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September 23 |
Organizational meeting (UCSD)
This meeting will take place exclusively over Zoom. |
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September 30 |
NO MEETING |
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October 7 |
Kiran Kedlaya (UCSD) (slides)
We describe several recent results on orders of abelian varieties over $\mathbb{F}_2$: every positive integer occurs as the order of an ordinary abelian variety over $\mathbb{F}_2$ (joint with E. Howe); every positive integer occurs infinitely often as the order of a simple abelian variety over $\mathbb{F}_2$; the geometric decomposition of the simple abelian varieties over $\mathbb{F}_2$ can be described explicitly (joint with T. D'Nelly-Warady); and the relative class number one problem for function fields is reduced to a finite computation (work in progress). All of these results rely on the relationship between isogeny classes of abelian varieties over finite fields and Weil polynomials given by the work of Weil and Honda-Tate. With these results in hand, most of the work is to construct algebraic integers satisfying suitable archimedean constraints. |
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October 14 |
Jeff Lagarias (Michigan)
This talk is expository. It describes the history of an exciting connection made by physicists between an unsolved problem in combinatorial design theory- the existence of maximal sets of $d^2$ complex equiangular lines in ${\mathbb C}^d$- rephrased as a problem in quantum information theory, and topics in algebraic number theory involving class fields of real quadratic fields. Work of my former student Gene Kopp recently uncovered a surprising, deep (unproved!) connection with the Stark conjectures. For infinitely many dimensions $d$ he predicts the existence of maximal equiangular sets, constructible by a specific recipe starting from suitable Stark units, in the rank one case. Numerically computing special values at $s=0$ of suitable L-functions then permits recovering the units numerically to high precision, then reconstructing them exactly, then testing they satisfy suitable extra algebraic identities to yield a construction of the set of equiangular lines. It has been carried out for $d=5, 11, 17$ and $23$. |
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October 21 |
Thomas Grubb (UCSD)
Let $X$ be a smooth, geometrically irreducible scheme over a finite field of characteristic $p > 0$. With respect to rigid cohomology, $p$-adic coefficient objects on $X$ come in two types: convergent $F$-isocrystals and the subcategory of overconvergent $F$-isocrystals. Overconvergent isocrystals are related to $\ell$-adic etale objects ($\ell\neq p$) via companions theory, and as such it is desirable to understand when an isocrystal is overconvergent. We show (under a geometric tameness hypothesis) that a convergent $F$-isocrystal $E$ is overconvergent if and only if its restriction to all smooth curves on $X$ is. The technique reduces to an algebraic setting where we use skeleton sheaves and crystalline companions to compare $E$ to an isocrystal which is patently overconvergent. Joint with Kiran Kedlaya and James Upton. |
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October 28 |
Rahul Dalal (Johns Hopkins) (paper)
Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $GL_2$. Like holomorphic modular forms, they are defined by having their real component be one of a particularly nice class (in this case, called quaternionic discrete series). We count quaternionic automorphic representations on the exceptional group $G_2$ by developing a $G_2$ version of the classical Eichler-Selberg trace formula for holomorphic modular forms. There are two main technical difficulties. First, quaternionic discrete series come in L-packets with non-quaternionic members and standard invariant trace formula techniques cannot easily distinguish between discrete series with real component in the same L-packet. Using the more modern stable trace formula resolves this issue. Second, quaternionic discrete series do not satisfy a technical condition of being "regular", so the trace formula can a priori pick up unwanted contributions from automorphic representations with non-tempered components at infinity. Applying some computations of Mundy, this miraculously does not happen for our specific case of quaternionic representations on $G_2$. Finally, we are only studying level-1 forms, so we can apply some tricks of Chenevier and Taïbi to reduce the problem to counting representations on the compact form of $G_2$ and certain pairs of modular forms. This avoids involved computations on the geometric side of the trace formula. |
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November 4 |
Linus Hamann (Princeton) (paper)
Given a prime $p$, a finite extension $L/\mathbb{Q}_{p}$, a connected $p$-adic reductive group $G/L$, and a smooth irreducible representation $\pi$ of $G(L)$, Fargues-Scholze recently attached a semisimple Weil parameter to such $\pi$, giving a general candidate for the local Langlands correspondence. It is natural to ask whether this construction is compatible with known instances of the correspondence after semisimplification. For $G = GL_{n}$ and its inner forms, Fargues-Scholze and Hansen-Kaletha-Weinstein show that the correspondence is compatible with the correspondence of Harris-Taylor/Henniart. We verify a similar compatibility for $G = GSp_{4}$ and its unique non-split inner form $G = GU_{2}(D)$, where $D$ is the quaternion division algebra over $L$, assuming that $L/\mathbb{Q}_{p}$ is unramified and $p > 2$. In this case, the local Langlands correspondence has been constructed by Gan-Takeda and Gan-Tantono. Analogous to the case of $GL_{n}$ and its inner forms, this compatibility is proven by describing the Weil group action on the cohomology of a local Shimura variety associated to $GSp_{4}$, using basic uniformization of abelian type Shimura varieties due to Shen, combined with various global results of Kret-Shin and Sorensen on Galois representations in the cohomology of global Shimura varieties associated to inner forms of $GSp_{4}$ over a totally real field. After showing the parameters are the same, we apply some ideas from the geometry of the Fargues-Scholze construction explored recently by Hansen, to give a more precise description of the cohomology of this local Shimura variety, verifying a strong form of the Kottwitz conjecture in the process. |
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November 18 |
Gabriel Dorfsman-Hopkins (UC Berkeley) (paper)
Let $X$ be a perfectoid space with tilt $X^\flat$. We build a natural map $\theta:\Pic X^\flat\to\lim\Pic X$ where the (inverse) limit is taken over the $p$-power map, and show that $\theta$ is an isomorphism if $R = \Gamma(X,\sO_X)$ is a perfectoid ring. As a consequence we obtain a characterization of when the Picard groups of $X$ and $X^\flat$ agree in terms of the $p$-divisibility of $\Pic X$. The main technical ingredient is the vanishing of higher derived limits of the unit group $R^*$, whence the main result follows from the Grothendieck spectral sequence. |
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December 2 |
James Upton (UC San Diego) (paper)
Let $X$ be a smooth, affine, geometrically connected curve over a finite field of characteristic $p > 2$. Let $\rho:\pi_1(X) \to \mathbb{C}^\times$ be a character of finite order $p^n$. If $\rho\neq 1$, then the Artin $L$-function $L(\rho,s)$ is a polynomial, and a theorem of Kramer-Miller states that its $p$-adic Newton polygon $\mathrm{NP}(\rho)$ is bounded below by a certain Hodge polygon $\mathrm{HP}(\rho)$ which is defined in terms of local monodromy invariants. In this talk we discuss the interaction between the polygons $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$. Our main result states that if $X$ is ordinary, then $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$ share a vertex if and only if there is a corresponding vertex shared by certain "local" Newton and Hodge polygons associated to each ramified point of $\rho$. As an application, we give a local criterion that is necessary and sufficient for $\mathrm{NP}(\rho)$ and $\mathrm{HP}(\rho)$ to coincide. This is joint work with Joe Kramer-Miller. |