# Algebra & Number Theory Seminar

Archives can be found here

**Fall 2020**

**ANT Zoominar**

**Fridays @ 3pm California Time**

**9045829721**--- password: seminar

For further information please contact

**Schedule** *(click dates for title and abstract)*

10/9 | James Cameron | UCLA |

10/16 | John McHugh | UCSC |

10/23 | U of Georgia | |

10/30 | Soumya Sankar | MSRI |

11/6 | Andrew Kobin | UCSC |

11/13 | Bregje Pauwels | U of Sydney |

11/20 | Clover May | UCLA |

11/27 | BLACK FRIDAY | (holiday; no talk) |

12/4 |

**Clover May** (UCLA)

**title**

abstract

**Bregje Pauwels** (U of Sydney)

**title**

abstract

**Andrew Kobin** (UCSC)

**Witt vectors, lifting problems, and moduli spaces of curves**

The ring of Witt vectors is an essential tool for understanding relationships between the worlds of characteristic 0 and characteristic p algebra and geometry. In this talk, I will describe how Witt vectors provide an elegant solution to the problem of lifting field extensions from characteristic p to characteristic 0. An important application is to the lifting problem for covers of curves. This situation motivated Garuti to define a projective scheme which compactifies the ring (scheme) of Witt vectors "equivariantly" (with respect to Witt vector addition). After describing Garuti's construction and its application to covers of curves, I will introduce a new compactification in the category of stacks that I am currently using to describe moduli stacks of curves in characteristic p.

**Soumya Sankar **(MSRI)

**Ordinarity of curves in positive characteristic**

A curve in characteristic p is called ordinary if the p-torsion of its Jacobian is as large as possible. One may ask, what is the probability that a given curve is ordinary? I will talk about various perspectives on this question as well as some heuristics about what one might expect. I will answer this question for some families of curves, namely Artin-Schreier and superelliptic curves.

**Changho Han** (U of Georgia)

*Arithmetic inflection locus for pluricanonical series on hyperelliptic curves*

Over the algebraic closed fields, one can find the number of inflection points of a linear series (of rank r and degree d) on a curve of genus g via the degree of the Wronskian. Over the real numbers, one can use topology to take the "oriented count" on the number of inflection points upto sign. Here, we explore the geometric meaning of a natural analogue for arbitrary fields via \(A^1\)-homotopy theory, especially finding the meaning of local degrees at an inflectionary point. No prior knowledge of \(A^1\)-homotopy theory is assumed. This is a joint work with Ethan Cotterill and Ignacio Darago.

**John McHugh** (UCSC)

**On the image of the trivial source ring in the ring of virtual characters of a finite group**

Let \(G\) be a finite group and let \(\mathcal{O}\) be a complete discrete valuation ring whose residue field \(\mathcal{O}/J(\mathcal{O})\) is algebraically closed of positive characteristic p. The character of a finitely generated \(\mathcal{O}\)-free \(\mathcal{O}G\)-module can be considered as an element of \(R_{\mathbb{C}}(G)\), the virtual character ring of \(G\). In particular, "taking characters" gives a map from the ring \(T_{\mathcal{O}}(G)\) of trivial source \(\mathcal{O}G\)-modules to \(R_{\mathbb{C}}(G)\). By a theorem of A. Dress, the image of this map is a subring of the ring of \(p\)-rational characters. We give a "detection theorem'' that answers the question: when is a \(p\)-rational character of \(G\) contained in the image of \(T_{\mathcal{O}}(G)\)? It turns out that when \(p\) is odd every \(p\)-rational character is contained in the image. This is not the case when \(p=2\): the quaternion group \(Q_{8}\) is a counterexample. We show that, in the \(p=2\) case, any other counterexample "comes from" \(Q_{8}\). The theories of \({biset functors}\) (introduced by S. Bouc) and \({fibered biset functors}\) (developed by R. Boltje and O. Coşkun) make this precise.

**James Cameron** (UCLA)

**Computing homological residue fields**

Homological residue fields give a notion of residue fields in tensor triangulated geometry. They are useful, but rather abstractly defined. In many tt categories there are tt residue fields, which are easy to understand. In this talk I will relate homological residue fields to the tt residue fields that exist in examples and describe a characterization of the homological residue fields in examples from algebra, topology, and representation theory.