# Seminars this semester

Oct 3 | Tue | Beth Romano (Cambridge) | |

13:00 | J11 | On the arithmetic of simple singularities of type E | |

Abstract:Given a simply laced Dynkin diagram, one can use Vinberg theory of graded Lie algebras to construct a family of algebraic curves. In the case when the diagram is of type $E_7$ or $E_8$, Jack Thorne and I have used the relationship between these families of curves and their associated Vinberg representations to gain information about integral points on the curves. In my talk, I’ll focus on the role Lie theory plays in the construction of the curves and in our proofs. |
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Oct 10 | Tue | Daniel Loughran (Manchester) | |

13:00 | J11 | Determinants as sums of two squares | |

Abstract:A classical theorem due independently to Landau and Ramanujan gives an asymptotic formula for the number of integers which can be written as a sum of two squares. We prove an analogous result for the determinant of a matrix using the spectral theory of automorphic forms. This is a special case of a more general result on a problem of Serre concerning specialisations of Brauer group elements on semisimple algebraic groups. This is joint work with Sho Tanimoto and Ramin Takloo-Bighash. |
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Oct 17 | Tue | Lassina Dembele (King's College London) | |

13:00 | J11 | On the compatibility between base change and Hecke action | |

Abstract:Let $F/E$ be a Galois extension of totally real number fields. In this talk, we will discuss the action of $Gal(F/E)$ on Hecke orbits of automorphic forms on $GL_2$. This reveals some compatibility between base change and Hecke action, which has several implications for Langlands functoriality. |
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Oct 24 | Tue | Henri Johnston (Exeter) | |

13:00 | J11 | The p-adic Stark conjecture at s=1 and applications | |

Abstract:Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The `p-adic Stark conjecture at s=1' relates the leading terms at s=1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to E/F. When E=F this is equivalent to Leopoldt’s conjecture for E at p and the ‘p-adic class number formula’ of Colmez. In this talk we discuss the p-adic Stark conjecture at s=1 and applications to certain cases of the equivariant Tamagawa number conjecture (ETNC). This is joint work with Andreas Nickel. |
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Nov 21 | Tue | Fredrik Stromberg (Nottingham) | |

13:00 | J11 | Spectral theory and Maass forms for noncongruence subgroups | |

Abstract:The spectral theory for congruence subgroups of the modular group is fairly well understood since Selberg and the development of the Selberg trace formula. In particular it is known that congruence subgroups has an infinite number of discrete eigenvalues (corresponding to Maass cusp forms) and there is extensive support towards Selberg’s conjecture that there are no small eigenvalues for congruence subgroups. In contrast to this setting, much less is known for noncongruence subgroups of the modular group even though these groups are clearly arithmetic. In fact, it can be shown that under certain circumstances small eigenvalues must exist. And even the existence of infinitely many “new” discrete eigenvalues is not known for these groups. The main obstacle for developing the spectral theory here is that there is in general no explicit formula for the scattering determinant. In this talk I will present sufficient conditions for an “odd” discrete spectrum to exist and I will also give experimental support for the conjecture that these conditions are also necessary. I will also present an experimental version of Turing’s method for certifying correctness of the spectral counting. |
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Nov 28 | Tue | Carl Wang-Erickson (Imperial) | |

13:00 | J11 | The rank of Mazur's Eisenstein ideal | |

Abstract:In his landmark 1976 paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and an Eisenstein series of weight 2 and prime level N. He proved a great deal about these congruences, and also posed some questions: how big is the space of cusp forms that are congruent to the Eisenstein series? How big is the extension generated by their coefficients? In joint work with Preston Wake, we give an answer to these questions in terms of cup products (and Massey products) in Galois cohomology. We will introduce these products and explain what algebraic number-theoretic information they encode. Time permitting, we may be able to indicate some partial generalisations to square-free level. |
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Dec 5 | Tue | Ariel Weiss (Sheffield) | |

13:00 | J11 | TBA | |

Dec 12 | Tue | David Spencer (Sheffield) | |

13:00 | J11 | TBA | |