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Coming from a number-theoretic background, I certainly care about $L$-functions and in particular automorphic ones. For automorphic forms on $SL_2(\mathbb{Z}) \backslash SL_2(\mathbb{R})$, $L$-function can be interpreted as the Mellin transform of Fourier expansion at the cusp.

If we look at automorphic forms on $SL_(n,\mathbb{Z}) \backslash SL(n,\mathbb{R})$, we still have a Fourier expansion in $n-1$ variables since it is periodic with respect to the super-diagonal unipotent group. This gives us Fourier coefficients $A(m_1,\cdots,m_{n-1})$. In forming the $L$-functions for $SL(n,\mathbb{R})$, we just look at all the Fourier coefficients $A(m,1,\cdots,1)$ and define $$L(s) = \sum_{m=1}^{\infty} A(m,1,\cdots,1)m^{-s}$$ But the most natural thing to do (comparing to the $SL_2(\mathbb{R})$ case) would be to form the multiple Dirichlet series instead $$\sum_{m_1,\cdots,m_{n-1}} \frac{A(m_1,\cdots,m_{n-1})}{m_1^{s_1} \cdots m_{n-1}^{s_{n-1}}}$$ It certainly seems that people care "less" about this multiple Dirichlet series: in the whole Langlands business we always take about L-parameters and stuff, which seems to imply that we track only the information of $L$-function but not the whole multiple Dirichlet series.

So here are my questions,

  • Is there any conceptual reason why we care more about $L$-functions rather than multiple Dirichlet series?
  • Of course, Fourier expansion is available when there is a cusp. For cocompact arithmetic quotients of $GL_n(\mathbb{R})$ (or other reductive groups in general), can one similarly define a multiple Dirichlet series that should incorporate the data of $L$-function?

Thank you.

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up vote 10 down vote accepted

It may help clarify things to work out a specific example, although the OP may know this. In case $n=3$, the double Dirichlet series evaluates as $$ \sum_{m,n=1}^{\infty} \frac{A_F(m,n)}{m^{w} n^s} = \frac{L(\overline{F},w) L(F, s)}{\zeta(s+w)}.$$ Here $\overline{F}$ is the contragredient of $F$. This is known as Bump's double Dirichlet series, and this is worked out in Section 6.6 of Goldfeld's book, Automorphic forms and $L$-functions for the group $GL(n,\mathbb{R})$. There are references there for generalizations.

It seems clear (to me) that the $GL_3$ automorphic $L$-function $L(F,s)$ (and its contragredient, and $\zeta$) are the fundamental objects, but also that there are many interesting Dirichlet series that one can construct from an automorphic form.

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2  
Thanks! I will get hold of the book later today to check the references. Meanwhile, is there such a product formula for $GL(n)$-forms? If that's the case I would agree that these multiple Dirichlet series contain no extra information comparing to $L$-functions and that $L$-functions are more fundamental. – user31415 Feb 2 at 18:18
    
I wasn't able to find an online copy of the paper referred to in Section 6.6 (by Bump and Friedberg), because it is in a conference proceedings. I don't think that it calculates the multiple Dirichlet series you wanted. I'm not aware of any product formula for it, but perhaps nobody has tried to evaluate it. – Matt Young Feb 3 at 2:22
    
Thanks for your answer! By looking at how the identity in $GL(3)$ case is proved I'm pretty convinced that the same method would work in the general case, up to unraveling what the series $\sum_{m=1}^{\infty} \frac{A_F (1,\cdots, 1, m, 1 \cdots, 1)}{m^s}$ means - I'll think about this in more details later. Also, Bump-Friedberg's paper seems to be of a different nature. They were able to write $L(s_1, \pi)L(s_2, \pi, \Lambda^2)$ as some Rankin-Selberg convolution - so they were generalizing the right hand side of the above formula. It does not look like their resulting integral is related.. – user31415 Feb 4 at 2:07
    
to what I am asking. – user31415 Feb 4 at 2:07

To address the conceptual question, the $L$-function essentially characterizes the automorphic representation and can be studied locally (associating local $L$-functions to local components of the global representation), so it seems to me there is little need (a priori) for using a more complicated Dirichlet series to study these representations.

On the other hand, multiple Dirichlet series typically do not have Euler products, so do not admit a local-global study, at least in a naive way (though see Bump's survey article). Of course, as Matt indicates, they are useful for studying $L$-functions.

Moreover, from an arithmetic point of view, $L$-functions are naturally related to varieties such as elliptic curves. As far as I know (though I am not an expert on multiple Dirichlet series), there is no direct connection between MDS and counting points on varieties.

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