Seminars this semester
Oct 8  Fri  Bodan Arsovski (Sheffield)  Number Theory seminar  
13:00  Meet  Limiting measures of supersingularities  
Abstract: In 2001, Gouvêa and Buzzard noticed several interesting phenomena in the distributions of the padic slopes of weight k, level Γ_0(Np) eigenforms, the two main ones being that the slopes, in almost all cases, are integers; and that the slopes, in almost all cases, are no larger than (k1)/(p+1), a much smaller bound than expected. In trying to explain the second phenomenon, Gouvêa made a conjecture that, after normalizing the slopes by dividing by k1 so that they lie in the interval [0,1], their distributions tend to the uniform measure on [0,1/(p+1)]∪[p/(p+1),1] as k tends to infinity. In particular, Gouvêa's conjecture implies that the slopes are concentrated away from the middle interval (1/(p+1),p/(p+1)). This conjecture can be seen as the padic version of an interesting twist on the Sato–Tate conjecture: while the Sato–Tate conjecture asks about the distribution of the (real) slopes of a fixed modular form for varying primes, here one is interested in the distributions of the (padic) slopes of varying modular forms for a fixed prime. In this talk we discuss a proof that the slopes are indeed concentrated away from the middle interval (1/(p+1),p/(p+1)) when p>3 is Γ_0(N)regular, which uses the padic local Langlands correspondence. 



Oct 15  Fri  Haluk Sengun (Sheffield)  Number Theory seminar  
13:00  Meet  Periods of mod p Bianchi modular forms and Selmer groups  
Abstract: The relationship between special values of Lfunctions of modular forms and Selmer group of modular padic Galois representations is a major theme in number theory. Given the developing mod p Langlands program, it is natural to ask whether there is some kind of mod p analogue of the above theme. Notice that mod p modular forms do not have associated Lfunctions! In this talk, I will report on ongoing work with Lewis Combes in which we formulate, and computationally test, a connection between Selmer groups of mod p Galois representations and mod p Bianchi modular forms. This is inspired by a speculation of Calegari and Venkatesh. 



Nov 12  Fri  Hiraku Atobe (Hokkaido University)  Number Theory seminar  
13:00  Meet  Local newforms for GL(n)  
Abstract: In 1981, JacquetPiatetskiiShapiroShalika established the theory of local newforms for irreducible generic representations of general linear groups over padic fields. In this talk, we extend their results to all irreducible representations. To do this, we introduce a new family of compact open subgroups indexed by certain tuples of nonnegative integers. For the proof, we define local RankinSelberg integrals for Speh representations. This is a joint work with Satoshi Kondo and Seidai Yasuda. 



Nov 19  Fri  Vaidehee Thatte (King's College London)  Number Theory seminar  
13:00  Meet  Arbitrary Valuation Rings and Wild Ramification  
Abstract: Classical ramification theory deals with complete discrete valuation fields k((X)) with perfect residue fields k. Invariants such as the Swan conductor capture important information about extensions of these fields. Many fascinating complications arise when we allow nondiscrete valuations and imperfect residue fields k. Particularly in positive residue characteristic, we encounter the mysterious phenomenon of the defect (or ramification deficiency). The occurrence of a nontrivial defect is one of the main obstacles to longstanding problems, such as obtaining resolution of singularities in positive characteristic. Degree p extensions of valuation fields are building blocks of the general case. In this talk, we will present a generalization of ramification invariants for such extensions and discuss how this leads to a better understanding of the defect. If time permits, we will briefly discuss their connection with some recent work (joint with K. Kato) on upper ramification groups. 



Nov 30  Tue  Narasimha Kumar (Indian Institute of Technology Hyderabad)  Number Theory seminar  
10:00  Meet  


Dec 3  Fri  Hanneke Wiersema (Cambridge)  Number Theory seminar  
13:00  Meet  


Dec 10  Fri  Sadiah Zahoor (Sheffield)  Number Theory seminar  
13:00  Meet  

