The notions of $H^0(\widehat{ D})$ and $h^0(\widehat{D})$ are not satisfactory

Question

Let $K$ be a number field with ring of integers $O_K$. Moreover consider an Arakelov divisor $\widehat{D}\in\overline{\operatorname{Div }(\operatorname {Spec }O_K)}$, namely $$D=\sum_{\mathfrak p\;\text{prime}}n_\mathfrak p\mathfrak p+\sum_\sigma\lambda_\sigma \sigma$$ where $\sigma$ is an archimedian place i.e. a complex embedding of $K$ up to conjugation. Clearly we have the proper notion of principal Arakelov divisor $\widehat{(f)}$ for any $f\in K^\ast$. The usual (and most natural) definition of $H^0(\widehat{D})$ is the following: $$H^0(\widehat{D})=\{f\in K^\ast:v_\mathfrak p(f)\ge -n_\mathfrak p \;\forall \mathfrak p,\; v_\sigma(f)\ge-\lambda_\sigma\; \forall \sigma\}$$ where $v_\sigma$ is the adequate normalized real valuation at archimedian places. The above definition of $H^0(\widehat{D})$ in my opinion emulates perfectly the notion of $H^0(X,\mathcal O_X(D))$ in the usual geometric setting. One can show that $H^0(\widehat{D})$ is a finite set and its "dimension" is defined simply as (a multiple of): $$h^0(\widehat{D})=\log(\#H^0(\widehat{D}))$$

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Now in this paper (van der Geer, Schoof- Effectivity of Arakelov Divisors and the Theta Divisor of a Number Field), the authors claim that

the above notions of $H^0(\widehat{D})$ and $h^0(\widehat{D})$ are not satisfactory,

and they suggest new definitions which lead to a "Riemann-Roch theorem". The point is that they don't explain why the usual objects $H^0(\widehat{D})$ and $h^0(\widehat{D})$ don't work, so I'd like to understand the motivation of their strong claim.

On the contrary it seems to me that the above objects are very useful to get the usual Riemann-Roch theorem for number fields which is due to Lang.

Answer 1

The Arakelov-theoretical analogue of Riemann's inequality is Minkowski's theorem. As Felipe Voloch's answer indicates, Tate's approach makes the Riemann-Roch theorem a consequence of the Poisson summation formula, and the self duality of the characteristic functions of (ultrametric) balls is crucial. Over archimedean fields, the characteristic functions of balls are no longer self-dual, but the Gaussian kernel is. It is thus natural, in the Arakelov-theory framework, to consider a mollified version of the $H^0$, as has been done by van der Geer and Schoof and, apparently independently, by Damian Rössler; see also Groenwegen. This has also various advantages:

• A better Minkowski-type lower bound;
• A clean version of Serre duality;
• Subadditivity properties in exact sequences.

This is amply explained in recent preprint of J-B. Bost, Theta invariants of euclidean lattices and infinite-dimensional hermitian vector bundles over arithmetic curves, https://arxiv.org/abs/1512.08946, where he uses them to study infinite dimensional analogues (that will crucially appear in forthcoming work).

Answer 2

This issue already comes up in Tate's proof of the functional equation for zeta functions. The functional equation should come out of some version of Riemann-Roch and, for function fields, it does. (What follows is very anachronistic!) Tate discovered that, if instead of the characteristic function of $H^0(\hat{D})$ one should use a fast decaying smooth function on $K$ (or ${\cal{O}}_K$) concentrated mostly on $H^0(\hat{D})$ (a fuzzy version of $H^0(\hat{D})$, if you will) and the relevant version of Riemann-Roch is a consequence of the Poisson summation formula. The only change occurs in the archimedian places because characteristic functions of balls are continuous in the non-archimedian case, but not in the archimedian case and using a smooth function resolves the analytic issues.

Of course, in some ways it is a matter of taste and also what your aims are. I also think the Riemann-Roch with $H^0(\hat{D})$ is neat but it hasn't had much usefulness.