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January 6 |
Organizational meeting (Zoom only) |
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January 13, online |
Tong Liu (Purdue)
In this talk, I will explain a theorem of Bhatt-Scholze: the equivalence between prismatic $F$-crystal and $\mathbb Z_p$-lattices inside crystalline representation, and how to extend this theorem to allow more general types of base ring like Tate algebra ${\mathbb Z}_p \langle t^{\pm 1}\rangle$. This is a joint work with Heng Du, Yong-Suk Moon and Koji Shimizu. This is a talk in integral $p$-adic Hodge theory. So in the pre-talk, I will explain the motivations and base ideas in integral $p$-adic Hodge theory. |
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January 20, online |
Claudius Heyer (Münster)
Recent advances in the theory of smooth mod $p$ representations of a $p$-adic reductive group $G$ involve more and more derived methods. It becomes increasingly clear that the proper framework to study smooth mod $p$ representations is the derived category $D(G)$. I will talk about smooth mod $p$ representations and highlight their shortcomings compared to, say, smooth complex representations of $G$. After explaining how the situation improves in the derived category, I will spend the remaining time on the left adjoint of the derived parabolic induction functor. |
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January 27, online |
Petar Bakic (Utah)
The theory of local theta correspondence is built up from two main ingredients: a reductive dual pair inside a symplectic group, and a Weil representation of its metaplectic cover. Exceptional correspondences arise similarly: dual pairs inside exceptional groups can be constructed using so-called Freudenthal Jordan algebras, while the minimal representation provides a suitable replacement for the Weil representation. The talk will begin by recalling these constructions. Focusing on a particular dual pair, we will explain how one obtains Howe duality for the correspondence in question. Finally, we will discuss applications of these results. The new work in this talk is joint with Gordan Savin. |
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February 3 |
Alex Smith (Stanford)
Take $E$ to be an elliptic curve over a number field whose four torsion obeys certain technical conditions. In this talk, we will outline a proof that 100% of the quadratic twists of $E$ have rank at most one. To do this, we will find the distribution of $2^k$-Selmer ranks in this family for every positive $k$. We will also show how are techniques may be applied to find the distribution of $2^k$-class groups of quadratic fields. The pre-talk will focus on the definition of Selmer groups. We will also give some context for the study of the arithmetic statistics of these groups. |
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February 10 |
Gabrielle De Micheli (UCSD)
The Tower variant of the Number Field Sieve (TNFS) is known to be asymptotically the most efficient algorithm to solve the discrete logarithm problem in finite fields of medium characteristics, when the extension degree is composite. A major obstacle to an efficient implementation of TNFS is the collection of algebraic relations, as it happens in dimensions greater than 2. This requires the construction of new sieving algorithms which remain efficient as the dimension grows. In this talk, I will present how we overcome this difficulty by considering a lattice enumeration algorithm which we adapt to this specific context. We also consider a new sieving area, a high-dimensional sphere, whereas previous sieving algorithms for the classical NFS considered an orthotope. Our new sieving technique leads to a much smaller running time, despite the larger dimension of the search space, and even when considering a larger target, as demonstrated by a record computation we performed in a 521-bit finite field GF(p^6). The target finite field is of the same form as finite fields used in recent zero-knowledge proofs in some blockchains. This is the first reported implementation of TNFS. In the pre-talk, I will briefly present the core ideas of the quadratic sieve algorithm and its evolution to the Number Field Sieve algorithm. |
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February 17 |
Aaron Pollack (UCSD)
I will report on joint work with Spencer Leslie where we define an analogue of the Cohen-Zagier Eisenstein series to the exceptional group $G_2$. Recall that the Cohen-Zagier Eisenstein series is a weight $3/2$ modular form whose Fourier coefficients see the class numbers of imaginary quadratic fields. We define a particular modular form of weight $1/2$ on $G_2$, and prove that its Fourier coefficients see (certain torsors for) the 2-torsion in the narrow class groups of totally real cubic fields. In particular: 1) we define a notion of modular forms of half-integral weight on certain exceptional groups, 2) we prove that these modular forms have a nice theory of Fourier coefficients, and 3) we partially compute the Fourier coefficients of a particular nice example on G_2. |
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February 24 |
Paulina Fust (Duisburg-Essen)
We prove that the continuous cohomology groups of a $p$-adic reductive group with coefficients in an admissible unitary $\mathbb{Q}_p$-Banach space representation $\Pi$ are finite-dimensional and compare them to certain Ext-groups. As an application of this result, we show that the continuous cohomology of $SL_2(\mathbb{Q}_p) $ with values in non-ordinary irreducible $\mathbb{Q}_p$-Banach space representations of $GL_2(\mathbb{Q}_p)$ vanishes. |
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March 3, 9:00AM, online |
Annie Carter (UCSD) (paper)
Introduced by Lehmer in 1933, the classical Mahler measure of a complex rational function $P$ in one or more variables is given by integrating $\log|P(x_1, \ldots, x_n)|$ over the unit torus. Lehmer asked whether the Mahler measures of integer polynomials, when nonzero, must be bounded away from zero, a question that remains open to this day. In this talk we generalize Mahler measure by associating it with a discrete dynamical system $f: \mathbb{C} \to \mathbb{C}$, replacing the unit torus by the $n$-fold Cartesian product of the Julia set of $f$ and integrating with respect to the equilibrium measure on the Julia set. We then characterize those two-variable integer polynomials with dynamical Mahler measure zero, conditional on a dynamical version of Lehmer's conjecture. |
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March 10, APM 7321 |
David Urbanik (Toronto) (slides) (paper)
Given a smooth projective family $f : X \to S$ defined over the ring of integers of a number field, the Shafarevich problem is to describe those fibres of f which have everywhere good reduction. This can be interpreted as asking for the dimension of the Zariski closure of the set of integral points of $S$, and is ultimately a difficult diophantine question about which little is known as soon as the dimension of $S$ is at least 2. Recently, Lawrence and Venkatesh gave a general strategy for addressing such problems which requires as input lower bounds on the monodromy of f over essentially arbitrary closed subvarieties of $S$. In this talk we review their ideas, and describe recent work which gives a fully effective method for computing these lower bounds. This gives a fully effective strategy for solving Shafarevich-type problems for essentially arbitrary families $f$. |