SERRE’S CONJECTURE AND ITS GENERALISATIONS Sample Clauses

SERRE’S CONJECTURE AND ITS GENERALISATIONS. A geometric Serre weight conjecture. In [DS17], Xxxxxxx and Xxxxxx consider mod p Xxxxxxx modular forms for F as sections of automorphic line bundles on Xxxxxxx modular varieties of level prime to p in characteristic p. For a mod p Xxxxxxx modular eigenform of arbitrary weight, the authors then asso- ciate a two-dimensional representation of the absolute Galois group of F , extending existing work on attaching Galois representations to Xxxxxxx modular forms which have parity hypotheses on the weights. Xxxxxxx and Xxxxxx formulate a conjecture describing the set of weights of the mod p Xxxxxxx modular forms from which Galois representation arise in terms of crystalline liftability. In this view, they further formulate a conjecture establishing the existence of a minimal (geometric) weight. This notion of minimality is defined using a partial ordering, inspired by work in [AG05] and [DK17]. The work of Diamond and Xxxxxx allows for the study of geometric modularity with respect to Xxxxxxx modular forms with partial weight one, as briefly men- tioned above. We will study this in Chapter 4. Next to their results on geometric modularity, Diamond and Xxxxxx postulate many connections, some conjectural, some proven, between algebraic modularity, geometric modularity and crystalline liftability, which we will study in Chapter 6.
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