numbers $alpha_{pi,1}(mathfrak{p}), ldots, alpha_{pi,n}(mathfrak{p})$; these are known as the Satake parameters of $pi$ at $mathfrak{p}$. It is known that the product of the Satake parameters has absolute value $1$ (more precisely, the product is the central character) if $pi$ is unramified at $mathfrak{p}$, and otherwise it is smaller.



          For $n = 1$, this means that the Ramanujan hypothesis holds. For $n = 2$, the best bound is $|alpha_{pi,1}(mathfrak{p})|, |alpha_{pi,2}(mathfrak{p})| leq 7/64$ (for $F = mathbb{Q}$, this is due to Kim and Sarnak; for arbitrary $F$, this is due to Blomer and Brumley). Similarly, for $n = 3$, we have the bound $5/14$, and for $n = 4$, we have the bound $9/22$.



          In general, it is not hard to show the inequality $|alpha_{pi,1}(mathfrak{p})|, ldots, |alpha_{pi,n}(mathfrak{p})| < 1/2$ (I think this goes back to Jacquet and Shalika). The best that is known now is the bound $frac{1}{2} - frac{1}{n^2 + 1}$, due to Luo, Rudnick, and Sarnak.



          Some good references are these two papers.






          share|cite|improve this answer





















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            Let $pi$ be a cuspidal unitary automorphic representation of $mathrm{GL}_n(mathbb{A}_F)$, where $mathbb{A}_F$ denotes the ring of adèles of an algebraic number field $F$. Then
            [L(s,pi) = sum_{substack{mathfrak{a} subset mathcal{O}_F \ mathfrak{a} neq {0}}} frac{lambda_{pi}(mathfrak{a})}{N_{F/mathbb{Q}}(mathfrak{a})^s}]
            converges absolutely for $Re(s) > 1$. For $mathfrak{a}$ a prime ideal $mathfrak{p}$, we have that
            [lambda_{pi}(mathfrak{p}) = alpha_{pi,1}(mathfrak{p}) + cdots + alpha_{pi,n}(mathfrak{p})]
            for some complex numbers $alpha_{pi,1}(mathfrak{p}), ldots, alpha_{pi,n}(mathfrak{p})$; these are known as the Satake parameters of $pi$ at $mathfrak{p}$. It is known that the product of the Satake parameters has absolute value $1$ (more precisely, the product is the central character) if $pi$ is unramified at $mathfrak{p}$, and otherwise it is smaller.



            For $n = 1$, this means that the Ramanujan hypothesis holds. For $n = 2$, the best bound is $|alpha_{pi,1}(mathfrak{p})|, |alpha_{pi,2}(mathfrak{p})| leq 7/64$ (for $F = mathbb{Q}$, this is due to Kim and Sarnak; for arbitrary $F$, this is due to Blomer and Brumley). Similarly, for $n = 3$, we have the bound $5/14$, and for $n = 4$, we have the bound $9/22$.



            In general, it is not hard to show the inequality $|alpha_{pi,1}(mathfrak{p})|, ldots, |alpha_{pi,n}(mathfrak{p})| < 1/2$ (I think this goes back to Jacquet and Shalika). The best that is known now is the bound $frac{1}{2} - frac{1}{n^2 + 1}$, due to Luo, Rudnick, and Sarnak.



            Some good references are these two papers.






            share|cite|improve this answer


























              1














              Let $pi$ be a cuspidal unitary automorphic representation of $mathrm{GL}_n(mathbb{A}_F)$, where $mathbb{A}_F$ denotes the ring of adèles of an algebraic number field $F$. Then
              [L(s,pi) = sum_{substack{mathfrak{a} subset mathcal{O}_F \ mathfrak{a} neq {0}}} frac{lambda_{pi}(mathfrak{a})}{N_{F/mathbb{Q}}(mathfrak{a})^s}]
              converges absolutely for $Re(s) > 1$. For $mathfrak{a}$ a prime ideal $mathfrak{p}$, we have that
              [lambda_{pi}(mathfrak{p}) = alpha_{pi,1}(mathfrak{p}) + cdots + alpha_{pi,n}(mathfrak{p})]
              for some complex numbers $alpha_{pi,1}(mathfrak{p}), ldots, alpha_{pi,n}(mathfrak{p})$; these are known as the Satake parameters of $pi$ at $mathfrak{p}$. It is known that the product of the Satake parameters has absolute value $1$ (more precisely, the product is the central character) if $pi$ is unramified at $mathfrak{p}$, and otherwise it is smaller.



              For $n = 1$, this means that the Ramanujan hypothesis holds. For $n = 2$, the best bound is $|alpha_{pi,1}(mathfrak{p})|, |alpha_{pi,2}(mathfrak{p})| leq 7/64$ (for $F = mathbb{Q}$, this is due to Kim and Sarnak; for arbitrary $F$, this is due to Blomer and Brumley). Similarly, for $n = 3$, we have the bound $5/14$, and for $n = 4$, we have the bound $9/22$.



              In general, it is not hard to show the inequality $|alpha_{pi,1}(mathfrak{p})|, ldots, |alpha_{pi,n}(mathfrak{p})| < 1/2$ (I think this goes back to Jacquet and Shalika). The best that is known now is the bound $frac{1}{2} - frac{1}{n^2 + 1}$, due to Luo, Rudnick, and Sarnak.



              Some good references are these two papers.






              share|cite|improve this answer
























                1












                1








                1






                Let $pi$ be a cuspidal unitary automorphic representation of $mathrm{GL}_n(mathbb{A}_F)$, where $mathbb{A}_F$ denotes the ring of adèles of an algebraic number field $F$. Then
                [L(s,pi) = sum_{substack{mathfrak{a} subset mathcal{O}_F \ mathfrak{a} neq {0}}} frac{lambda_{pi}(mathfrak{a})}{N_{F/mathbb{Q}}(mathfrak{a})^s}]
                converges absolutely for $Re(s) > 1$. For $mathfrak{a}$ a prime ideal $mathfrak{p}$, we have that
                [lambda_{pi}(mathfrak{p}) = alpha_{pi,1}(mathfrak{p}) + cdots + alpha_{pi,n}(mathfrak{p})]
                for some complex numbers $alpha_{pi,1}(mathfrak{p}), ldots, alpha_{pi,n}(mathfrak{p})$; these are known as the Satake parameters of $pi$ at $mathfrak{p}$. It is known that the product of the Satake parameters has absolute value $1$ (more precisely, the product is the central character) if $pi$ is unramified at $mathfrak{p}$, and otherwise it is smaller.



                For $n = 1$, this means that the Ramanujan hypothesis holds. For $n = 2$, the best bound is $|alpha_{pi,1}(mathfrak{p})|, |alpha_{pi,2}(mathfrak{p})| leq 7/64$ (for $F = mathbb{Q}$, this is due to Kim and Sarnak; for arbitrary $F$, this is due to Blomer and Brumley). Similarly, for $n = 3$, we have the bound $5/14$, and for $n = 4$, we have the bound $9/22$.



                In general, it is not hard to show the inequality $|alpha_{pi,1}(mathfrak{p})|, ldots, |alpha_{pi,n}(mathfrak{p})| < 1/2$ (I think this goes back to Jacquet and Shalika). The best that is known now is the bound $frac{1}{2} - frac{1}{n^2 + 1}$, due to Luo, Rudnick, and Sarnak.



                Some good references are these two papers.






                share|cite|improve this answer












                Let $pi$ be a cuspidal unitary automorphic representation of $mathrm{GL}_n(mathbb{A}_F)$, where $mathbb{A}_F$ denotes the ring of adèles of an algebraic number field $F$. Then
                [L(s,pi) = sum_{substack{mathfrak{a} subset mathcal{O}_F \ mathfrak{a} neq {0}}} frac{lambda_{pi}(mathfrak{a})}{N_{F/mathbb{Q}}(mathfrak{a})^s}]
                converges absolutely for $Re(s) > 1$. For $mathfrak{a}$ a prime ideal $mathfrak{p}$, we have that
                [lambda_{pi}(mathfrak{p}) = alpha_{pi,1}(mathfrak{p}) + cdots + alpha_{pi,n}(mathfrak{p})]
                for some complex numbers $alpha_{pi,1}(mathfrak{p}), ldots, alpha_{pi,n}(mathfrak{p})$; these are known as the Satake parameters of $pi$ at $mathfrak{p}$. It is known that the product of the Satake parameters has absolute value $1$ (more precisely, the product is the central character) if $pi$ is unramified at $mathfrak{p}$, and otherwise it is smaller.



                For $n = 1$, this means that the Ramanujan hypothesis holds. For $n = 2$, the best bound is $|alpha_{pi,1}(mathfrak{p})|, |alpha_{pi,2}(mathfrak{p})| leq 7/64$ (for $F = mathbb{Q}$, this is due to Kim and Sarnak; for arbitrary $F$, this is due to Blomer and Brumley). Similarly, for $n = 3$, we have the bound $5/14$, and for $n = 4$, we have the bound $9/22$.



                In general, it is not hard to show the inequality $|alpha_{pi,1}(mathfrak{p})|, ldots, |alpha_{pi,n}(mathfrak{p})| < 1/2$ (I think this goes back to Jacquet and Shalika). The best that is known now is the bound $frac{1}{2} - frac{1}{n^2 + 1}$, due to Luo, Rudnick, and Sarnak.



                Some good references are these two papers.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 27 at 10:30









                Peter Humphries

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                2,38511022






























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