Timeline for Hodge conjecture as the equality of arithmetic and algebraic weights of motivic L-functions
Current License: CC BY-SA 4.0
6 events
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| Oct 27, 2023 at 7:15 | comment | added | David Loeffler | 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). | |
| Oct 26, 2023 at 23:53 | comment | added | KStar | 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. | |
| Oct 26, 2023 at 23:53 | comment | added | KStar | 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. | |
| Oct 26, 2023 at 23:52 | comment | added | KStar | 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. | |
| Oct 26, 2023 at 23:50 | comment | added | KStar | 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 $w_{alg}$. | |
| Oct 26, 2023 at 7:22 | history | answered | David Loeffler | CC BY-SA 4.0 |