kcl.ac.uk to request the password. Meetings take place online, Wednesdays (except the week of June 10th, talk rescheduled to June 12th), 1600–1700 London time (exceptionally, 1400–1500 on June 17th). Meeting link: https://zoom.us/j/91638038954.
Password for the meeting will be sent out to the London number theory mailing lists (sign up is here: http://wwwf.imperial.ac.uk/~buzzard/LNTS/lnts.html), if you don't want to sign up or you mislaid the message, email j.newtonkcl.ac.uk to request the password.
There will also be a study group on the weight part of Serre's conjecture 1400–1530, more details here.
22/4/20 Tiago Jardim Da Fonseca (Oxford)
Title: On Fourier coefficients of Poincaré series
Abstract: Poincaré series are among the first examples of holomorphic and weakly holomorphic modular forms. They are useful in many analytical questions, but their Fourier coefficients seem hard to grasp algebraically. In this talk, I will discuss the arithmetic nature of Fourier coefficients of Poincaré series by characterizing them as cohomological invariants (periods).
29/4/20 Ila Varma (Toronto)
Title: Malle's Conjecture for octic D4-fields.
Abstract: We consider the family of normal octic fields with Galois group D4, ordered by their discriminants. In forthcoming joint work with Arul Shankar, we verify the strong form of Malle's conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove this and related results.
6/5/20 Chris Lazda (Warwick)
Title: A Néron–Ogg–Shafarevich criterion for K3 surfaces
Abstract: The naive analogue of the Néron–Ogg–Shafarevich criterion fails for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified etale cohomology groups, but which do not admit good reduction over K. Assuming potential semi-stable reduction, I will show how to correct this by proving that a K3 surface has good reduction if and only if its second cohomology is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain “canonical reduction” of X. This is joint work with B. Chiarellotto and C. Liedtke.
13/5/20 Chantal David (Concordia)
Title: Non-vanishing cubic Dirichlet $L$-functions at $s = 1/2$
Abstract: Click here.
20/5/20 Rong Zhou (Imperial)
Title: Independence of $l$ for Frobenius conjugacy classes attached to abelian varieties.
Abstract: Let $A$ be an abelian variety over a number field $E\subset \mathbb{C}$ and let $v$ be a place of good reduction lying over a prime $p$. For a prime $l\neq p$, a theorem of Deligne implies that upon making a finite extension of $E$, the Galois representation on the $l$-adic Tate module factors as $\rho_l:\Gamma_E\rightarrow G_A(\mathbb{Q}_l)$, where $G_A$ is the Mumford-Tate group of $A$. We prove that the conjugacy class of $\rho_l(Frob_v)$ is defined over $\mathbb{Q}$ and independent of $l$. This is joint work with Mark Kisin.
27/5/20 Matteo Tamiozzo (Imperial)
Title: Bloch–Kato special value formulas for Hilbert modular forms
Abstract: The Bloch–Kato conjectures predict a relation between arithmetic invariants of a motive and special values of the associated $L$-function.
We will outline a proof of (the $p$-part of) one inequality in the relevant special value formula for Hilbert modular forms of parallel weight two, in analytic rank at most one.
03/6/20 Yunqing Tang (Paris-Saclay)
Title: Picard ranks of reductions of K3 surfaces over global fields
Abstract: For a K3 surface $X$ over a number field with potentially good reduction everywhere, we prove that there are infinitely many primes modulo which the reduction of $X$ has larger geometric Picard rank than that of the generic fiber $X$. A similar statement still holds true for ordinary K3 surfaces with potentially good reduction everywhere over global function fields. In this talk, I will present the proofs via the (arithmetic) intersection theory on good integral models (and its special fibers) of $\mathrm{GSpin}$ Shimura varieties. These results are generalizations of the work of Charles on exceptional isogenies between reductions of a pair of elliptic curves. This talk is based on joint work with Ananth Shankar, Arul Shankar, and Salim Tayou and with Davesh Maulik and Ananth Shankar.
12/6/20 Vesselin Dimitrov (Toronto) please note the change of date
Title: p-adic Eisenstein series, arithmetic holonomicity criteria, and irrationality of the 2-adic $\zeta(5)$
Slides: here.
Abstract: In this exposition of a joint work in progress with Frank Calegari and Yunqing Tang, I will explain a new arithmetic criterion for a formal function to be holonomic, and how it revives an approach to the arithmetic nature of special values of L-functions. The new consequence to be proved in this talk is the irrationality of the 2-adic version of $\zeta(5)$ (of Kubota-Leopoldt). But I will also draw a parallel to a work of Zudilin, and try to leave some additional open ends where the holonomicity theorem could be useful. The ingredients of the irrationality proof are Calegari's p-adic counterpart of the Apery-Beukers method, which is based on the theory of overconvergent p-adic modular forms (IMRN, 2005) taking its key input from Buzzard's theorem on p-adic analytic continuation (JAMS, 2002), and a Diophantine approximation method of Andre enhanced to a power of the modular curve $X_0(2)$. The overall argument, as we shall discuss, turns out to bear a surprising affinity to a recent solution of the Schinzel-Zassenhaus conjecture on the orbits of Galois around the unit circle.
17/6/20 Yifeng Liu (Yale) 2pm this week
Title: Beilinson-Bloch conjecture and arithmetic inner product formula
Abstract: In this talk, we study the Chow group of the motive associated to a tempered global $L$-packet $\pi$ of unitary groups of even rank with respect to a CM extension, whose global root number is $-1$. We show that, under some restrictions on the ramification of $\pi$, if the central derivative $L'(1/2,\pi)$ is nonvanishing, then the $\pi$-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson--Bloch conjecture for Chow groups and L-functions (which generalizes the B-SD conjecture). Moreover, assuming the modularity of Kudla's generating functions of special cycles, we explicitly construct elements in a certain $\pi$-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative $L'(1/2,\pi)$ and local doubling zeta integrals. This confirms the conjectural arithmetic inner product formula proposed by me a decade ago. This is a joint work with Chao Li.
08/7/20 Jared Weinstein (Boston University)
Title: Partial Frobenius structures, Tate’s conjecture, and BSD over function fields.
Slides: here.
Abstract: Tate’s conjecture predicts that Galois-invariant classes in the $l$-adic cohomology of a variety are explained by algebraic cycles. It is known to imply the conjecture of Birch and Swinnerton-Dyer (BSD) for elliptic curves over function fields. When the variety, now assumed to be in characteristic p, admits a “partial Frobenius structure”, there is a natural extension of Tate’s conjecture. Assuming this conjecture, we get not only BSD, but the following result: the top exterior power of the Mordell-Weil group of an elliptic curve is spanned by a “Drinfeld-Heegner” point. This is a report on work in progress.