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Seminars and Colloquia

Mathematics

Arithmetic jet spaces of abelian schemes

Tue, May 15, 2018, 03:00 PM to 04:00 PM at Seminar Room 41, 3rd Floor, Main Building

Dr. Arnab Saha
ISI, Bangalore

Arithmetic jet spaces are constructed in analogy with jet spaces in differential algebra. In differential algebra, given a variety $\"X\"$ defined over a ring $\"R\"$ with a derivation $\"\\\\partial\"$ on $\"R\"$ , the jet space associated to $\"X\"$ should be thought of as the natural space that has a derivation on it lifting the derivation $\"\\\\partial\"$ on $\"R\"$ . This has led to various applications in differential geometry as well as in diophantine geometry over the function rings. In the case of arithmetic, one replaces the derivation with the notion of the lift of Frobenius to construct the analogue called the arithmetic jet spaces. This leads to various applications in arithmetic. After introducing the main concepts, we will talk about the construction of a canonical filtered $\"F\"$ -isocrystal $\"\\\\mathbb{H}\"$ associated to an abelian scheme, admitting a map to the the Hodge sequence. It also leads to new $\"p\"$ -adic differential modular forms. This is joint work with Jim Borger.