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January 7 |
Joshua Lam (Harvard University)
I will discuss the Attractor Conjecture for Calabi-Yau varieties, which was formulated by Moore in the nineties, highlighting the difference between Calabi-Yau varieties with and without Shimura moduli. In the Shimura case, I show that the conjecture holds and gives rise to an explicit parametrization of CM points on certain Shimura varieties; in the case without Shimura moduli, I’ll present counterexamples to the conjecture using unlikely intersection theory. Part of this is joint work with Arnav Tripathy. |
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January 14 |
Aranya Lahiri (Indiana University) (paper)
Locally analytic representations of $p$-adic analytic groups have played a crucial role in many areas of arithmetic and representation theory (including in $p$-adic local Langlands program) since their introduction by Schneider and Teitelbaum. In this talk we will briefly review some aspects of the theory of locally analytic representations. Then, for a locally analytic representation $V$ of $GL_2(F)$ we will construct a coefficient system attached to the Bruhat-Tits tree of $Gl_2(F)$. Finally we will use this coefficient system to construct a resolution for locally analytic principal series of $GL_2(F)$. |
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January 21 |
Kwun Angus Chung (University of Michigan) (paper)
Classically over the rational numbers, the exponential and logarithm series converge $p$-adically within some open disc of $\mathbb{C}_p$. For function fields, exponential and logarithm series arise naturally from Drinfeld modules, which are objects constructed by Drinfeld in his thesis to prove the Langlands conjecture for $\mathrm{GL}_2$ over function fields. For a "finite place" $v$ on such a curve, one can ask if the exp and log possess similar $v$-adic convergence properties. For the most basic case, namely that of the Carlitz module over $\mathbb{F}_q[T]$, this question has been long understood. In this talk, we will show the $v$-adic convergence for Drinfeld-(Hayes) modules on elliptic curves and a certain class of hyperelliptic curves. As an application, we are then able to obtain a formula for the $v$-adic $L$-value $L_v(1,\Psi)$ for characters in these cases, analogous to Leopoldt's formula in the number field case. |
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January 28 |
Ashwin Iyengar (King's College, London) (slides)
I will give a brief historical motivation for the Iwasawa main conjecture, and then I will talk about a construction of a $p$-adic $L$-function in families over the extended eigencurve, and how to formulate a two-variable Iwasawa main conjecture. If time permits, I will state some open questions about this family of functions. |
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February 4 |
Naomi Sweeting (Harvard University)
Given an elliptic curve E, Kolyvagin used CM points on modular curves to construct a system of classes valued in the Galois cohomology of the torsion points of E. Under the conjecture that not all of these classes vanish, he gave a description for the Selmer group of E. This talk will report on recent work proving new cases of Kolyvagin's conjecture. The methods follow in the footsteps of Wei Zhang, who used congruences between modular forms to prove Kolyvagin's conjecture under some technical hypotheses. We remove many of these hypotheses by considering congruences modulo higher powers of p. The talk will explain the difficulties associated with higher congruences of modular forms and how they can be overcome. I will also provide an introduction to the conjecture and its consequences, including a 'converse theorem': algebraic rank one implies analytic rank one. |
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February 11 |
Allechar Serrano Lopez (University of Utah)
A generalization of Mazur's theorem states that there are 26 possibilities for the torsion subgroup of an elliptic curve over a quadratic extension of $\mathbb{Q}$. If $G$ is one of these groups, we count the number of elliptic curves of bounded naive height whose torsion subgroup is isomorphic to $G$ in the case of imaginary quadratic fields. |
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February 18 |
Zuhair Mullath (University of Massachusetts, Amherst) (paper)
To a cuspidal automorphic representation of GSp(4) over $\mathbb Q$, one can associate a compatible system of Galois representations $\{\rho_p\}_{p \; \mathrm{prime}}$. For $p$ sufficiently large, the deformation theory of the mod-$p$ reduction $\overline \rho_p$ is expected to be unobstructed -- meaning the universal deformation ring is a power series ring. The global obstructions to deforming $\overline \rho_p$ is controlled by certain adjoint Bloch-Kato Selmer groups, which are expected to be trivial for $p$ large enough. I will talk about some recent results showing that there are no local obstructions to the deformation theory of $\overline \rho_p$ for almost all $p$. This is joint work with M. Broshi, C. Sorensen, and T. Weston. |
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February 25 |
Tim Trudgian (UNSW Canberra at ADFA)
Sadly, I won’t have time to prove the Riemann hypothesis in this talk. However, I do hope to outline recent work in a record partial-verification of RH. This is joint work with Dave Platt, in Bristol, UK. |
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March 4 |
Soumya Sankar (The Ohio State University)
The classical problem of counting elliptic curves with a rational N-isogeny can be phrased in terms of counting rational points on certain moduli stacks of elliptic curves. Counting points on stacks poses various challenges, and I will discuss these along with a few ways to overcome them. I will also talk about the theory of heights on stacks developed in recent work of Ellenberg, Satriano and Zureick-Brown and use it to count elliptic curves with an $N$-isogeny for certain $N$. The talk assumes no prior knowledge of stacks and is based on joint work with Brandon Boggess. |
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March 11 |
Organizational meeting (UCSD)
Organizational meeting to plan for next quarter. No talk. |