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Title: On the locally reducible part of the eigencurve
Author: Arsovski, Bodan
ISNI:       0000 0004 7659 0908
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2019
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This thesis studies crystalline Galois representations, which are certain constructions in p-adic Hodge theory whose most interesting property is that they arise as representations associated with modular forms. The main theorems proved in this thesis are partial results towards a conjecture about crystalline representations which dates back to computational observations by Buzzard and Gouvêa. Among the curiosities they noticed when computing the p-adic slopes of modular forms (i.e. the p-adic valuations of their pth coefficients) was that, for a certain class of so-called Γ_0(N)-regular primes when the level N is coprime to p, the p-adic slopes of eigenforms of level Γ_0(N) are always integers. Since, at least when p > 2, the reductions modulo p of the Galois representations associated with such modular forms are always reducible, this eventually pointed to a more general conjecture that the "locally reducible" eigenforms-i.e. those eigenforms of level coprime to p which have associated Galois representations whose reductions modulo p are reducible-always have integer slopes. In fact, while all representations associated with the aforementioned modular forms are crystalline, the class of all crystalline representations is larger and constructed entirely locally via p-adic Hodge theory (and therefore does not need any of the global structure coming from the world of modular forms). Thus the main conjecture tackled in this thesis is an entirely local statement saying that the slopes of locally reducible crystalline representations of even weight are always integers. The main result we prove in this thesis is that this conjecture is true when the slope is less than (p-1)/2. We additionally classify the reductions modulo p for a large class of crystalline representations.
Supervisor: Buzzard, Kevin ; Caraiani, Ana Sponsor: Imperial College
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral