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I have read Connes' survey article http://www.alainconnes.org/docs/rhfinal.pdf and I am somewhat familiar with his classic paper on the trace formula: http://www.alainconnes.org/docs/selecta.ps

Very roughly speaking the idea is to describe a dictionary which translates the concepts and techniques used in the proof of the analogue of the Riemann Hypothesis for function fields. This translation uses various techniques from tropical geometry and topos theory. At first I was hopeful I might understand the key issues with this translation, since I have some experience with the theory of Grothendieck toposes (or topoi). Nevertheless I have been lost when it comes to understanding precisely what the remaining problems are. As already briefly discussed in this thread: Riemann hypothesis via absolute geometry in the proof of the Riemann hypothesis for a curve $X$ over $F_p$ the surface $X \times X$ plays an important role. According to a new preprint of Connes / Consani there is now a suitable analogy for the surface $X \times X$ which is given by a topos called the "scaling site", cf. https://arxiv.org/abs/1603.03191

I would like to know what are the issues that are left open to complete the analogy with the proof of RH in the case of a curve over $F_p$?

I have read Connes' survey article http://www.alainconnes.org/docs/rhfinal.pdf and I am somewhat familiar with his classic paper on the trace formula: http://www.alainconnes.org/docs/selecta.ps

Very roughly speaking the idea is to describe a dictionary which translates the concepts and techniques used in the proof of the analogue of the Riemann Hypothesis for function fields. This translation uses various techniques from tropical geometry and topos theory. At first I was hopeful I might understand the key issues with this translation, since I have some experience with the theory of Grothendieck toposes (or topoi). Nevertheless I have been lost when it comes to understanding precisely what the remaining problems are. As already briefly discussed in this thread: Riemann hypothesis via absolute geometry in the proof of the Riemann hypothesis for a curve $X$ over $F_p$ the surface $X \times X$ plays an important role. According to a new preprint of Connes / Consani there is now a suitable analogy for the surface $X \times X$ which is given by a topos called the "scaling site", cf. https://arxiv.org/abs/1603.03191

I would like to know what are the issues that are left open to complete the analogy with the proof of RH in the case of a curve over $F_p$?

I have read Connes' survey article www.alainconnes.org/docs/rhfinal.pdf and I am somewhat familiar with his classic paper on the trace formula: www.alainconnes.org/docs/selecta.ps

Very roughly speaking the idea is to describe a dictionary which translates the concepts and techniques used in the proof of the analogue of the Riemann Hypothesis for function fields. This translation uses various techniques from tropical geometry and topos theory. At first I was hopeful I might understand the key issues with this translation, since I have some experience with the theory of Grothendieck toposes (or topoi). Nevertheless I have been lost when it comes to understanding precisely what the remaining problems are. As already briefly discussed in this thread: Riemann hypothesis via absolute geometry in the proof of the Riemann hypothesis for a curve $X$ over $F_p$ the surface $X \times X$ plays an important role. According to a new preprint of Connes / Consani there is now a suitable analogy for the surface $X \times X$ which is given by a topos called the "scaling site", cf. https://arxiv.org/abs/1603.03191

I would like to know what are the issues that are left open to complete the analogy with the proof of RH in the case of a curve over $F_p$?

I have read Connes' survey article http://www.alainconnes.org/docs/rhfinal.pdf and I am somewhat familiar with his classic paper on the trace formula: http://www.alainconnes.org/docs/selecta.ps

Very roughly speaking the idea is to describe a dictionary which translates the concepts and techniques used in the proof of the analogue of the Riemann Hypothesis for function fields. This translation uses various techniques from tropical geometry and topos theory. At first I was hopeful I might understand the key issues with this translation, since I have some experience with the theory of Grothendieck toposes (or topoi). Nevertheless I have been lost when it comes to understanding precisely what the remaining problems are. As already briefly discussed in this thread: Riemann hypothesis via absolute geometry in the proof of the Riemann hypothesis for a curve $X$ over $F_p$ the surface $X \times X$ plays an important role. According to a new preprint of Connes / Consani there is now a suitable analogy for the surface $X \times X$ which is given by a topos called the "scaling site", cf. https://arxiv.org/abs/1603.03191

I would like to know what are the issues that are left open to complete the analogy with the proof of RH in the case of a curve over $F_p$?