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A note on some Diophantine inequalities over adelic curves
Paolo Dolce
Without assuming the Northcott property we provide an upper bound on the number of "big solutions" of a special system of Diophantine inequalities over proper adelic curves. This s…
Intersection matrices for the minimal regular model of and applications to the Arakelov canonical sheaf
Paolo Dolce, Pietro Mercuri
Let be an integer coprime to such that and let be the genus of the modular curve . We compute the intersection matrices relative to s…
On the generalisation of Roth's theorem
Paolo Dolce, Francesco Zucconi
We present two possible generalisations of Roth's approximation theorem on proper adelic curves, assuming some technical conditions on the behavior of the logarithmic absolute valu…
Numerical equivalence of -divisors and Shioda-Tate formula for arithmetic varieties
Paolo Dolce, Roberto Gualdi
Let be an arithmetic variety over the ring of integers of a number field , with smooth generic fiber . We give a formula that relates the dimension of the first Arakelo…
Adelic geometry on arithmetic surfaces II: completed adeles and idelic Arakelov intersection theory
Weronika Czerniawska, Paolo Dolce
We work with completed adelic structures on an arithmetic surface and justify that the construction under consideration is compatible with Arakelov geometry. The ring of completed…