BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Rujie Yang (Stony Brook) DTSTART:20201119T233000Z DTEND:20201120T003000Z DTSTAMP:20260422T212746Z UID:UCSBsga/1 DESCRIPTION:Title: Decomposition theorem for semisimple local systems\nby Rujie Yang (Sto ny Brook) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract \nIn complex algebraic geometry\, the decomposition theorem asserts that s emisimple geometric objects remain semisimple after taking direct images u nder proper algebraic maps. This was conjectured by Kashiwara and is prove d by Mochizuki and Sabbah in a series of very long papers via harmonic ana lysis and D-modules. In this talk\, I would like to explain a simpler proo f in the case of semisimple local systems using a more geometric approach. This is joint work in progress with Chuanhao Wei.\n LOCATION:https://researchseminars.org/talk/UCSBsga/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Alice Pozzi (Imperial College London) DTSTART:20201203T233000Z DTEND:20201204T003000Z DTSTAMP:20260422T212746Z UID:UCSBsga/2 DESCRIPTION:Title: Derivatives of Hida families and rigid meromorphic cocycles\nby Alice Pozzi (Imperial College London) as part of UCSB Seminar on Geometry and Ar ithmetic\n\n\nAbstract\nA rigid meromorphic cocycle is a class in the firs t cohomology of the group ${\\rm SL}_2(\\mathbb{Z}[1/p])$ acting on the no n-zero rigid meromorphic functions on the Drinfeld $p$-adic upper half pla ne by Mobius transformation. Rigid meromorphic cocycles can be evaluated a t points of real multiplication\, and their values conjecturally lie in th e ring class field of real quadratic fields\, suggesting striking analogie s with the classical theory of complex multiplication.\n\nIn this talk\, w e discuss the relation between the derivatives of certain $p$-adic familie s of Hilbert modular forms and rigid meromorphic cocycles. We explain how the study of congruences between cuspidal and Eisenstein families allows u s to show the algebraicity of the values of a certain rigid meromorphic co cycle at real multiplication points.\n\nThis is joint work with Henri Darm on and Jan Vonk.\n LOCATION:https://researchseminars.org/talk/UCSBsga/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Jaclyn Lang (Oxford) DTSTART:20201210T190000Z DTEND:20201210T200000Z DTSTAMP:20260422T212746Z UID:UCSBsga/3 DESCRIPTION:Title: Eisenstein congruences at prime-square level and an application to class n umbers\nby Jaclyn Lang (Oxford) as part of UCSB Seminar on Geometry an d Arithmetic\n\n\nAbstract\nIn Mazur's seminal work on the Eisenstein idea l\, he showed that when $N$ and $p > 3$ are primes\, there is a weight $2$ cusp form of level $N$ congruent to the unique weight $2$ Eisenstein seri es of level $N$ if and only if $N = 1$ mod $p$. Calegari--Emerton\, Lecout urier\, and Wake--Wang-Erickson have work that relates these cuspidal-Eise nstein congruences to the $p$-part of the class group of $\\mathbb{Q}(N^{1 /p})$. Calegari observed that when $N = -1$ mod $p$\, one can use Galois c ohomology and some ideas of Wake--Wang-Erickson to show that $p$ divides t he class group of $\\mathbb{Q}(N^{1/p})$. He asked whether there is a way to directly construct the relevant degree $p$ everywhere unramified extens ion of $\\mathbb{Q}(N^{1/p})$ in this case. After discussing some of this background\, I will report of work in progress with Preston Wake in which we give a positive answer to this question using cuspidal-Eisenstein congr uences at prime-square level.\n LOCATION:https://researchseminars.org/talk/UCSBsga/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Bharath Palvannan (NCTS\, Taiwan) DTSTART:20201217T233000Z DTEND:20201218T003000Z DTSTAMP:20260422T212746Z UID:UCSBsga/4 DESCRIPTION:Title: Congruence ideal associated to Yoshida lifts\nby Bharath Palvannan (NC TS\, Taiwan) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstr act\nThis talk will be a report of work in progress with Ming-Lun Hsieh. I n analogy with the study of the congruences involving Hecke eigenvalues as sociated to Eisenstein series\, we study congruences involving p-adic fami lies of Hecke eigensystems associated to the space of Yoshida lifts of two Hida families. Our goal is to show that under suitable assumptions\, the characteristic ideal of the dual Selmer group associated to the Rankin--Se lberg product of the Hida families is contained in the corresponding congr uence ideal. We also discuss connections between pseudo-cyclicity of the d ual Selmer group and higher codimension Iwasawa theory.\n LOCATION:https://researchseminars.org/talk/UCSBsga/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Olivier Martin (Stony Brook) DTSTART:20201210T233000Z DTEND:20201211T003000Z DTSTAMP:20260422T212746Z UID:UCSBsga/5 DESCRIPTION:Title: The degree of irrationality of most abelian g-folds is at least 2g\nby Olivier Martin (Stony Brook) as part of UCSB Seminar on Geometry and Arit hmetic\n\n\nAbstract\nThe degree of irrationality of a complex projective n-dimensional variety X is the minimal degree of a dominant rational map f rom X to n-dimensional projective space. It is a birational invariant that measures how far X is from being rational. Accordingly\, one expects the computation of this invariant in general to be a difficult problem. Alzati and Pirola showed in 1993 that the degree of irrationality of any abelian g-fold is at least g+1 using inequalities on holomorphic length. Tokunaga and Yoshihara later proved that this bound is sharp for abelian surfaces and Yoshihara asked for examples of abelian surfaces with degree of irrati onality at least 4. Recently\, Chen and Chen-Stapleton showed that the deg ree of irrationality of any abelian surface is at most 4. In this work\, I provide the first examples of abelian surfaces with degree of irrationali ty 4. In fact\, I show that most abelian g-folds have degree of irrational ity at least 2g. We will present the proof of the case g=2 and indicate ho w to obtain the result in general. The argument rests on Mumford's theorem on rational equivalence of zero-cycles on surfaces with p_g>0 along with (new?) results on integral Hodge classes on self-products of abelian varie ties.\n LOCATION:https://researchseminars.org/talk/UCSBsga/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Sam Mundy (Columbia University) DTSTART:20210108T230000Z DTEND:20210109T000000Z DTSTAMP:20260422T212746Z UID:UCSBsga/6 DESCRIPTION:Title: Cuspidal $p$-adic deformations of critical Eisenstein series for $G_2$ \nby Sam Mundy (Columbia University) as part of UCSB Seminar on Geometry a nd Arithmetic\n\n\nAbstract\nIn this talk\, I will explain how (under cert ain standard conjectures of Arthur) Urban's eigenvariety allows us to $p$- adically deform\, in generically cuspidal families\, critical $p$-stabiliz ations of certain maximal parabolic Eisenstein series for $G_2$. This has consequences for the symmetric cube Bloch--Kato conjecture.\n LOCATION:https://researchseminars.org/talk/UCSBsga/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Botong Wang (University of Wisconsin-Madison) DTSTART:20210115T230000Z DTEND:20210116T000000Z DTSTAMP:20260422T212746Z UID:UCSBsga/7 DESCRIPTION:Title: The algebraic geometry of posets\nby Botong Wang (University of Wiscon sin-Madison) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstr act\nNatural poset (partially ordered set) structure arises in polyhedral fans\, Coxeter groups\, and hyperplane arrangements. In nice cases\, there exist stratified algebraic varieties realizing the above poset structures \, namely toric varieties\, Schubert varieties\, and matroid Schubert vari eties respectively. I will discuss how to prove combinatorial results usin g the corresponding algebraic varieties. I will also talk about how to ext end the above arguments beyond geometry\, for example\, the singular Hodge theory of arbitrary matroids. The last part is joint work with Tom Braden \, June Huh\, Jacob Matherne and Nick Proudfoot.\n LOCATION:https://researchseminars.org/talk/UCSBsga/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Ziyang Gao (CNRS/Jussieu) DTSTART:20210122T190000Z DTEND:20210122T200000Z DTSTAMP:20260422T212746Z UID:UCSBsga/8 DESCRIPTION:Title: Bound on the number of rational points on curves\nby Ziyang Gao (CNRS/ Jussieu) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract\ nMazur conjectured\, after Faltings’s proof of the Mordell conjecture\, that the number of rational points on a curve is bounded from above by the genus\, the degree of the number field and the Mordell-Weil rank. In this talk I will explain the proof of this conjecture. This is joint work with Vesselin Dimitrov and Philipp Habegger.\n LOCATION:https://researchseminars.org/talk/UCSBsga/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Nick Addington (University of Oregon) DTSTART:20210129T230000Z DTEND:20210130T000000Z DTSTAMP:20260422T212746Z UID:UCSBsga/9 DESCRIPTION:Title: Moduli spaces of sheaves on moduli spaces of sheaves\nby Nick Addingto n (University of Oregon) as part of UCSB Seminar on Geometry and Arithmeti c\n\n\nAbstract\nIt often happens that if M is a moduli space of vector bu ndles on a curve C\, then C is also a moduli space of vector bundles on M\ , where the bundles on M come from taking "wrong-way slices" of the the un iversal bundle on M x C. This story starts in the '70s and is due to Naras imhan and Ramanan\, Newstead\, and others. Reede and Zhang recently obser ved that a similar result holds for some Hilbert schemes of points\, and f or certain moduli spaces of rank-0 sheaves on K3 surfaces. I will discuss joint work with my student Andrew Wray\, showing that it holds for moduli spaces of high-rank sheaves on K3 surfaces. Techniques include the Quill en metric on determinant line bundles and twistor families of hyperkaehler manifolds.\n LOCATION:https://researchseminars.org/talk/UCSBsga/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Ziquan Yang (Harvard University) DTSTART:20210205T230000Z DTEND:20210206T000000Z DTSTAMP:20260422T212746Z UID:UCSBsga/10 DESCRIPTION:Title: Finiteness and the Tate Conjecture in Codimension 2 for K3 Squares\nb y Ziquan Yang (Harvard University) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract\nTwo years ago\, via a refined CM lifting theory \, Ito-Ito-Koshikawa proved the Tate Conjecture for squares of K3 surfaces over finite fields by reducing to Tate's theorem on the endomorphisms of abelian varieties. I will explain a different proof\, which is based on a twisted version of Fourier-Mukai transforms between K3 surfaces. In partic ular\, I do not use Tate's theorem after assuming some known properties of individual K3's. The main purpose of doing so is to illustrate Tate's ins ight on the connection between the Tate conjecture and the positivity resu lts in algebraic geometry for codimension 2 cycles\, through some "geometr y in cohomological degree 2".\n LOCATION:https://researchseminars.org/talk/UCSBsga/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Jiuya Wang (Duke University) DTSTART:20210212T230000Z DTEND:20210213T000000Z DTSTAMP:20260422T212746Z UID:UCSBsga/11 DESCRIPTION:Title: Pointwise Bound for $\\ell$-torsion of Class Groups\nby Jiuya Wang (D uke University) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAb stract\n$\\ell$-torsion conjecture states that $\\ell$-torsion of the clas s group $|\\text{Cl}_K[\\ell]|$ for every number field $K$ is bounded by $ \\text{Disc}(K)^{\\epsilon}$. It follows from a classical result of Brauer -Siegel\, or even earlier result of Minkowski that the class number $|\\te xt{Cl}_K|$ of a number field $K$ are always bounded by $\\text{Disc}(K)^{1 /2+\\epsilon}$\, therefore we obtain a trivial bound\n$\\text{Disc}(K)^{1/ 2+\\epsilon}$ on $|\\text{Cl}_K[\\ell]|$. We will talk about recent works on breaking the trivial bound for $\\ell$-torsion of class groups in some cases based on the work of Ellenberg-Venkatesh. We will also mention sever al questions following this line.\n LOCATION:https://researchseminars.org/talk/UCSBsga/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Smith (MIT) DTSTART:20210219T230000Z DTEND:20210220T000000Z DTSTAMP:20260422T212746Z UID:UCSBsga/12 DESCRIPTION:Title: Selmer groups and a Cassels-Tate pairing for finite Galois modules\nb y Alexander Smith (MIT) as part of UCSB Seminar on Geometry and Arithmetic \n\n\nAbstract\nI will discuss some new results on the structure of Selmer groups of finite Galois modules over global fields. Tate's definition of the Cassels-Tate pairing can be extended to a pairing on such Selmer group s with little adjustment\, and many of the fundamental properties of the C assels-Tate pairing can be reproved with new methods in this setting. I wi ll also give a general definition of the theta/Mumford group and relate it to the structure of the Cassels-Tate pairing\, generalizing work of Poone n and Stoll.\n LOCATION:https://researchseminars.org/talk/UCSBsga/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Peng Zhou (UC Berkeley) DTSTART:20210226T230000Z DTEND:20210227T000000Z DTSTAMP:20260422T212746Z UID:UCSBsga/13 DESCRIPTION:Title: Derived Equivalences from Variation of Lagrangian Skeletons\nby Peng Zhou (UC Berkeley) as part of UCSB Seminar on Geometry and Arithmetic\n\n\ nAbstract\nA Lagrangian skeleton is a singular Lagrangian in a symplectic manifold\, such that it has a tubular neighborhood as Weinstein manifold. One can associate a category (wrapped Fukaya category) to a Lagrangian ske leton\, and study when does the category remain invariant as the Lagrangia n varies. Many categories in mirror symmetry and representation theory can be described using such categories on Lagrangian skeletons\, and it’s i nteresting to see how variation of skeleton induces derived equivalences b etween categories. I will begin with definition and basic examples\, no pr ior knowledge of wrapped Fukaya category is needed. Some of the results ar e based on works arXiv:1804.08928\, arXiv:2011.03719\, arXiv:2011.06114 (joint with Jesse Huang).\n LOCATION:https://researchseminars.org/talk/UCSBsga/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Kazuto Ota (Osaka University) DTSTART:20210312T230000Z DTEND:20210313T000000Z DTSTAMP:20260422T212746Z UID:UCSBsga/14 DESCRIPTION:Title: On Iwasawa theory for CM elliptic curves at inert primes\nby Kazuto O ta (Osaka University) as part of UCSB Seminar on Geometry and Arithmetic\n \n\nAbstract\nIn this talk\, I will report on recent work on the anticyclo tomic Iwasawa theory for CM elliptic curves at inert primes. A key result is a structure theorem of local units predicted by Rubin\, which leads to new developments in supersingular Iwasawa theory\, an instance: a Bertolin i-Darmon-Prasanna style formula for Rubin's p-adic L-function. I will also discuss the BDP formula and applications. This is joint work with Ashay B urungale and Shinichi Kobayashi.\n LOCATION:https://researchseminars.org/talk/UCSBsga/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Jie Lin (Universitat Duiburg-Essen) DTSTART:20210319T180000Z DTEND:20210319T190000Z DTSTAMP:20260422T212746Z UID:UCSBsga/15 DESCRIPTION:Title: Periods and L-values of automorphic motives\nby Jie Lin (Universitat Duiburg-Essen) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbs tract\nIn this talk\, we will first introduce a conjecture of Deligne on s pecial values of L-functions. This conjecture generalizes the famous resul t by Euler on the Riemann zeta values at positive even integers\, and pred icts a relation between motivic L-values and geometric periods. We will th en explain an approach towards this conjecture for automorphic motives and summarize some recent progress (joint with H. Grobner and M. Harris)\n LOCATION:https://researchseminars.org/talk/UCSBsga/15/ END:VEVENT BEGIN:VEVENT SUMMARY:David Nguyen (UCSB) DTSTART:20210305T230000Z DTEND:20210306T000000Z DTSTAMP:20260422T212746Z UID:UCSBsga/16 DESCRIPTION:Title: Distribution of the ternary divisor function in arithmetic progressions\nby David Nguyen (UCSB) as part of UCSB Seminar on Geometry and Arithme tic\n\n\nAbstract\nThe ternary divisor function\, denoted $\\tau_3(n)$\, c ounts the number of ways to write a natural number $n$ as an ordered produ ct of three positive integers. Thus\, $\\sum_{n=1}^\\infty \\tau_3(n) n^{- s} = \\zeta^3(s).$ Given two coprime positive integers $a$ and $q$\, we st udy the distribution of $\\tau_3$ in arithmetic progressions $n \\equiv a (\\text{mod } q).$ The distribution of $\\tau_3$ in arithmetic progression s has a rich history and has applications to the distribution of prime num bers and moments of Dirichlet $L$-functions. We show that $\\tau_3$ is equ idistributed on average for moduli $q$ up to $X^{2/3}$\, extending the ind ividual estimate of Friedlander and Iwaniec (1985). We will also discuss a n averaged variance of $\\tau_3$ in arithmetic progressions related to a r ecent conjecture of Rodgers and Soundararajan (2018) about asymptotic of t his variance. One of the key inputs to this asymptotic come from a modifie d additive correlation sum of $\\tau_3.$\n LOCATION:https://researchseminars.org/talk/UCSBsga/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Isabel Vogt (University of Washington) DTSTART:20210402T220000Z DTEND:20210402T230000Z DTSTAMP:20260422T212746Z UID:UCSBsga/17 DESCRIPTION:Title: Brill--Noether theory over the Hurwitz space\nby Isabel Vogt (Univers ity of Washington) as part of UCSB Seminar on Geometry and Arithmetic\n\n\ nAbstract\nLet C be a curve of genus g. A fundamental problem in the theor y of algebraic curves is to understand maps of C to projective space of di mension r of degree d. When the curve C is general\, the moduli space of s uch maps is well-understood by the main theorems of Brill--Noether theory. However\, in nature\, curves C are often encountered already equipped wi th a map to some projective space\, which may force them to be special in moduli. The simplest case is when C is general among curves of fixed gona lity. Despite much study over the past three decades\, a similarly comple te picture has proved elusive in this case. In this talk\, I will discuss joint work with Eric Larson and Hannah Larson that completes such a pictur e\, by proving analogs of all of the main theorems of Brill--Noether theor y in this setting. In the course of our degenerative argument\, we'll exp loit a close relationship with the combinatorics of the affine symmetric g roup.\n LOCATION:https://researchseminars.org/talk/UCSBsga/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Huachen Chen (UCSB) DTSTART:20210409T220000Z DTEND:20210409T230000Z DTSTAMP:20260422T212746Z UID:UCSBsga/18 DESCRIPTION:Title: Brill-Noether loci for K3 categories\nby Huachen Chen (UCSB) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract\nI will discuss B rill-Noether loci in moduli spaces of stable objects in K3 categories. The question we are interested is whether these Brill-Noether loci are of exp ected dimensions. I will explain how the symplectic structures on the modu li spaces guarantee a large class of Brill-Noether loci has dimensions as expected. This is based on joint work with Arend Bayer and Qingyuan Jiang. \n LOCATION:https://researchseminars.org/talk/UCSBsga/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Tony Feng (MIT) DTSTART:20210416T220000Z DTEND:20210416T230000Z DTSTAMP:20260422T212746Z UID:UCSBsga/19 DESCRIPTION:Title: Higher Siegel-Weil formulas over function fields\nby Tony Feng (MIT) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract\nThe Sieg el-Weil formula relates the integral of a theta function along a classical group H to a special value of a Siegel-Eisenstein series on another group G. Kudla proposed an "arithmetic analogue" of the Siegel-Weil formula\, r elating intersection numbers of special cycles on Shimura varieties for H to the first derivative at a special value of a Siegel-Eisenstein series o n G. We study a function field analogue of this problem in joint work with Zhiwei Yun and Wei Zhang. We define special cycles on moduli stacks of un itary shtukas\, construct associated virtual fundamental classes\, and rel ate their degrees to the derivatives to *all* orders of Siegel-Eisenstein series. The results can be seen as “higher derivative” analogues of th e Kudla-Rapoport Conjecture. A key to the proof is a categorification of l ocal density formulas for Fourier coefficients of Eisenstein series\, and a parallel categorification of the degrees of virtual fundamental classes of special cycles\, in terms of a global variant of Springer theory.\n LOCATION:https://researchseminars.org/talk/UCSBsga/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Gill Moss (University of Utah) DTSTART:20210423T221000Z DTEND:20210423T231000Z DTSTAMP:20260422T212746Z UID:UCSBsga/20 DESCRIPTION:Title: Moduli of Langlands parameters\nby Gill Moss (University of Utah) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract\nThe local L anglands correspondence connects representations of p-adic groups to Langl ands parameters\, which are certain representations of Galois groups of lo cal fields. In recent work with Dat\, Helm\, and Kurinczuk\, we have shown that Langlands parameters\, when viewed through the right lens\, occur na turally within a moduli space over Z[1/p]\, and we can say some things abo ut the geometry of this moduli space. This geometry should be reflected in the representation theory of p-adic groups\, on the other side of the loc al Langlands correspondence. The "local Langlands in families" conjecture describes the moduli space of Langlands parameters in terms of the center of the category of representations of the p-adic group-- it was establishe d for GL(n) in 2018. The goal of the talk is to give an overview of this p icture\, including current work in-progress\, with some discussion of the relation with recent work of Zhu and Fargues-Scholze.\n LOCATION:https://researchseminars.org/talk/UCSBsga/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Litt (University of Georgia) DTSTART:20210430T220000Z DTEND:20210430T230000Z DTSTAMP:20260422T212746Z UID:UCSBsga/21 DESCRIPTION:Title: The tropical section conjecture\nby Daniel Litt (University of Georgi a) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract\nGroth endieck's section conjecture predicts that for a curve X of genus at least 2 over an arithmetically interesting field (say\, a number field or p-adi c field)\, the étale fundamental group of X encodes all the information a bout rational points on X. In this talk I will formulate a tropical analog ue of the section conjecture and explain how to use methods from low-dimen sional topology and moduli theory to prove many cases of it. As a byproduc t\, I'll construct many examples of curves for which the section conjectur e is true\, in interesting ways. For example\, I will explain how to prove the section conjecture for the generic curve\, and for the generic curve with a rational divisor class\, as well as how to construct curves over p- adic fields which satisfy the section conjecture for geometric reasons. Th is is joint work with Wanlin Li\, Nick Salter\, and Padma Srinivasan.\n LOCATION:https://researchseminars.org/talk/UCSBsga/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Yukako Kezuka (Max Planck Institute) DTSTART:20210507T180000Z DTEND:20210507T190000Z DTSTAMP:20260422T212746Z UID:UCSBsga/22 DESCRIPTION:Title: Tamagawa number divisibility of central $L$-values\nby Yukako Kezuka (Max Planck Institute) as part of UCSB Seminar on Geometry and Arithmetic\ n\n\nAbstract\nIn this talk\, I will study the conjecture of Birch and Swi nnerton-Dyer for elliptic curves $E$ of the form $x^3+y^3=N$ for cube-free positive integers $N$. They are cubic twists of the Fermat elliptic curv e $x^3+y^3=1$\, and admit complex multiplication by the ring of integers o f $\\mathbb{Q}(\\sqrt{-3})$. First\, I will study the $p$-adic valuation o f the algebraic part of their central $L$-values\, and exhibit a curious r elation between the $3$-part of the Tate--Shafarevich group of $E$ and the number of prime divisors of $N$ which are inert in $\\mathbb{Q}(\\sqrt{-3 })$. In the second part of the talk\, I will study in more detail the case s when $N=2p$ or $2p^2$ for an odd prime number $p$ congruent to $2$ or $5 $ modulo $9$. For these curves\, we establish the $3$-part of the Birch--S winnerton-Dyer conjecture and a relation between the ideal class group of $\\mathbb{Q}(\\sqrt[3]{p})$ and the $2$-Selmer group of $E$\, which can be used to study non-triviality of the $2$-part of their Tate--Shafarevich g roup. The second part of this talk is joint work with Yongxiong Li.\n LOCATION:https://researchseminars.org/talk/UCSBsga/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Adel Betina (University of Vienna) DTSTART:20210514T180000Z DTEND:20210514T190000Z DTSTAMP:20260422T212746Z UID:UCSBsga/23 DESCRIPTION:Title: Eisenstein intersection points on the Hilbert Eigenvariety\nby Adel B etina (University of Vienna) as part of UCSB Seminar on Geometry and Arith metic\n\n\nAbstract\nIn this talk\, we will report a joint work with M. Di mitrov and S.C. Shih in which we study the local geometry of the Hilbert Eigenvariety at an intersection point between an Eisenstein component and the cuspidal locus. As applications\, we show the non-vanishing of certain Katz $p$-adic $L$-functions at $s=0$ and give a new proof of the Gross-St ark conjecture in the rank one case.\n LOCATION:https://researchseminars.org/talk/UCSBsga/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Boya Wen (Princeton University) DTSTART:20210521T220000Z DTEND:20210521T230000Z DTSTAMP:20260422T212746Z UID:UCSBsga/24 DESCRIPTION:Title: A Gross-Zagier Formula for CM cycles over Shimura Curves\nby Boya Wen (Princeton University) as part of UCSB Seminar on Geometry and Arithmetic \n\n\nAbstract\nIn this talk I will introduce my thesis work (in preparati on) to prove a Gross-Zagier formula for CM cycles over Shimura curves. The formula connects the global height pairing of special cycles in Kuga vari eties over Shimura curves with the derivatives of the L-functions associat ed to weight-2k modular forms. As a key original ingredient of the proof\, I will introduce some harmonic analysis on local systems over graphs\, in cluding an explicit construction of Green's function\, which we apply to c ompute some local intersection numbers.\n LOCATION:https://researchseminars.org/talk/UCSBsga/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Jon Aycock (University of Oregon) DTSTART:20210528T220000Z DTEND:20210528T230000Z DTSTAMP:20260422T212746Z UID:UCSBsga/25 DESCRIPTION:Title: Overconvergent Differential Operators for Hilbert Modular Forms\nby J on Aycock (University of Oregon) as part of UCSB Seminar on Geometry and A rithmetic\n\n\nAbstract\nIn the 1970's\, Katz constructed p-adic L-functio ns for CM fields by relating the values of the Dedekind zeta function to t he values of certain nearly holomorphic Eisenstein series. Crucial in his construction was the action of the Maass--Shimura differential operators. Katz's p-adic interpolation of these differential operators is only define d over the ordinary locus\, which leads to a restriction on what p are all owed. Recently\, this restriction has been lifted in the case of quadratic imaginary fields by Andreatta and Iovita using an "overconvergent" analog of the Maass--Shimura operator for elliptic modular forms. We will give a n overview of the theory of overconvergent Hilbert modular forms before co nstructing an "overconvergent" analog of the Maass--Shimura operator for t his setting.\n LOCATION:https://researchseminars.org/talk/UCSBsga/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Anthony Várilly-Alvarado (Rice University) DTSTART:20210604T220000Z DTEND:20210604T230000Z DTSTAMP:20260422T212746Z UID:UCSBsga/26 DESCRIPTION:Title: From Merel's theorem to Brauer groups of K3 surfaces\nby Anthony Vár illy-Alvarado (Rice University) as part of UCSB Seminar on Geometry and Ar ithmetic\n\n\nAbstract\nOver number fields\, the Brauer group of a K3 surf ace behaves similarly to the subgroup of points of finite order of an elli ptic curve. In 1996\, Merel showed that the order of the torsion subgroup of an elliptic curve E/K is bounded by a constant depending only on the d egree of the extension [K:Q]. I will discuss an analogous conjecture in t he context of Brauer groups of K3 surfaces\, and the evidence we have accu mulated so far for it.\n LOCATION:https://researchseminars.org/talk/UCSBsga/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Yilong Zhang (Ohio State University) DTSTART:20210611T220000Z DTEND:20210611T230000Z DTSTAMP:20260422T212746Z UID:UCSBsga/27 DESCRIPTION:Title: Hilbert schemes of skew lines on cubic threefolds\nby Yilong Zhang (O hio State University) as part of UCSB Seminar on Geometry and Arithmetic\n \n\nAbstract\nOn a smooth cubic threefold Y\, a pairs of skew lines determ ines an irreducible component H of the Hilbert scheme of Y. We will show t hat the component H is smooth and is isomorphic to the blow-up of the 2nd symmetric product of Fano surface of lines on Y along the diagonal. This w ork is based on the work on Hilbert schemes of skew lines on projective sp aces by Chen-Coskun-Nollet in 2011. Moreover\, I'll explain the relation o f the component H to the compactification of locus of vanishing cycles on hyperplane sections of Y and the stable moduli space considered by Altavil la-Petkovic-Rota.\n LOCATION:https://researchseminars.org/talk/UCSBsga/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Salim Tayou (Harvard) DTSTART:20211028T233000Z DTEND:20211029T010000Z DTSTAMP:20260422T212746Z UID:UCSBsga/28 DESCRIPTION:Title: Equidistribution of Hodge loci\nby Salim Tayou (Harvard) as part of U CSB Seminar on Geometry and Arithmetic\n\n\nAbstract\nGiven a polarized va riation of Hodge structures\, the Hodge locus is a countable union of prop er algebraic subvarieties where extra Hodge classes appear. In this talk\, I will explain a general equidistribution theorem for these Hodge loci an d explain several applications: equidistribution of higher codimension Noe ther-Lefschetz loci\, equidistribution of Hecke translates of a curve in t he moduli space of abelian varieties and equidistribution of some families of CM points in Shimura varieties. The results of this talk are joint wor k with Nicolas Tholozan.\n LOCATION:https://researchseminars.org/talk/UCSBsga/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Cristian Popescu (UCSD) DTSTART:20211104T233000Z DTEND:20211105T010000Z DTSTAMP:20260422T212746Z UID:UCSBsga/29 DESCRIPTION:Title: An equivariant Tamagawa number formula for Drinfeld modules and beyond\nby Cristian Popescu (UCSD) as part of UCSB Seminar on Geometry and Arit hmetic\n\n\nAbstract\nTo a Galois extension of characteristic p global fie lds and a suitable Drinfeld module\,\none can associate an equivariant\, c haracteristic p valued\, rigid analytic Goss-type L-function. \nI will dis cuss the construction of this L-function and the statement and proof of a (Tamagawa number) formula for \nits special value at 0\, which generalize s to the Galois equivariant setting Taelman's celebrated class-number form ula\, \nproved in 2012. Next\, I will show how this formula implies a perf ect analog of the Brumer-Stark conjecture for Drinfeld modules.\nIf time p ermits\, I will discuss the very recent extension of the above formula to the much larger category of t-modules\n(t-motives)\, as well as its applic ations to the development of an Iwasawa theory for Drinfeld modules. The l ecture is based\non several recent joint works with N. Green\, J. Ferrara and Z. Higgins.\n LOCATION:https://researchseminars.org/talk/UCSBsga/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Isabella Negrini (McGill University) DTSTART:20211203T003000Z DTEND:20211203T020000Z DTSTAMP:20260422T212746Z UID:UCSBsga/30 DESCRIPTION:Title: A Shimura-Shintani correspondence for rigid analytic cocycles\nby Isa bella Negrini (McGill University) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract\nIn their paper Singular moduli for real quadrati c fields: a rigid analytic approach\, Darmon and Vonk introduced rigid mer omorphic cocycles\, i.e. elements of $H^1(\\mathrm{SL}_2(\\mathbb{Z}[1/p]) \,\\mathcal{M}^\\times)$ where $\\mathcal{M}^\\timesx$ is the multiplicati ve group of rigid meromorphic functions on the $p$-adic upper-half plane. Their values at RM points belong to narrow ring class fields of real quadr atic fiends and behave analogously to CM values of modular functions on $\ \mathrm{SL}_2(\\mathbb{Z})\\backslash\\mathbb{H}$. In this talk I will pre sent some progress towards developing a Shimura-Shintani correspondence in this setting.\n LOCATION:https://researchseminars.org/talk/UCSBsga/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Kazim Buyukboduk (UC Dublin) DTSTART:20220120T173000Z DTEND:20220120T183000Z DTSTAMP:20260422T212746Z UID:UCSBsga/31 DESCRIPTION:Title: Arithmetic of $\\theta$-critical p-adic L-functions\nby Kazim Buyukbo duk (UC Dublin) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAb stract\nIn joint work with Denis Benois\, we give an étale construction o f Bellaïche's p-adic L-functions about $\\theta$-critical points on the C oleman–Mazur eigencurve. I will discuss applications of this constructio n towards leading term formulae in terms of p-adic regulators on what we c all the thick Selmer groups\, which come attached to the infinitesimal def ormation at the said \\theta-critical point along the eigencurve\, and an exotic ($\\Lambda$-adic) $\\mathcal{L}$-invariant. Besides our interpolati on of the Beilinson–Kato elements about this point\, the key input to pr ove the interpolative properties of this p-adic L-function is a new p-adic Hodge-theoretic "eigenspace-transition via differentiation" principle.\n LOCATION:https://researchseminars.org/talk/UCSBsga/31/ END:VEVENT BEGIN:VEVENT SUMMARY:Andres Fernandez Herrero (Cornell University) DTSTART:20220128T003000Z DTEND:20220128T020000Z DTSTAMP:20260422T212746Z UID:UCSBsga/32 DESCRIPTION:Title: Intrinsic construction of moduli spaces via affine Grassmannians\nby Andres Fernandez Herrero (Cornell University) as part of UCSB Seminar on G eometry and Arithmetic\n\n\nAbstract\nModuli spaces arise as a geometric w ay of classifying objects of interest in algebraic geometry. For example\, there exists a quasiprojective moduli space that parametrizes stable vect or bundles on a smooth projective curve C. In order to further understand the geometry of this space\, Mumford constructed a compactification by add ing a boundary parametrizing semistable vector bundles. If the smooth curv e C is replaced by a higher dimensional projective variety X\, then one ca n compactify the moduli problem by allowing vector bundles to degenerate t o an object known as a "torsion-free sheaf". Gieseker and Maruyama constru cted moduli spaces of semistable torsion-free sheaves on such a variety X. More generally\, Simpson proved the existence of moduli spaces of semista ble pure sheaves supported on smaller subvarieties of X. All of these cons tructions use geometric invariant theory (GIT).\n\nFor a projective variet y X\, the moduli problem of coherent sheaves on X is naturally parametrize d by an algebraic stack M\, which is a geometric object that naturally enc odes the notion of families of sheaves. In this talk I will explain a GIT- free construction of the moduli space of Gieseker semistable pure sheaves which is intrinsic to the moduli stack M. This approach also yields a Hard er-Narasimhan stratification of the unstable locus of the stack. Our main technical tools are the theory of Theta-stability introduced by Halpern-Le istner\, and some recent techniques developed by Alper\, Halpern-Leistner and Heinloth. In order to apply these results\, one needs to prove some mo notonicity conditions for a polynomial numerical invariant on the stack. W e show monotonicity by defining a higher dimensional analogue of the affin e grassmannian for pure sheaves. I will also explain some applications of these ideas to other moduli problems. This talk is based on joint work wit h Daniel Halpern-Leistner and Trevor Jones\, as well as work with Tomas Go mez and Alfonso Zamora.\n LOCATION:https://researchseminars.org/talk/UCSBsga/32/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Disegni (BGU) DTSTART:20220203T173000Z DTEND:20220203T190000Z DTSTAMP:20260422T212746Z UID:UCSBsga/33 DESCRIPTION:Title: Theta cycles\nby Daniel Disegni (BGU) as part of UCSB Seminar on Geom etry and Arithmetic\n\n\nAbstract\nFor any elliptic curve E over Q\, an ex plicit construction yields a point P in E(Q) that is canonical\, in the fo llowing sense: (*) P is non-torsion <=> the group E(Q) and all p^\\infty-S elmer groups of E have rank 1.\nI will discuss a partial generalization of this picture to higher-rank motives M enjoying a `conjugate-symplectic’ symmetry\; examples arise from symmetric products of elliptic curves. The construction of the “canonical algebraic cycle on M"\, based on works o f Kudla and Y. Liu\, uses theta series valued in Chow groups of Shimura va rieties\, and it relies on two very different modularity conjectures. Assu ming those\, I will present a version of the " => " part of (*)\, whose pr oof uses recent advances in the theory of Euler systems.\n LOCATION:https://researchseminars.org/talk/UCSBsga/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Stevan Gajovic (University of Groningen) DTSTART:20220210T173000Z DTEND:20220210T190000Z DTSTAMP:20260422T212746Z UID:UCSBsga/34 DESCRIPTION:Title: Quadratic Chabauty for integral points and p-adic heights on even degree hyperelliptic curves\nby Stevan Gajovic (University of Groningen) as p art of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract\nIn this talk \, we show how to construct p-adic (Coleman-Gross) heights on even degree hyperelliptic curves\, more precisely\, its local component above p. Using heights\, we construct a locally analytic function that we use to compute integral points on certain even degree hyperelliptic curves whose Jacobia n has Mordell-Weil rank over Q equal to the genus. This is joint work with Steffen Müller.\n LOCATION:https://researchseminars.org/talk/UCSBsga/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Sarah Frei (Rice University) DTSTART:20220225T003000Z DTEND:20220225T020000Z DTSTAMP:20260422T212746Z UID:UCSBsga/35 DESCRIPTION:Title: Reduction of Brauer classes on K3 surfaces\nby Sarah Frei (Rice Unive rsity) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract\nF or a very general polarized K3 surface over the rational numbers\, it is a consequence of the Tate conjecture that the Picard rank jumps upon reduct ion modulo any prime. This jumping in the Picard rank is countered by a dr op in the size of the Brauer group. In this talk\, I will report on joint work with Brendan Hassett and Anthony Várilly-Alvarado\, in which we cons ider the following: Given a non-trivial Brauer class on a very general pol arized K3 surface over Q\, how often does this class become trivial upon r eduction modulo various primes? This has implications for the rationality of reductions of cubic fourfolds as well as reductions of twisted derived equivalent K3 surfaces.\n LOCATION:https://researchseminars.org/talk/UCSBsga/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Chan-Ho Kim (Korea Institute for Advanced Study) DTSTART:20220422T003000Z DTEND:20220422T020000Z DTSTAMP:20260422T212746Z UID:UCSBsga/36 DESCRIPTION:Title: A structural refinement of Birch and Swinnerton-Dyer conjecture\nby C han-Ho Kim (Korea Institute for Advanced Study) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract\nWe discuss a new application of (a part of) the Iwasawa main conjecture to the non-triviality of Kato's Koly vagin systems and a structural refinement of Birch and Swinnerton-Dyer con jecture. If time permits\, a certain relation with Heegner point Kolyvagin systems will be discussed.\n LOCATION:https://researchseminars.org/talk/UCSBsga/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Xin Wan (Chinese Academy of Sciences) DTSTART:20220429T003000Z DTEND:20220429T020000Z DTSTAMP:20260422T212746Z UID:UCSBsga/37 DESCRIPTION:Title: Iwasawa theory and Bloch-Kato conjecture for unitary groups\nby Xin W an (Chinese Academy of Sciences) as part of UCSB Seminar on Geometry and A rithmetic\n\n\nAbstract\nwe discuss a new way to study Iwasawa theory and Eisenstein congruences on unitary groups of general signature using p-adic functional equation\, and deduce from it consequences on Bloch-Kato conje cture in the ordinary case. We also discuss work in progress with Castella and Liu to generalize this to finite slope cases.\n LOCATION:https://researchseminars.org/talk/UCSBsga/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Waqar Shah (UCSB) DTSTART:20220929T233000Z DTEND:20220930T010000Z DTSTAMP:20260422T212746Z UID:UCSBsga/38 DESCRIPTION:Title: Zeta elements for Shimura Varieties I: Generalities\nby Waqar Shah (U CSB) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract\nA w ell-established technique towards understanding Selmer groups of Galois re presentations is the construction of an Euler system. One may ask if such systems can be created for Galois representations that arise in the cohomo logy of a given Shimura variety. For such purposes\, it is customary to ut ilize push-forwards of fundamental cycles or Eisenstein classes from sub-S himura varieties\, and to then establish norm relations between the push-f orwarded classes involving certain Hecke operators which compute appropria te automorphic L-factors.\n\nIn this talk\, I will motivate how classical Euler systems such as Kato's Siegel units and Kolyvagin's Heegner points m ay be viewed through an axiomatic lens and build up to a general framework in which norm relations for higher dimensional Shimura varieties may be s ystematically studied. I will highlight some of the computational challeng es that arise in higher dimensions and outline a theory of double coset de compositions due to Lansky that allows one to overcome these challenges. I n the next talk\, I'll apply these techniques to concrete examples of arit hmetic interest\, some old and some new.\n LOCATION:https://researchseminars.org/talk/UCSBsga/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Waqar Shah (UCSB) DTSTART:20221006T233000Z DTEND:20221007T010000Z DTSTAMP:20260422T212746Z UID:UCSBsga/39 DESCRIPTION:Title: Zeta elements for Shimura varieties II: Examples\nby Waqar Shah (UCSB ) as part of UCSB Seminar on Geometry and Arithmetic\n\n\nAbstract\nIn thi s second talk\, I will begin by reviewing a generalization of the double c oset decomposition recipe for parahoric subgroups originally due to Lansky and illustrate the recipe in some simple cases. Paired with the machinery of zeta elements and mixed Hecke correspondences that I described in my p revious talk\, this decomposition recipe yields a powerful and effective m ethod for studying norm relation problems for classes constructed using th e push-forward formalism that would otherwise be too cumbersome to study d irectly. I will illustrate this method in two key situations of arithmetic interest that were recently studied using alternate methods: Shimura vari eties of GU(1\,2n-1) & GSp_4. I will also highlight several key technical advantages of this approach over the earlier ones in these cases. Time per mitting\, I will discuss a new example of an Euler system for the Galois r epresentations arising from GU(2\,2) Shimura varieties.\n LOCATION:https://researchseminars.org/talk/UCSBsga/39/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrew Graham (Université Paris-Saclay) DTSTART:20221201T173000Z DTEND:20221201T190000Z DTSTAMP:20260422T212746Z UID:UCSBsga/40 DESCRIPTION:Title: A unipotent circle action on nearly overconvergent modular forms\nby Andrew Graham (Université Paris-Saclay) as part of UCSB Seminar on Geomet ry and Arithmetic\n\n\nAbstract\nRecent work of Howe shows that the action of the Atkin--Serre operator on p-adic modular forms can be reinterpreted as a $\\widehat{\\mathbb{G}}_m$ action on the Katz Igusa tower. By p-adic Fourier theory\, this gives an action of continuous functions on $\\mathb b{Z}_p$ on sections of the Igusa tower (p-adic modular forms). In this tal k I will explain how one can ``overconverge'' this action\, i.e. show that the subspace of nearly overconvergent modular forms is stable under the a ction of locally analytic functions on $\\mathbb{Z}_p$. This recovers (but is more general than) the construction of Andreatta--Iovita and has appli cations to the construction of p-adic L-functions. (Joint work with Vincen t Pilloni and Joaquin Rodrigues).\n LOCATION:https://researchseminars.org/talk/UCSBsga/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Florian Sprung (Arizona State University) DTSTART:20230217T003000Z DTEND:20230217T020000Z DTSTAMP:20260422T212746Z UID:UCSBsga/41 DESCRIPTION:Title: On characteristic power series of dual signed Selmer groups\nby Flori an Sprung (Arizona State University) as part of UCSB Seminar on Geometry a nd Arithmetic\n\n\nAbstract\nIn joint work with Jishnu Ray\, we relate the cardinality of the p-primary part of the Bloch-Kato Selmer group over Q a ttached to a modular form at a non-ordinary prime p to the constant term o f the characteristic power series of the signed Selmer groups over the cyc lotomic Zp-extension of Q. This generalizes a result of Vigni and Longo in the ordinary case. In the case of elliptic curves\, such results follow f rom earlier works by Greenberg\, Kim\, the second author\, and Ahmed–Lim \, covering both the ordinary and most of the supersingular case.\n LOCATION:https://researchseminars.org/talk/UCSBsga/41/ END:VEVENT END:VCALENDAR