As of spring 2020, this site is dynamically generated from researchseminars.org, which see for other seminars worldwide (or for this seminar listed in your local timezone).
|
September 28 |
Organizational meeting (no Zoom) |
|
October 5 |
Aaron Pollack (UC San Diego)
Quaternionic modular forms (QMFs) on the split exceptional group G_2 are a special class of automorphic functions on this group, whose origin goes back to work of Gross-Wallach and Gan-Gross-Savin. While the group G_2 does not possess any holomorphic modular forms, the quaternionic modular forms seem to be able to be a good substitute. In particular, QMFs on G_2 possess a semi-classical Fourier expansion and Fourier coefficients, just like holomorphic modular forms on Shimura varieties. I will explain the proof that the cuspidal QMFs of even weight at least 6 admit an arithmetic structure: there is a basis of the space of all such cusp forms, for which every Fourier coefficient of every element of this basis lies in the cyclotomic extension of Q. |
|
October 19 |
Christian Klevdal (UC San Diego)
In this talk, we study a Tannakian category of admissible pairs, which arise naturally when one is comparing etale and de Rham cohomology of p-adic formal schemes. Admissible pairs are parameterized by local Shimura varieties and their non-minuscule generalizations, which admit period mappings to de Rham affine Grassmannians. After reviewing this theory, we will state a result characterizing the basic admissible pairs that admit CM in terms of transcendence of their periods. This result can be seen as a p-adic analogue of a theorem of Cohen and Shiga-Wolfhart characterizing CM abelian varieties in terms of transcendence of their periods. All work is joint with Sean Howe. |
|
November 2 |
Kiran Kedlaya (UC San Diego)
The Fargues-Fontaine curve associated to an algebraically closed nonarchimedean field of characteristic $p$ is a fundamental geometric object in $p$-adic Hodge theory. Via the tilting equivalence it is related to the Galois theory of finite extensions of Q_p; it also occurs in Fargues's program to geometrize the local Langlands correspondence for such fields. Recently, Peter Dillery and Alex Youcis have proposed using a related object, the "affine cone" over the aforementioned curve, to incorporate some recent insights of Kaletha into Fargues's program. I will summarize what we do and do not yet know, particularly about vector bundles on this and some related spaces (all joint work in progress with Dillery and Youcis). |
|
November 9, 12:40PM |
Robin Zhang (MIT)
The class number formula describes the behavior of the Dedekind zeta function at s = 0 and s = 1. The Stark and Gross conjectures extend the class number formula, describing the behavior of Artin L-functions and p-adic L-functions at s = 0 and s = 1 in terms of units. The Harris–Venkatesh conjecture describes the residue of Stark units modulo p, giving a modular analogue to the Stark and Gross conjectures while also serving as the first verifiable part of the broader conjectures of Venkatesh, Prasanna, and Galatius. In this talk, I will draw an introductory picture, formulate a unified conjecture combining Harris–Venkatesh and Stark for weight one modular forms, and describe the proof of this in the imaginary dihedral case. |
|
November 9 |
Shou-Wu Zhang (Princeton)
In this talk, we consider a conjecture by Gross and Kudla that relates the derivatives of triple-product L-functions for three modular forms and the height pairing of the Gross—Schoen cycles on Shimura curves. Then, we sketch a proof of a generalization of this conjecture for Hilbert modular forms in the spherical case. This is a report of work in progress with Xinyi Yuan and Wei Zhang, with help from Yifeng Liu. |
|
November 16, online |
Tony Feng (UC Berkeley)
The Breuil-Mezard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" that should govern congruences between mod p automorphic forms on a reductive group G. Most of the progress thus far has been concentrated on the case G = GL_2, which has several special features. I will talk about joint work with Bao Le Hung on a new approach to the Breuil-Mezard Conjecture, which applies for arbitrary groups (and in particular, in arbitrary rank). It is based on the intuition that the Breuil-Mezard conjecture is analogous to homological mirror symmetry. |
|
November 30, APM 7218 and online |
Finley McGlade (UC San Diego)
The classical Spezialschar is the subspace of the space of holomorphic modular forms on $\mathrm{Sp}_4(\mathbb{Z})$ whose Fourier coefficients satisfy a particular system of linear equations. An equivalent characterization of the Spezialschar can be obtained by combining work of Maass, Andrianov, and Zagier, whose work identifies the Spezialschar in terms of a theta-lift from $\widetilde{\mathrm{SL}_2}$. Inspired by work of Gan-Gross-Savin, Weissman and Pollack have developed a theory of modular forms on the split adjoint group of type D_4. In this setting we describe an analogue of the classical Spezialschar, in which Fourier coefficients are used to characterize those modular forms which arise as theta lifts from holomorphic forms on $\mathrm{Sp}_4(\mathbb{Z})$. |
|
December 7, APM 7218 |
Jon Aycock (UC San Diego)
The concept of p-adic families of automorphic forms has far reaching applications in number theory. In this talk, we will discuss one of the first examples of such a family, built from the Eisenstein series, before allowing this to inform a construction of a family on an exceptional group of type G_2. |