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Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions:

Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with spectral parameters $\{\mu_j\}$ and $\{\nu_k\}$. We say that $L(s)$ is of algebraic type if either every $\mu_j$ and $\nu_k$ is in $\mathbb{Z}$, or every $\mu_j$ and $\nu_k$ is in $\frac{1}{2}+\mathbb{Z}$. The integer $w_{alg}=2\max\{0,\nu_1,\dots,\nu_{d_2}\}$ is called the algebraic weight of the $L$-function.

Definition 4.2. Suppose $L(s) = \sum a_n n^{-s}$ is an analytic $L$-function. We say that $L(s)$ is of arithmetic type if there exists $w_{ar}\in\mathbb{Z}$ and a number field $F$ such that $a_nn^{w_{ar}/2}\in\mathcal{O}_F$ for all $n$. The smallest such $F$ is called the field of coefficients, and the smallest such $w_{ar}$ is called the arithmetic weight of the $L$-function.

The statement I am curious to know more about is the following given at the beginning of page 13 of the linked paper:

Furthermore, we have the Hodge conjecture: $w_{alg} = w_{ar}$.

Firstly, is this the same Hodge conjecture as the usual millennium problem asking whether the dimension of the space of algebraic cycles equals the dimension of a cohomology group? If so, how does one explain this supposed connection between the usual Hodge conjecture and $L$-functions?

My rough understanding of the equivalence between this statement and the usual way the Hodge conjecture is stated is the following:

The unresolved Hodge conjecture is about algebraic varieties and any algebraic variety $X$ does indeed have an $L$-function. The values of the $L$-function at certain points along the real axis (related to algebraic weight) appear to capture information about the dimension of cohomology groups of $X$. In contrast, the coefficients of the Dirichlet series (related to arithmetic weight) appear to capture information about algebraic subvarieties of $X$. So to say that arithmetic and algebraic weights are equal might amount to saying that the dimension of the space of algebraic cycles equals the dimension of a cohomology group?

The relationship between $w_{alg}$ and cohomology appears potentially easier to explain: for any variety, the motive computing its $k$-th cohomology has weight $k$, and hence, per the claim of the authors of the linked paper in section 4.1, the attached $L$-function will also have weight $k$. However, as far as I am aware, this says nothing about the dimension of this cohomology space, let alone the space of Hodge cycles inside of it.

For context, I became aware of this equivalence from this great YouTube video, however, the relationship to the Hodge conjecture seems to also be unknown to the creator of the video, and thus I was curious to find out more as it would seem to suggest that all 3 unresolved millennium problems about pure mathematics (not considering P vs. NP) are in some way related to $L$-functions.

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I don't think this statement is equivalent to the Hodge conjecture, but it does follow from a certain natural generalisation of the Hodge conjecture. See this blog post for further discussion:

https://www.galoisrepresentations.com/2013/07/03/effective-motives/

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    $\begingroup$ Sorry, could you elaborate on how it would follow from a generalisation of the Hodge conjecture? Having read the blog post, and discussing it with a friend of mine, I'm still a bit uncertain. I don't see a full implication, especially for L-functions not derived from motives. Weight parity complicates matters, but assuming problem A has a positive solution (which it does under generalised Hodge), consider a motive of algebraic weight $w_{alg}$. $\endgroup$
    – KStar
    Commented Oct 26, 2023 at 23:50
  • $\begingroup$ Applying a suitable Tate twist, which changes both algebraic and arithmetic weights by the same amounts, we can assume $w_{alg}=0$ (ignoring technicalities in that we can only change $w_{alg}$ by even amounts due to the Tate motive having weight 2). So, it seems one obtains $w_{ar}\geq w_{alg}$ this way with a lot of asterisks. $\endgroup$
    – KStar
    Commented Oct 26, 2023 at 23:52
  • $\begingroup$ I am uncertain how exactly the quantities in the definition of algebraic weight relate to Hodge-Tate weight, but assuming $w_{alg}=0$ implies the Hodge-Tate weights are nonnegative, then problem A seems to imply the motive is effective and hence have nonnegative arithmetic weight. It seems we can obtain $w_{ar}\leq w_{alg}$ in a similar way. By twisting, we may assume $w_{ar}=0$, so the L-function has (algebraic) integer coefficients. $\endgroup$
    – KStar
    Commented Oct 26, 2023 at 23:53
  • $\begingroup$ This implies the Euler factors have integer coefficients, which are the characteristic polynomials of Frobenius, so by Problem C we get effectiveness and nonnegative algebraic weight. Something else I am wondering about is that the paper's definition of algebraic weight seems to always be nonnegative, but motives can have negative weight, so I am wondering if this discrepancy might stem from normalisation? If so, I am unsure why their claim about the Hodge conjecture following from $w_{alg}=w_{ar}$ is necessarily correct if you have to renormalise. $\endgroup$
    – KStar
    Commented Oct 26, 2023 at 23:53
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    $\begingroup$ MO comments are not the place for discussions. Moreover, your argument seems to have gone some way off-piste – "assuming $w_{\mathrm{alg}} = 0$ implies the HT weights are non-negative" is almost the opposite of the truth ($w_{\mathrm{alg}}$ is the average of the HT weights, so if $w_{\mathrm{alg}} = 0$, then the HT weights cannot all be non-negative unless they are all 0). $\endgroup$
    – David Loeffler
    Commented Oct 27, 2023 at 7:15

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