The correct determinant exponent of the weight $k$-operator for defining Hecke operators/adelizing modular forms

Question

For $g \in \operatorname{SL}_2(\mathbb R)$, and $\mathbb H$ the upper half plane, and $k\geq 1$ an integer, the weight $k$-operator on functions $f: \mathbb H \rightarrow \mathbb C$ is defined by

$$f[g](z) = f(g.z) j(g,z)^{-k}$$

where $j(g,z) = (cz+d)^{-1}$, $g = \begin{pmatrix} a & b \\ c & d\end{pmatrix}$.

In order to define Hecke operators, or to adelize modular forms, or to identify modular forms as functions on $\operatorname{GL}_2(\mathbb R)^+$, it is necessary to extend this definition to $g \in \operatorname{GL}_2^+(\mathbb R)$. In A First Course in Modular Forms, in Chapter 5.1 Diamond and Shurman set

$$f[g](z) = f(g.z)j(g,z)^{-k} \det(g)^{k-1}$$

In Automorphic Forms and Representations, in Chapter 1.4 Bump sets

$$f[g](z) = f(g.z)j(g,z)^{-k} \det(g)^{k/2}$$

Which exponent of the determinant is better to use, and why? If we adelize a Hecke eigenform for $\operatorname{SL}_2(\mathbb Z)$ and look at the corresponding automorphic representation $\pi = \otimes_p \pi_p$, which normalization is better to define Hecke operators with, if we want the classical Hecke operator $T_p$ to coincide naturally with an action of the spherical Hecke algebra $\mathscr H(\operatorname{GL}_2(\mathbb Q), \operatorname{GL}_2(\mathbb Z_p))$ on the local component $\pi_p$?

Recall that to adelize a modular form $f$ of $\operatorname{SL}_2(\mathbb Z)$ of some given weight, we would first identify $f$ with a function $\phi$ on $\operatorname{GL}_2^+(\mathbb R)$ by setting

$$\phi(g) = f[g](i)$$

and then we would define an automorphic form $\varphi$ on $\operatorname{GL}_2(\mathbb Q) \backslash \operatorname{GL}_2(\mathbb A)$ by using the decomposition $\operatorname{GL}_2(\mathbb A) = \operatorname{GL}_2(\mathbb Q) \operatorname{GL}_2^+(\mathbb R)K$ for $K$ a suitable compact subgroup, writing $g = \alpha g_{\infty}k$, and setting $\varphi(g) = \phi(g_{\infty})$.

Answer 1

This is a question which has no "right" answer.

A posh interpretation of the choice of exponent is that a Hecke eigenform $f$ determines an equivalence class of irreducible representations $\Pi = \bigotimes'_v \Pi_v$ of $GL_2(\mathbb{A}_\mathbb{Q})$, differing by twists by powers of the character $g \mapsto \|\det(g) \|$, and the power of $\det$ that you put in the action of $GL_2^+(\mathbb{Q})$ determines which twist you get.

The normalisation that Diamond and Shurman use is the one that makes the eigenvalue of the double coset $\begin{pmatrix} p & 0 \\ 0 & 1 \end{pmatrix}$ on $\Pi_p$ correspond to the Fourier coefficient $a_p(f)$; while Bump's normalisation makes it correspond to $a_p(f) / p^{(k/2-1)}$.

From the perspective of the analytic theory of automorphic forms, Bump's choice is the "obviously right" one, since it makes $\Pi$ be unitary. Then you can find $\Pi$ as a subrepresentation of $L^2(GL_2(\mathbb{Q}) \backslash GL_2(\mathbb{A}))$ and the analytic theory works as it should. Since Bump's text emphasises the analytic theory of automorphic forms, this is the convention he chooses (and Paul Garrett's comment seems to be coming from the same viewpoint).

On the other hand, from the viewpoint of the algebraic theory (Galois representations, special values of L-functions, etc), the factor $k/2$ is extremely inconvenient, particularly when $k$ is odd. The $a_p$'s all lie in some common finite extension of $\mathbb{Q}$, but the eigenvalue of $\Pi_p$ has been multiplied by $p$ to a half-integer power; so the extension of $\mathbb{Q}$ generated by the eigenvalues is not finite, and correspondingly the critical values of the $L$-series are at half-integers rather than integers, meaning that the $L$-series of $\Pi$ cannot correspond to a motive. With Diamond and Shurman's normalisation, the finite part of $\Pi$ is definable over a number field, and its $L$-series is motivic. (This would work with any integer power of $\det$, but $\det^{k-1}$ is the minimal one which makes the Hecke eigenvalues algebraic integers.)

So the Diamond-Shurman normalisation is better for the algebraic theory, and the Bump normalisation for the analytic one.

There is a great quote on this (attributed to Deligne, but possibly apocryphal): "Langlands is very convinced he knows what the square root of $p$ is. I have never been so sure."