BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Victoria Cantoral DTSTART:20200727T163000Z DTEND:20200727T170000Z DTSTAMP:20260422T225755Z UID:POINT/1 DESCRIPTION:Title: Th e Mumford—Tate conjecture implies the algebraic Sato—Tate conjecture\nby Victoria Cantoral as part of POINT: New Developments in Number Theo ry\n\n\nAbstract\nThe famous Mumford-Tate conjecture asserts that\, for ev ery prime number $\\ell$\, Hodge cycles are $\\mathbb{Q}_{\\ell}$-linear c ombinations of Tate cycles\, through Artin's comparisons theorems between Betti and étale cohomology. The algebraic Sato-Tate conjecture\, introduc ed by Serre and developed later by Banaszak and Kedlaya\, is a powerful to ol in order to prove new instances of the generalized Sato-Tate conjecture . This previous conjecture is related with the equidistribution of Frobeni us traces.\n\nOur main goal is to prove that the Mumford-Tate conjecture f or an abelian variety A implies the algebraic Sato-Tate conjecture for A. The relevance of this result lies mainly in the fact that the list of know n cases of the Mumford-Tate conjecture was up to now a lot longer than the list of known cases of the algebraic Sato-Tate conjecture. This is a join t work with Johan Commelin.\n\nIf you like to attend the talk\, please reg ister here: https://umich.zoom.us/meeting/register/tJAufuqtqDksG9fEmjTbWHM 4QOEUad6Ke-DE.\n LOCATION:https://researchseminars.org/talk/POINT/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Rahul Dalal DTSTART:20200727T170000Z DTEND:20200727T173000Z DTSTAMP:20260422T225755Z UID:POINT/2 DESCRIPTION:Title: St atistics of Automorphic Representations through Simplified Trace Formulas< /a>\nby Rahul Dalal as part of POINT: New Developments in Number Theory\n\ n\nAbstract\nAutomorphic representations encode information about a broad range of interesting mathematical objects. They are very difficult to stud y individually so it is often good to study them in families instead. The Arthur-Selberg trace formula is a powerful tool for this. For certain very nice families (discrete series at infinity)\, the invariant and stable ve rsions of the trace formula take on a simpler form\, allowing us to much m ore easily prove distributional results. I will discuss some of these resu lts and the techniques used for the required trace formula computations.\n \nIf you like to attend the talk\, please register here: https://umich.zoo m.us/meeting/register/tJAufuqtqDksG9fEmjTbWHM4QOEUad6Ke-DE.\n LOCATION:https://researchseminars.org/talk/POINT/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Karol Koziol (University of Michigan) DTSTART:20200824T130000Z DTEND:20200824T133000Z DTSTAMP:20260422T225755Z UID:POINT/5 DESCRIPTION:Title: Su persingular representations of p-adic reductive groups.\nby Karol Kozi ol (University of Michigan) as part of POINT: New Developments in Number T heory\n\n\nAbstract\nThe representation theory of p-adic reductive groups plays an extremely important role in modern number theory. Namely\, the l ocal Langlands conjectures predict that (packets of) irreducible complex r epresentations of p-adic reductive groups (such as $\\mathrm{GL}_n(\\mathb b{Q}_p)$\, $\\mathrm{GSp}_{2n}(\\mathbb{Q}_p)$\, etc.) should be parametri zed by certain representations of the Weil-Deligne group (a variant of the usual absolute Galois group). A special role in this hypothetical corres pondence is held by the supercuspidal representations\, which generically are expected to correspond to irreducible objects on the Galois side\, and which serve as building blocks for all irreducible representations. Moti vated by recent advances in the mod-$p$ local Langlands program (i.e.\, wi th mod-$p$ coefficients instead of complex coefficients)\, I will give an overview of what is known about supersingular representations of $p$-adic reductive groups\, which are the "mod-$p$ coefficients" analogs of supercu spidal representations. This is joint work with Florian Herzig and Marie- France Vigneras.\n\nPlease register for the talks on August 24 here: \nhtt ps://virginia.zoom.us/meeting/register/tJMkc-uorT8iHdOXRaBkci8wHoKUkqiXaq- E\n LOCATION:https://researchseminars.org/talk/POINT/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Garen Chiloyan (University of Connecticut) DTSTART:20200824T133000Z DTEND:20200824T140000Z DTSTAMP:20260422T225755Z UID:POINT/6 DESCRIPTION:Title: A Classification of Isogeny-Torsion Graphs of Elliptic Curves Defined over t he Rationals\nby Garen Chiloyan (University of Connecticut) as part of POINT: New Developments in Number Theory\n\n\nAbstract\nAn isogeny graph is a nice visualization of the isogeny class of an elliptic curve. A theor em of Kenku shows sharp bounds on the number of distinct isogenies that a rational elliptic curve can have (in particular\, every isogeny graph has at most 8 vertices). In this talk\, we give a complete classification of t he torsion subgroups over $\\mathbb{Q}$ that can occur in each vertex of a given isogeny graph of elliptic curves defined over the rationals. This i s joint work with \\'Alvaro Lozano-Robledo.\n\nPlease register for the tal ks on August 24 here: \nhttps://virginia.zoom.us/meeting/register/tJMkc-uo rT8iHdOXRaBkci8wHoKUkqiXaq-E\n LOCATION:https://researchseminars.org/talk/POINT/6/ END:VEVENT BEGIN:VEVENT SUMMARY:André Macedo (University of Reading) DTSTART:20200909T000000Z DTEND:20200909T003000Z DTSTAMP:20260422T225755Z UID:POINT/7 DESCRIPTION:Title: Lo cal-global principles for norm equations\nby André Macedo (University of Reading) as part of POINT: New Developments in Number Theory\n\n\nAbst ract\nGiven an extension L/K of number fields\, we say that the Hasse norm principle (HNP) holds if every non-zero element of K which is a norm ever ywhere locally is in fact a global norm from L. If L/K is cyclic\, the ori ginal Hasse norm theorem states that the HNP holds. More generally\, there is a cohomological description (due to Tate) of the obstruction to the HN P for Galois extensions. In this talk\, I will present work developing exp licit methods to study this principle for non-Galois extensions as well as some key applications in extensions whose normal closure has Galois group A_n or S_n. I will additionally discuss the geometric interpretation of t his concept and how it relates to the weak approximation property for norm varieties. If time permits\, I will also present some recent developments on the statistics of the HNP\n LOCATION:https://researchseminars.org/talk/POINT/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Elisa Bellah (University of Oregon) DTSTART:20200909T003000Z DTEND:20200909T010000Z DTSTAMP:20260422T225755Z UID:POINT/8 DESCRIPTION:Title: No rm Form Equations and Linear Divisibility Sequences\nby Elisa Bellah ( University of Oregon) as part of POINT: New Developments in Number Theory\ n\n\nAbstract\nFinding integer solutions to norm form equations is a class ic Diophantine problem. Using the units of the associated coefficient ring \, we can produce sequences of solutions to these equations. It turns out that these solutions can be written as tuples of linear homogeneous recurr ence sequences\, each with characteristic polynomial equal to the minimal polynomial of our unit. We show that for certain families of norm forms\, these sequences are linear divisibility sequences.\n LOCATION:https://researchseminars.org/talk/POINT/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Jon Aycock (University of Oregon) DTSTART:20200921T163000Z DTEND:20200921T170000Z DTSTAMP:20260422T225755Z UID:POINT/9 DESCRIPTION:Title: Fa milies of Differential Operators for Overconvergent Hilbert Modular Forms< /a>\nby Jon Aycock (University of Oregon) as part of POINT: New Developmen ts in Number Theory\n\n\nAbstract\nWe construct differential operators for families of overconvergent Hilbert modular forms by interpolating the Gau ss--Manin connection on strict neighborhoods of the ordinary locus. This i s related to work done by Harron and Xiao and by Andreatta and Iovita in t he case of modular forms and has applications in particular to p-adic L-fu nctions of CM fields.\n LOCATION:https://researchseminars.org/talk/POINT/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Neelam Saikia (Indian Institute of Technology Guwahati) DTSTART:20200921T170000Z DTEND:20200921T173000Z DTSTAMP:20260422T225755Z UID:POINT/10 DESCRIPTION:Title: Z eros of $p$-adic hypergeometric series\nby Neelam Saikia (Indian Insti tute of Technology Guwahati) as part of POINT: New Developments in Number Theory\n\n\nAbstract\nLet $p$ be an odd prime. McCarthy initiated a study of hypergeometric functions in the $p$-adic setting. This function can be understood as $p$-adic analogue of Gauss' hypergeometric function\, and so me kind of generalisation of Greene's hypergeometric function over finite fields. In this talk we investigate arithmetic properties of certain famil ies of McCarthy's hypergeometric functions. In particular\, we explicitly discuss all the possible values of these functions. Moreover\, we discuss zeros of these functions.\n LOCATION:https://researchseminars.org/talk/POINT/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Seoyoung Kim (Queen's University) DTSTART:20200810T163000Z DTEND:20200810T170000Z DTSTAMP:20260422T225755Z UID:POINT/11 DESCRIPTION:Title: F rom the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture\nby Seoyoung Kim (Queen's University) as part of POINT: New Developments in N umber Theory\n\n\nAbstract\nLet $E$ be an elliptic curve over $\\mathbb{Q} $ with discriminant\, and let $a_p$ be the Frobenius trace for each prime p. In 1965\, Birch and Swinnerton-Dyer formulated a conjecture which impli es\n\n$\\lim\\limits_{x \\rightarrow \\infty} \\frac{1}{\\log x} \\sum_{p< x} \\frac{a_p\\log p}{p}=-r+\\frac{1}{2}\,$\n\nwhere $r$ is the order of the zero of the $L$-function of $E$ at $s=1$\, which is predicted to be th e Mordell-Weil rank of $E(\\mathbb{Q})$. We show that if the above limit e xits\, then the limit equals $-r+\\frac{1}{2}$\, and study the connections to Riemann hypothesis for $E$. We also relate this to Nagao's conjecture. This is a recent joint work with M. Ram Murty.\n\nPlease register for the talks on August 10 here:\nhttps://fordham.zoom.us/meeting/register/tJwpde 2srTgqHdYG6NMu5WmgzPiDnNJQMTsM\n LOCATION:https://researchseminars.org/talk/POINT/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Sachi Hashimoto (Boston University) DTSTART:20200810T170000Z DTEND:20200810T173000Z DTSTAMP:20260422T225755Z UID:POINT/12 DESCRIPTION:Title: C omputing rational points on databases of curves\nby Sachi Hashimoto (B oston University) as part of POINT: New Developments in Number Theory\n\n\ nAbstract\nFor a curve of genus at least $2$\, we know from Faltings's the orem that its set of rational points is finite. A major challenge is to pr ovably determine\, for a given curve\, this set of rational points. One pr omising method is the Chabauty-Coleman method\, which uses $p$-adic (Colem an) integrals to compute a finite set of p-adic points on the curve includ ing the rational points. We will discuss computations using the Chabauty-C oleman method to provably determine rational point sets for databases of c ertain genus $3$ superelliptic curves. This is joint work with Maria de Fr utos Fernandez and Travis Morrison.\n\nPlease register for the talks on Au gust 10 here:\nhttps://fordham.zoom.us/meeting/register/tJwpde2srTgqHdYG6N Mu5WmgzPiDnNJQMTsM\n LOCATION:https://researchseminars.org/talk/POINT/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Julie Desjardins (University of Toronto Mississauga) DTSTART:20201005T130000Z DTEND:20201005T133000Z DTSTAMP:20260422T225755Z UID:POINT/13 DESCRIPTION:Title: D ensity of rational points on a family of del Pezzo surface of degree 1 \nby Julie Desjardins (University of Toronto Mississauga) as part of POINT : New Developments in Number Theory\n\n\nAbstract\nLet $k$ be a number fie ld and $X$ an algebraic variety over $k$. We want to study the set of $k$- rational points $X(k)$. For example\, is $X(k)$ empty? If not\, is it dens e with respect to the Zariski topology? Del Pezzo surfaces are classified by their degrees $d$ (an integer between 1 and 9). Manin and various autho rs proved that for all del Pezzo surfaces of degree $d>1$\, $X(k)$ is dens e provided that the surface has a $k$-rational point (that lies outside a specific subset of the surface for $d=2$). For $d=1$\, the del Pezzo surfa ce always has a rational point. However\, we don't know if the set of rati onal points is Zariski-dense. In this talk\, I present a result that is jo int with Rosa Winter in which we prove the density of rational points for a specific family of del Pezzo surfaces of degree 1 over $k$.\n LOCATION:https://researchseminars.org/talk/POINT/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Mingming Zhang (Oklahoma State University) DTSTART:20201005T133000Z DTEND:20201005T140000Z DTSTAMP:20260422T225755Z UID:POINT/14 DESCRIPTION:Title: M ahler Measure and its behavior under iteration\nby Mingming Zhang (Okl ahoma State University) as part of POINT: New Developments in Number Theor y\n\n\nAbstract\nFor an algebraic number $\\alpha$ we denote by $M(\\alpha )$ the Mahler measure of $\\alpha$. As $M(\\alpha)$ is again an algebraic number (indeed\, an algebraic integer)\, $M(\\cdot)$ is a self-map on $\\o verline{\\mathbb{Q}}$\, and therefore defines a dynamical system. The $\\m athit{orbit}$ $\\mathit{size}$ of $\\alpha$\, denoted $\\# \\mathcal{O}_M( \\alpha)$\, is the cardinality of the forward orbit of $\\alpha$ under $M$ . In this talk\, we will start by introducing the definition of Mahler mea sure\, briefly discuss results on the orbit sizes of algebraic numbers wi th degree at least 3 and non-unit norm\, then we will turn our focus to th e behavior of algebraic units\, which are of interest in Lehmer's problem. We will mention the results regarding algebraic units of degree 4 and dis cuss that if $\\alpha$ is an algebraic unit of degree $d\\geq 5$ such that the Galois group of the Galois closure of $\\mathbb{Q}(\\alpha)$ contains $A_d$\, then the orbit size must be 1\, 2 or $\\infty$. Furthermore\, we will show that there exists units with orbit size larger than 2! This is j oint work with Paul Fili and Lucas Pottmeyer.\n LOCATION:https://researchseminars.org/talk/POINT/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Allechar Serrano López (University of Utah) DTSTART:20201102T180000Z DTEND:20201102T183000Z DTSTAMP:20260422T225755Z UID:POINT/17 DESCRIPTION:Title: C ounting elliptic curves with prescribed torsion over imaginary quadratic f ields\nby Allechar Serrano López (University of Utah) as part of POIN T: New Developments in Number Theory\n\n\nAbstract\nA generalization of Ma zur's theorem\, proved by Kamienny\, states that there are 26 possibilitie s for the torsion subgroup of an elliptic curve over quadratic extensions of the rational numbers. We prove that if $G$ is isomorphic to one of thes e subgroups then the elliptic curves up to height $X$ whose torsion is iso morphic to $G$ is on the order of $X^{\\frac{1}{d}}$ where $d>1$.\n LOCATION:https://researchseminars.org/talk/POINT/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Ashwin Iyengar (King's College London) DTSTART:20201102T173000Z DTEND:20201102T180000Z DTSTAMP:20260422T225755Z UID:POINT/18 DESCRIPTION:Title: m od p spectral Hecke algebras\nby Ashwin Iyengar (King's College London ) as part of POINT: New Developments in Number Theory\n\n\nAbstract\nIn th is talk I will discuss work in progress (for $\\textnormal{GL}_2(\\mathbb{ Q}_p)$) on describing the mod $p$ derived Hecke algebra attached to a Serr e weight\, as well as the mod $p$ spectral Hecke algebra attached to the c orresponding crystalline deformation ring. These objects should act compat ibly on the cohomology of arithmetic groups. I will describe these Hecke a lgebras and their actions in more detail.\n LOCATION:https://researchseminars.org/talk/POINT/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Ayreena Bakhtawar (La Trobe University) DTSTART:20201117T090000Z DTEND:20201117T093000Z DTSTAMP:20260422T225755Z UID:POINT/19 DESCRIPTION:Title: C ontribution to uniform Diophantine approximation via continued fractions\nby Ayreena Bakhtawar (La Trobe University) as part of POINT: New Devel opments in Number Theory\n\n\nAbstract\nDiophantine approximation is a bra nch of number theory which is concerned with the question of how well can an irrational number be approximated by a rational?\nOne of the major ingr edients to study problems in Diophantine approximation is continued fracti on expansion as they provide quick and efficient way for finding good rati onal approximations to irrational numbers.\nI will discuss the relationshi p between Diophantine approximation and the theory of continued fractions. And along the way I will talk about some measure theoretic results includ ing the landmark results of Dirichlet (1842)\, Khintchine (1924)\, and Jar nik (1931) theorems to the questions in continued fractions. These enable us to improve the classical results by using continued fractions.\n LOCATION:https://researchseminars.org/talk/POINT/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Katharina Müller (University Göttingen) DTSTART:20201117T093000Z DTEND:20201117T100000Z DTSTAMP:20260422T225755Z UID:POINT/20 DESCRIPTION:Title: T he split prime $\\mathbb{Z}_p$-extension of imaginary quadratic fields \nby Katharina Müller (University Göttingen) as part of POINT: New Devel opments in Number Theory\n\n\nAbstract\nLet $K$ be an imaginary quadratic field and $p$ a rational prime that splits into $p_1$ and $p_2$. Then ther e is a unique $\\mathbb{Z}_p$ extension that is only ramified at one of th e primes above $p$. We will shift this extension by an abelian extension o ver $L/ K$ to $L_{\\infty}$. Let $M$ be the maximal $p$-abelian $p_1$-rami fied extension of $L_{\\infty}$. Generalizing work of Schneps we will show that $Gal(M/L_{\\infty})$ is a finitely generated $\\mathbb{Z}_p$-module. If time allows we will also discuss the main conjecture for these extensi ons. Part of this talk is joint work with Vlad Crisan.\n LOCATION:https://researchseminars.org/talk/POINT/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Maria Fox (University of Oregon) DTSTART:20201021T000000Z DTEND:20201021T003000Z DTSTAMP:20260422T225755Z UID:POINT/21 DESCRIPTION:Title: S upersingular loci in moduli spaces of abelian varieties\nby Maria Fox (University of Oregon) as part of POINT: New Developments in Number Theory \n\n\nAbstract\nGiven a moduli space of abelian varieties in characteristi c $p$\, for example the reduction modulo $p$ of a modular curve\, it is na tural to ask: what points in this moduli space parametrize supersingular a belian varieties? These points define the supersingular locus of the modul i space. In this talk\, we'll see several examples of moduli spaces of abe lian varieties\, and we'll discuss the geometry of their supersingular loc i.\n LOCATION:https://researchseminars.org/talk/POINT/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Ian Gleason (University of California\, Berkeley) DTSTART:20201021T003000Z DTEND:20201021T010000Z DTSTAMP:20260422T225755Z UID:POINT/22 DESCRIPTION:Title: O n the geometric connected components of local Shimura varieties\nby Ia n Gleason (University of California\, Berkeley) as part of POINT: New Deve lopments in Number Theory\n\n\nAbstract\nThrough the recent introduction o f the theory of diamonds\, P. Scholze was able to define local versions of Shimura varieties. These are rigid-analytic spaces that generalize the ge neric fiber of a Rapoport-Zink space. It is widely expected that the cohom ology of these interesting spaces realizes instances of the Langlands corr espondence. In this talk we describe the geometric connected components of these moduli spaces and relate it to local class field theory.\n LOCATION:https://researchseminars.org/talk/POINT/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Arpit Bansal (Jawaharlal Nehru University) DTSTART:20210201T173000Z DTEND:20210201T180000Z DTSTAMP:20260422T225755Z UID:POINT/23 DESCRIPTION:Title: L arge sieve with square moduli for Z[i].\nby Arpit Bansal (Jawaharlal N ehru University) as part of POINT: New Developments in Number Theory\n\n\n Abstract\nThe large sieve inequality is of fundamental importance in analy tic number theory. Its theory started with Linnik’s investigation of the least quadratic non-residue modulo primes on average. These days\, there is a whole zoo of large sieve inequalities in all kind of contexts (for nu mber fields\, function fields\, automorphic forms\, etc.). The large sieve with restricted sets of moduli $q \\in \\mathbb{Z}$\, in particular with square moduli\, were investigated by L. Zhao and S. Baier. Moreover\, the large sieve with square moduli has found many applications\, in particular \, in questions regarding elliptic curves. The large sieve for additive ch aracters was extended to number fields by Huxley. In my talk\, I will give a summary of the classical large sieve with square moduli and present new extenstions to number field $\\mathbb{Q}[i]$ which have recently been est ablished in joint work with Stephan Baier.\n LOCATION:https://researchseminars.org/talk/POINT/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Liem Nguyen (Louisiana State University) DTSTART:20210119T203000Z DTEND:20210119T210000Z DTSTAMP:20260422T225755Z UID:POINT/24 DESCRIPTION:Title: O n the Values and Spectrum of Weil Sum of Binomials\nby Liem Nguyen (Lo uisiana State University) as part of POINT: New Developments in Number The ory\n\n\nAbstract\nThe Weil sum of an additive character $\\mu$ over a fin ite field $F$ is defined to be $W_{F\,s}(a)=\\sum_{x \\in F} \\mu(x^s-ax)$ where $s$ is an integer coprime to $|F^*|$. The Weil spectrum counts dist inct values of the Weil sum as $a$ runs through the invertible elements in the finite field. Determining the values of these sums and the size of it s spectrum give answers to long-standing problems in cryptography\, coding and information theory. In this talk\, we prove a special case of the Van ishing Conjecture of Helleseth ($1971$) on the presence of zero in the Wei l spectrum. We then propose a new conjecture on when the Weil spectrum con tains at least five elements\, and prove it for a certain class of Weil su m.\n\nTo join the talks on January 19th\, please register here: \n\nhttps ://ucsd.zoom.us/meeting/register/tJIuf-6uqjoiH9eVbtLN9y1G9l0qEBmbDRmV\n LOCATION:https://researchseminars.org/talk/POINT/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Edmund Karasiewicz (Ben-Gurion University) DTSTART:20210201T180000Z DTEND:20210201T183000Z DTSTAMP:20260422T225755Z UID:POINT/25 DESCRIPTION:Title: T he Twisted Satake Transform and the Casselman-Shalika Formula\nby Edmu nd Karasiewicz (Ben-Gurion University) as part of POINT: New Developments in Number Theory\n\n\nAbstract\nThe Fourier coefficients of automorphic fo rms are an important object of study due to their connection to $L$-functi ons. In the adelic framework\, constructions of $L$-functions involving Fo urier coefficients (e.g. Langland-Shahidi and Rankin-Selberg methods) natu rally lead to spherical Whittaker functions on $p$-adic groups. Thus we wo uld like to understand these spherical Whittaker functions to better under stand $L$-functions.\n\nCasselman-Shalika determined a formula for the sph erical Whittaker functions\, and basic algebraic manipulations reveal that their formula can be more succinctly expressed in terms of characters of the Langlands dual group. We will describe a new proof of the Casselman-Sh alika formula that provides a conceptual explanation of the appearance of characters.\n LOCATION:https://researchseminars.org/talk/POINT/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Jeremy Booher (University of Canterbury) DTSTART:20210119T210000Z DTEND:20210119T213000Z DTSTAMP:20260422T225755Z UID:POINT/26 DESCRIPTION:Title: I nvariants in Towers of Curves over Finite Fields\nby Jeremy Booher (Un iversity of Canterbury) as part of POINT: New Developments in Number Theor y\n\n\nAbstract\nA $Z_p$ tower of curves in characteristic $p$ is a sequen ce $C_0\, C_1\, C_2\, ...$ of smooth projective curves over a perfect fiel d of characteristic $p$ such that $C_n$ is a branched cover of $C_{n-1}$ a nd $C_n$ is a branched Galois $Z/(p^n)$-cover of $C_0$. For nice examples of $Z_p$ towers\, the growth of the genus is stable: for sufficiently lar ge $n$\, the genus of $C_n$ is a quadratic polynomial in $p^n$. In charac teristic $p$\, there are additional curve invariants like the a-number whi ch are poorly understood. They describe the group-scheme structure of the $p$-torsion of the Jacobian. I will discuss work in progress with Bryden Cais studying these invariants and suggesting that their growth is also s table in genus stable $Z_p$ towers. This is a new kind of Iwasawa theory for function fields.\n\nTo join the talks on January 19th\, please registe r here: \n\nhttps://ucsd.zoom.us/meeting/register/tJIuf-6uqjoiH9eVbtLN9y1 G9l0qEBmbDRmV\n LOCATION:https://researchseminars.org/talk/POINT/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Jef Laga (University of Cambridge) DTSTART:20210215T130000Z DTEND:20210215T133000Z DTSTAMP:20260422T225755Z UID:POINT/27 DESCRIPTION:Title: S elmer groups of some families of genus 3 curves and abelian surfaces\n by Jef Laga (University of Cambridge) as part of POINT: New Developments i n Number Theory\n\n\nAbstract\nManjul Bhargava and Arul Shankar have deter mined the average size of the $n$-Selmer group of the family of all ellipt ic curves over $\\mathbb{Q}$ ordered by height\, for $n$ at most $5$. In t his talk we will consider a family of nonhyperelliptic genus $3$ curves\, and bound the average size of the $2$-Selmer group of their Jacobians. Thi s implies that a majority of curves in this family have relatively few rat ional points. We also consider a family of abelian surfaces which are not principally polarized and obtain similar results. The proof is a combinati on of the theory of simple singularities\, graded Lie algebras and orbit-c ounting techniques.\n LOCATION:https://researchseminars.org/talk/POINT/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Clifton (Emory University) DTSTART:20210215T133000Z DTEND:20210215T140000Z DTSTAMP:20260422T225755Z UID:POINT/28 DESCRIPTION:Title: A n exponential bound for exponential diffsequences\nby Alexander Clifto n (Emory University) as part of POINT: New Developments in Number Theory\n \n\nAbstract\nA theorem of van der Waerden states that for any positive in teger $r$\, if you partition $\\mathbf{N}$ into $r$ disjoint subsets\, the n one of them will contain arbitrarily long arithmetic progressions. It is natural to ask what other arithmetic structures are preserved when partit ioning $\\mathbf{N}$ into a finite number of disjoint sets and to pose qua ntitative questions about these. We consider $D$-diffsequences\, introduce d by Landman and Robertson\, which are increasing sequences in which the c onsecutive differences all lie in some given set $D$. Here\, we consider t he case where $D$ consists of all powers of $2$ and define $f(k)$ to be th e smallest $n$ such that partitioning $\\{1\,2\,\\cdots\,n\\}$ into $2$ su bsets guarantees the presence of a $D$-diffsequence of length $k$ containe d entirely within one subset. We establish that $f(k)$ grows exponentially .\n LOCATION:https://researchseminars.org/talk/POINT/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Maryam Khaqan (Emory University) DTSTART:20210303T003000Z DTEND:20210303T010000Z DTSTAMP:20260422T225755Z UID:POINT/29 DESCRIPTION:Title: E lliptic Curves and Thompson's Sporadic Group\nby Maryam Khaqan (Emory University) as part of POINT: New Developments in Number Theory\n\n\nAbstr act\nMoonshine began as a series of numerical coincidences connecting fini te groups to modular forms. It has since evolved into a rich theory that s heds light on the underlying structures that these coincidences reflect.\n \n\nWe prove the existence of one such structure\, a module for the Thomps on group\, whose graded traces are specific half-integral weight weakly ho lomorphic modular forms. We then proceed to use this module to study the r anks of certain\nfamilies of elliptic curves. In particular\, this serves as an example of moonshine being used to answer questions in number theory .\n LOCATION:https://researchseminars.org/talk/POINT/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Teppei Takamatsu (University of Tokyo) DTSTART:20210303T010000Z DTEND:20210303T013000Z DTSTAMP:20260422T225755Z UID:POINT/30 DESCRIPTION:Title: O n finiteness of twisted forms of hyperkähler varieties\nby Teppei Tak amatsu (University of Tokyo) as part of POINT: New Developments in Number Theory\n\n\nAbstract\nFor a finite field extension $L/K$ and a variety $X$ over $K$\,\nlet $Tw_{L/K} (X)$ be the set of isomorphism classes of varie ties $Y$\nover $K$ which are isomorphic to $X$ after the base change to $L $ (i.e.\nthe set of twisted forms of $X$ via $L/K$). In this talk\, we pro ve the\nfiniteness of $Tw_{L/K}$ for K3 surfaces of characteristic away fr om 2\nand hyperkähler varieties of characteristic 0. This work is a\ngene ralization of Cattaneo-Fu's work on real forms of hyperkähler\nvarieties. We also give an application to the finiteness of derived\nequivalent twis ted forms of hyperkähler varieties.\n LOCATION:https://researchseminars.org/talk/POINT/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Bella Tobin (Oklahoma State University) DTSTART:20210601T200000Z DTEND:20210601T203000Z DTSTAMP:20260422T225755Z UID:POINT/33 DESCRIPTION:Title: R eduction of post-critically finite polynomials\nby Bella Tobin (Oklaho ma State University) as part of POINT: New Developments in Number Theory\n \n\nAbstract\nPost-critically finite maps are described as dynamical analo gs of CM Abelian Varieties. A CM abelian varieties over a number field $K$ has everywhere good reduction in some finite extension $L/K$. This motiva tes us to ask the question: do PCF maps have good reduction? We can use a particular family of maps\, dynamical Belyi polynomials\, to provide nece ssary and sufficient conditions for a PCF polynomial of degree $d$ to have potential good reduction at a prime $p$. This is joint work with Jacqueli ne Anderson and Michelle Manes.\n LOCATION:https://researchseminars.org/talk/POINT/33/ END:VEVENT BEGIN:VEVENT SUMMARY:Shaoyun Yi (University of South Carolina) DTSTART:20210601T203000Z DTEND:20210601T210000Z DTSTAMP:20260422T225755Z UID:POINT/34 DESCRIPTION:Title: O n counting cuspidal automorphic representations of GSp(4)\nby Shaoyun Yi (University of South Carolina) as part of POINT: New Developments in Nu mber Theory\n\n\nAbstract\nThere are some well-known classical equidistrib ution results like Sato-Tate conjecture for elliptic curves and equidistri bution of Hecke eigenvalues of elliptic cusp forms. In this talk\, we will discuss a similar equidistribution result for a family of cuspidal automo rphic representations for GSp(4). We formulate our theorem explicitly in t erms of the number of cuspidal automorphic representations for GSp(4) with certain conditions at the local places. To count the number of these cusp idal automorphic representations\, we will explore the connection between Siegel cusp forms of degree 2 and cuspidal automorphic representations of GSp(4). This is a joint work with Manami Roy and Ralf Schmidt.\n LOCATION:https://researchseminars.org/talk/POINT/34/ END:VEVENT BEGIN:VEVENT SUMMARY:Francesco Battistoni (Universite de Framche-Comte) DTSTART:20210615T123000Z DTEND:20210615T130000Z DTSTAMP:20260422T225755Z UID:POINT/35 DESCRIPTION:Title: O n elliptic curves over $\\mathbb{Q}(T)$ and their ranks\nby Francesco Battistoni (Universite de Framche-Comte) as part of POINT: New Development s in Number Theory\n\n\nAbstract\nWe consider elliptic curves over $\\math bb{Q}(T)$ admitting Weierstrass model with coefficients being polynomials of small degree\, so that they are rational elliptic surfaces. In joint wo rk with Sandro Bettin and Christophe Delaunay\, we apply Nagao's formula i n order to detect the value of their ranks: this approach is orthogonal to other geometric investigations\, and gives the values of the ranks by loo king at purely algebraic properties like the factorization of some integer polynomials. We also prove that\, whenever restricting to some specific f amilies of curves\, the generic curve in these families has rank $0$.\n LOCATION:https://researchseminars.org/talk/POINT/35/ END:VEVENT BEGIN:VEVENT SUMMARY:Geoffrey Akers (CUNY Graduate Center) DTSTART:20210615T130000Z DTEND:20210615T133000Z DTSTAMP:20260422T225755Z UID:POINT/36 DESCRIPTION:Title: O n a universal deformation ring that is a discrete valuation ring\nby G eoffrey Akers (CUNY Graduate Center) as part of POINT: New Developments in Number Theory\n\n\nAbstract\nWe consider a crystalline universal deformat ion ring $R$ of an $n$-dimensional\, mod $p$ Galois representation whose s emisimplification is the direct sum of two non-isomorphic absolutely irred ucible representations. Under some hypotheses\, we obtain that $R$ is a di screte valuation ring. The method examines the ideal of reducibility of $R $\, which is used to construct extensions of representations in a Selmer g roup with specified dimension. This can be used to deduce modularity of r epresentations.\n LOCATION:https://researchseminars.org/talk/POINT/36/ END:VEVENT BEGIN:VEVENT SUMMARY:Ross Paterson (University of Glasgow) DTSTART:20210628T163000Z DTEND:20210628T170000Z DTSTAMP:20260422T225755Z UID:POINT/37 DESCRIPTION:Title: A verage Ranks of Elliptic Curves after $p$-Extension\nby Ross Paterson (University of Glasgow) as part of POINT: New Developments in Number Theor y\n\n\nAbstract\nAs $E$ varies among elliptic curves defined over the rati onal numbers\, a theorem of Bhargava and Shankar shows that the average ra nk of the Mordell--Weil group $E(\\mathbb{Q})$ is bounded. If we fix a nu mber field $K$\, it is natural to then ask: is the average rank of $E(K)$ also bounded in this family? Moreover\, how does the average rank of $E(K )$ depend on $K$?\nThis talk will discuss recent progress on these questio ns for a restricted set of $K$.\n LOCATION:https://researchseminars.org/talk/POINT/37/ END:VEVENT BEGIN:VEVENT SUMMARY:Damián Gvirtz (University College London) DTSTART:20210628T170000Z DTEND:20210628T173000Z DTSTAMP:20260422T225755Z UID:POINT/38 DESCRIPTION:Title: T he potential Hilbert Property for Enriques surfaces\nby Damián Gvirtz (University College London) as part of POINT: New Developments in Number Theory\n\n\nAbstract\nWhen does an algebraic variety have "many" rational points? A possible formalisation of this notion is the (weak) Hilbert Prop erty for algebraic varieties\, a generalisation of Hilbert's classical irr educibility theorem. I will report on joint work in progress with G. Mezze dimi about a conjecture due to Campana and Corvaja-Zannier which concerns this property in the case of Enriques surfaces.\n LOCATION:https://researchseminars.org/talk/POINT/38/ END:VEVENT BEGIN:VEVENT SUMMARY:Tung Nguyen (Western University) DTSTART:20220215T213000Z DTEND:20220215T220000Z DTSTAMP:20260422T225755Z UID:POINT/39 DESCRIPTION:Title: F ekete polynomials\, quadratic residues\, and arithmetics\nby Tung Nguy en (Western University) as part of POINT: New Developments in Number Theor y\n\n\nAbstract\nFekete polynomials play an important role in the study of special values of L-functions of quadratic fields. While their analytic p roperties are well-studied in the literature\, little is known about their arithmetics. In this talk\, we will discuss some surprising arithmetical properties of these polynomials. In particular\, we will see that special values of Fekete polynomials contain some rich information about the class numbers of quadratic fields. Furthermore\, their Galois groups seem to f ollow a rather simple pattern. Time permitting\, I will discuss some rece nt progress on generalized Fekete polynomials. This is based on joint work with Jan Minac and Nguyen Duy Tan.\n\nRegister for the next session of ND NT Round 5 on February 15 using the following links:\n\nhttps://uwmadison. zoom.us/meeting/register/tJMkc-2hqDojG9yua5n-EkWxonnHyGeeMJkJ\n LOCATION:https://researchseminars.org/talk/POINT/39/ END:VEVENT BEGIN:VEVENT SUMMARY:James Upton (University of California San Diego) DTSTART:20220215T220000Z DTEND:20220215T223000Z DTSTAMP:20260422T225755Z UID:POINT/40 DESCRIPTION:Title: N ewton Polygons of Artin-Schreier Coverings Curves\nby James Upton (Uni versity of California San Diego) as part of POINT: New Developments in Num ber Theory\n\n\nAbstract\nLet $X$ be a smooth\, affine\, geometrically con nected curve over a finite field of characteristic $p > 2$. Let $C/X$ be a finite Galois covering of degree p. A theorem of Kramer-Miller states tha t the p-adic Newton polygon NP($C$) is bounded below by a certain Hodge po lygon HP($C$) which is defined in terms of local monodromy invariants of $ C/X$. Our main result is a local criterion that is necessary and sufficien t for NP($C$) and HP($C$) to coincide. Time permitting\, we will discuss s ome further results concerning the interaction of these two polygons. This is joint work with Joe Kramer-Miller.\n\nRegister for the next session of NDNT Round 5 on February 15 using the following links:\n\nhttps://uwmadis on.zoom.us/meeting/register/tJMkc-2hqDojG9yua5n-EkWxonnHyGeeMJkJ\n LOCATION:https://researchseminars.org/talk/POINT/40/ END:VEVENT BEGIN:VEVENT SUMMARY:Yuki Yamamoto (The University of Tokyo) DTSTART:20220228T113000Z DTEND:20220228T120000Z DTSTAMP:20260422T225755Z UID:POINT/41 DESCRIPTION:Title: C omparing Bushnell-Kutzko and Sécherre's constructions of types for $\\mat hrm{GL}_{N}$ and its inner forms with Yu's construction\nby Yuki Yamam oto (The University of Tokyo) as part of POINT: New Developments in Number Theory\n\n\nAbstract\nLet $F$ be a non-archimedean local field\, $A$ be a central simple $F$-algebra\, and $G$ be the multiplicative group of $A$. To construct types for supercuspidal representations of $G$\, simple type s by Sécherre and Yu's construction are already known. In this talk\, we compare these constructions. In particular\, we show essentially tame su percuspidal representations of $G$ defined by Bushnell-Henniart are nothin g but tame supercuspidal representations defined by Yu. This is a joint w ork with Arnaud Mayeux.\n LOCATION:https://researchseminars.org/talk/POINT/41/ END:VEVENT BEGIN:VEVENT SUMMARY:Lasse Grimmelt (University of Oxford) DTSTART:20220228T120000Z DTEND:20220228T123000Z DTSTAMP:20260422T225755Z UID:POINT/42 DESCRIPTION:Title: G oldbach's Problem with Almost Twin Primes\nby Lasse Grimmelt (Universi ty of Oxford) as part of POINT: New Developments in Number Theory\n\n\nAbs tract\nThe Binary Goldbach and Twin Prime conjectures have shaped the deve lopment of Analytic Number Theory in the the last century in fundamental m anner. Among the strongest approximative results that we know are a power saving bound on the exceptional set of the first conjecture\, proved by Mo ntgomery and Vaughan\, and Chen’s result on almost Twin Primes. \n\nIn t his talk I present joint work with J. Teräväinen that aims to combine th e two just mentioned results. More precisely\, we show a power saving exce ptional set bound for sums of two primes $p_1\, p_2$ such that $p_1+2$ has at most 2\, $p_2+2$ at most 3 prime divisors. This improves previous resu lts of this type in both strength of saving and number of prime divisors o f the shifted primes. Our proof uses a wide range of techniques. I will gi ve a sketch of how they play together and what the limitations are.\n LOCATION:https://researchseminars.org/talk/POINT/42/ END:VEVENT BEGIN:VEVENT SUMMARY:Sarah Arpin (University of Colorado Boulder) DTSTART:20220321T150000Z DTEND:20220321T153000Z DTSTAMP:20260422T225755Z UID:POINT/43 DESCRIPTION:Title: A dding Level Structure to Supersingular Elliptic Curve Isogeny Graphs\n by Sarah Arpin (University of Colorado Boulder) as part of POINT: New Deve lopments in Number Theory\n\n\nAbstract\nSupersingular elliptic curves hav e seen a resurgence in the past decade with new post-quantum cryptographic applications. In this talk\, we will discover why and how these curves ar e used in new cryptographic protocol. Supersingular elliptic curve isogeny graphs can be endowed with additional level structure. We will look at th e level structure graphs and the corresponding picture in a quaternion alg ebra.\n LOCATION:https://researchseminars.org/talk/POINT/43/ END:VEVENT BEGIN:VEVENT SUMMARY:Weitong Wang (Harvard University) DTSTART:20220321T153000Z DTEND:20220321T160000Z DTSTAMP:20260422T225755Z UID:POINT/44 DESCRIPTION:Title: T he bad part of the class group\nby Weitong Wang (Harvard University) a s part of POINT: New Developments in Number Theory\n\n\nAbstract\nThe Cohe n-Lenstra-Martinet Heuristics provide us with basic ideas on the statistic al results class groups of number fields. We can apply the heuristics to t he $p$-Sylow subgroups of the class groups when $p$ does not divide the or der of the Galois group. In this talk\, however\, I will first present an algebraic result on the structure of the $p$-Sylow subgroups of the class groups when $p$ divides the Galois group. Then I will show that some of th ese primes can give us statistical results which show qualitative differen ce from primes not dividing the order of the Galois group.\n LOCATION:https://researchseminars.org/talk/POINT/44/ END:VEVENT BEGIN:VEVENT SUMMARY:Andrew Kobin (Emory University) DTSTART:20220404T213000Z DTEND:20220404T220000Z DTSTAMP:20260422T225755Z UID:POINT/45 DESCRIPTION:Title: Z eta functions and decomposition spaces\nby Andrew Kobin (Emory Univers ity) as part of POINT: New Developments in Number Theory\n\n\nAbstract\nZe ta functions show up everywhere in math these days. While some work in the past has brought homotopical methods into the theory of zeta functions\, there is in fact a lesser-known zeta function that is native to homotopy t heory. Namely\, every suitably finite decomposition space (aka 2-Segal spa ce) admits an abstract zeta function as an element of its incidence algebr a. In this talk\, I will show how many 'classical' zeta functions from num ber theory and algebraic geometry can be realized in this homotopical fram ework\, and briefly advertise some work in progress with Bogdan Krstic tow ards a motivic version of the above story.\n LOCATION:https://researchseminars.org/talk/POINT/45/ END:VEVENT BEGIN:VEVENT SUMMARY:Tyler Billingsley (St. Olaf College) DTSTART:20220404T220000Z DTEND:20220404T223000Z DTSTAMP:20260422T225755Z UID:POINT/46 DESCRIPTION:Title: R ecent Work on Specialization of Elliptic Surfaces\nby Tyler Billingsle y (St. Olaf College) as part of POINT: New Developments in Number Theory\n \n\nAbstract\nThe study of cubic equations in two variables with at least one rational solution\, i.e. the theory of elliptic curves\, is a central area of study in modern number theory. The properties of specialization of families of elliptic curves\, called elliptic surfaces\, is an area of cu rrent research\, in part because specialization was used by Elkies to prod uce the current record for the highest known Mordell-Weil rank of an ellip tic curve over $\\mathbb Q$. In this talk\, we will discuss a brief histor y of and some recent developments in working effectively with specializati on maps\, and in particular determining when they are injective.\n LOCATION:https://researchseminars.org/talk/POINT/46/ END:VEVENT BEGIN:VEVENT SUMMARY:Mar Curcó Iranzo (Utrecht University) DTSTART:20220419T183000Z DTEND:20220419T190000Z DTSTAMP:20260422T225755Z UID:POINT/47 DESCRIPTION:Title: R ational Torsion of Generalised Modular Jacobians\nby Mar Curcó Iranzo (Utrecht University) as part of POINT: New Developments in Number Theory\ n\n\nAbstract\nWe consider the generalized Jacobian $J_0(N)_m$ of the modu lar curve $X_0(N)$ of level $N$\, with respect to the modulus $m$ consisti ng of all cusps on the modular curve. When $N=p^(r)q^(s)$\, for $p$ and $q $ odd prime numbers\, we determine the group structure of the rational tor sion of the Jacobian $J_0(N)_m$ up to 2-primary\, $p$-primary and $q$-prim ary torsion. Our results extend known results for squarefree levels and fo r prime power levels. Our proofs use their techniques\, as well as results concerning the study of the rational points on the modular Jacobian and o f the rational divisor class group of $X_0(N)$.\n\nRegister here: https:// umanitoba.zoom.us/meeting/register/u5csf-CtqzstEtfTaoLl6L8gnaVIJnGVB49w\n LOCATION:https://researchseminars.org/talk/POINT/47/ END:VEVENT BEGIN:VEVENT SUMMARY:Kelly Emmrich (Colorado State University) DTSTART:20220419T190000Z DTEND:20220419T193000Z DTSTAMP:20260422T225755Z UID:POINT/48 DESCRIPTION:Title: N on-splitting of the Hilbert exact sequence via a principal version of the Chebotarev density theorem\nby Kelly Emmrich (Colorado State Universit y) as part of POINT: New Developments in Number Theory\n\n\nAbstract\nLet $K/k$ be a Galois extension of number fields and $C$ a conjugacy class of $\\text{Gal}(K/k)$. In this talk\, we will investigate the density of prim e ideals of $k$ which factor as the product of principal ideals in $K$ and have their associated Frobenius class equal to $C$. From this density we will determine a method for verifying the nonsplitting of the Hilbert exac t sequence.\n\nRegister here: https://umanitoba.zoom.us/meeting/register/u 5csf-CtqzstEtfTaoLl6L8gnaVIJnGVB49w\n LOCATION:https://researchseminars.org/talk/POINT/48/ END:VEVENT END:VCALENDAR