BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Richard Hain (Duke University) DTSTART;VALUE=DATE-TIME:20210510T140000Z DTEND;VALUE=DATE-TIME:20210510T150000Z DTSTAMP;VALUE=DATE-TIME:20240328T084015Z UID:DioGal2021/1 DESCRIPTION:Title: Weighted completion of Galois groups and rational points\nby Richar d Hain (Duke University) as part of Rational Points and Galois Representat ions\n\n\nAbstract\nThe goal of this talk is to explain how one can use we ighted\nand relative completion to generalize some recent results (arXiv:2 010.07331) of Li\, Litt\,\nSalter and Srinivasan. In particular\, we will explain how algebraic\ncompletions of mapping class groups and their arith metic analogues can\nbe used to give examples\, for each $g > 3$\, $n \\ge 0$\, $r > 0$\, of a smooth\nprojective curve of genus $g > 3$ over a fini tely generated field $K$ of\nchar 0 where $\\#C(K) = n$\, $\\mathrm{Pic}^1 (C)(K)$ is non-empty and $\\mathrm{Pic}^0(C)(K)$\ncontains a free abelian subgroup of rank $n + r - 1$.\n\nThe talk will begin with a review of how one uses weighted (unipotent)\ncompletion of the Galois group of the funct ion field of a moduli\nspaces of curves (plus other structure) to study ra tional points of\nthe the universal curve over its generic point. If there is sufficient\ntime\, I will explain how this leads to a theory of charac teristic\nclasses of rational points.\n\nREFERENCES:\n\n[HM2003] R. Hain\, M. Matsumoto: Weighted completion of Galois groups\n and Galois a ctions on the fundamental group of P^1-{0\,1\,infty}.\n Compositio Math. 139 (2003)\, 119--167.\n arXiv:math/0006158\n\n[H2011] R. Hain: Rational points of universal curves\, J. Amer. Math. Soc. 24 (2011)\ ,\n 709--769.\n https://www.ams.org/journals/jams/2011-24- 03/S0894-0347-2011-00693-0/home.html\n\n[LLSP] W. Li\, D. Litt\, N. Salt er\, P. Srinivasan: Surface bundles and the section\n conjecture\, arXiv:2010.07331.\n\n[W2019] T. Watanabe: Rational points of universal c urves in positive characteristics\,\n Trans. Amer. Math. Soc. 372 (2019)\, 7639--7676.\n https://www.ams.org/journals/tran/2019-372- 11/S0002-9947-2019-07842-X/\n LOCATION:https://researchseminars.org/talk/DioGal2021/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Padmavathi Srinivasan (University of Georgia) DTSTART;VALUE=DATE-TIME:20210510T153000Z DTEND;VALUE=DATE-TIME:20210510T163000Z DTSTAMP;VALUE=DATE-TIME:20240328T084015Z UID:DioGal2021/2 DESCRIPTION:Title: Towards a unified theory of canonical heights on abelian varieties\ nby Padmavathi Srinivasan (University of Georgia) as part of Rational Poin ts and Galois Representations\n\n\nAbstract\np-adic heights have been a ri ch source of explicit functions vanishing on rational points on a curve. I n this talk\, we will outline a new construction of canonical p-adic heigh ts on abelian varieties from p-adic adelic metrics\, using p-adic Arakelov theory developed by Besser. This construction closely mirrors Zhang's con struction of canonical real valued heights from real-valued adelic metrics . We will use this new construction to give direct explanations (avoiding p-adic Hodge theory) of the key properties of height pairings needed for t he quadratic Chabauty method for rational points. This is joint work in pr ogress with Amnon Besser and Steffen Mueller.\n LOCATION:https://researchseminars.org/talk/DioGal2021/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Kirsten Wickelgren (Duke University) DTSTART;VALUE=DATE-TIME:20210510T170000Z DTEND;VALUE=DATE-TIME:20210510T180000Z DTSTAMP;VALUE=DATE-TIME:20240328T084015Z UID:DioGal2021/3 DESCRIPTION:Title: Colloquium Presentation: zeta functions and a quadratic enrichment\ nby Kirsten Wickelgren (Duke University) as part of Rational Points and Ga lois Representations\n\n\nAbstract\nThe beautiful Weil conjectures connect the Betti numbers of a complex variety whose defining equations can be re duced mod p to the number of solutions mod p. We will discuss these connec tions\, introduce A1-homotopy theory\, and an analogue in A1-homotopy theo ry. The new work in this talk is joint with Margaret Bilu\, Wei Ho\, Padma vathi Srinivasan\, and Isabel Vogt.\n LOCATION:https://researchseminars.org/talk/DioGal2021/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Francesca Bianchi (University of Groningen) DTSTART;VALUE=DATE-TIME:20210511T133000Z DTEND;VALUE=DATE-TIME:20210511T143000Z DTSTAMP;VALUE=DATE-TIME:20240328T084015Z UID:DioGal2021/4 DESCRIPTION:Title: p-adic heights on Jacobians of genus 2 curves and applications\nby Francesca Bianchi (University of Groningen) as part of Rational Points and Galois Representations\n\n\nAbstract\nWe describe an algorithmic construc tion of a p-adic height on the Jacobian of a genus 2 curve over the ration als (here p is not necessarily of good reduction). In particular\, the foc us will be on the local component at p of the height\, which is defined in terms of some p-adic sigma/theta functions.\n\nThese local heights differ from those in the Coleman--Gross construction in a crucial way\; neverthe less\, in some cases we can prove a suitable comparison theorem. Thus\, we can use our heights as an alternative to the Coleman--Gross heights in so me instances of the quadratic Chabauty method. The application given in th is talk concerns the rational points on some quite special genus 4 hyperel liptic curves.\n\nThis talk is partly based on joint work with Enis Kaya a nd Steffen Müller.\n LOCATION:https://researchseminars.org/talk/DioGal2021/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Betts (Harvard University) DTSTART;VALUE=DATE-TIME:20210511T145000Z DTEND;VALUE=DATE-TIME:20210511T155000Z DTSTAMP;VALUE=DATE-TIME:20240328T084015Z UID:DioGal2021/5 DESCRIPTION:Title: Weights of Coleman functions and effective Chabauty--Kim\nby Alexan der Betts (Harvard University) as part of Rational Points and Galois Repre sentations\n\n\nAbstract\nThe Chabauty--Kim method is a technique for stud ying the rational points on a curve X using motivic properties of quotient s U of the fundamental group of X. For specific quotients U\, the method h as been made effective in work of Coleman and later by Balakrishnan--Dogra \, in the sense that it provides an explicit upper bound on the number of rational points. In this talk\, I will discuss a recent project in which I extend these effective results to all quotients U\, and give some applica tions (joint work with David Corwin\, in progress) towards uniformity resu lts for higher genus curves. A significant part of the proof\, which I wil l discuss in more detail\, lies in defining a notion of "weight" for Colem an analytic functions\, and showing\, following arguments of Balakrishnan- -Dogra\, that the number of zeroes of a non-zero Coleman analytic function can be bounded in terms of its weight.\n LOCATION:https://researchseminars.org/talk/DioGal2021/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Benjamin Matschke (Boston University) DTSTART;VALUE=DATE-TIME:20210511T161000Z DTEND;VALUE=DATE-TIME:20210511T163000Z DTSTAMP;VALUE=DATE-TIME:20240328T084015Z UID:DioGal2021/6 DESCRIPTION:Title: A general S-unit equation solver and tables of elliptic curves over num ber fields\nby Benjamin Matschke (Boston University) as part of Ration al Points and Galois Representations\n\n\nAbstract\nIn this talk we presen t work in progress on a new highly optimized\nsolver for general and const raint S-unit equations over number fields.\nIt has diophantine application s including asymptotic Fermat theorems\,\nSiegel's method for computing in tegral points\, and most strikingly for\ncomputing large tables of ellipti c curves over number fields with good\nreduction outside given sets of pri mes S. For the latter\, we improved\non the method of Koutsianas (Parshin\ , Shafarevich\, Elkies).\n LOCATION:https://researchseminars.org/talk/DioGal2021/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Jackson Morrow (Université de Montréal) DTSTART;VALUE=DATE-TIME:20210511T174000Z DTEND;VALUE=DATE-TIME:20210511T184000Z DTSTAMP;VALUE=DATE-TIME:20240328T084015Z UID:DioGal2021/7 DESCRIPTION:Title: Progress on Mazur's Program B -- a horizontal perspective\nby Jacks on Morrow (Université de Montréal) as part of Rational Points and Galois Representations\n\n\nAbstract\nIn this talk\, I will discuss recent progr ess on "Mazur's Program B" --- the problem of classifying all possibilitie s for the image of Galois for an elliptic curve over $\\mathbb{Q}$. I will focus on the horizontal perspective of Mazur's Program B\, which strives to classify the composite (non-prime power) images of Galois for an ellipt ic curve over $\\mathbb{Q}$. In particular\, I will introduce the notion o f an entanglement of division fields\, give a group theoretic characteriza tion of an entanglement\, and describe two sets of joint work. The first i s with Harris Daniels where we classify all infinite families of elliptic curves over $\\mathbb{Q}$ which have an "unexplained" entanglement between their $p$ and $q$ division fields where $p\,q$ are distinct primes\, and the second is with Harris Daniels and Álvaro Lozano-Robledo where we prov e several results on elliptic curves (and more generally\, principally pol arized abelian varieties) over $\\mathbb{Q}$ when the entanglement occurs over an abelian extension.\n LOCATION:https://researchseminars.org/talk/DioGal2021/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Abbey Bourdon (Wake Forest University) DTSTART;VALUE=DATE-TIME:20210512T140000Z DTEND;VALUE=DATE-TIME:20210512T150000Z DTSTAMP;VALUE=DATE-TIME:20240328T084015Z UID:DioGal2021/8 DESCRIPTION:Title: Families of Sporadic Points on Modular Curves\nby Abbey Bourdon (Wa ke Forest University) as part of Rational Points and Galois Representation s\n\n\nAbstract\nA closed point $x$ on a curve $C$ is sporadic if there ar e only finitely many points of degree at most deg($x$). In the case where $C$ is the modular curve $X_1(N)$\, a non-cuspidal sporadic point correspo nds to an elliptic curve with a point of order $N$ defined over a number f ield of unusually low degree. In this talk\, we will focus on sporadic poi nts arising from $\\mathbb{Q}$-curves\, which are elliptic curves isogenou s to their Galois conjugates. In particular\, our investigations are inspi red by the following question: Are there only finitely many non-CM $\\math bb{Q}$-curves which produce sporadic points on any modular curve of the form $X_1(N)$? I will show that an affirmative answer to this questio n would imply Serre's Uniformity Conjecture and discuss partial progress i n the case of sporadic points of odd degree. This is joint work with Filip Najman.\n LOCATION:https://researchseminars.org/talk/DioGal2021/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Will Sawin (Columbia University) DTSTART;VALUE=DATE-TIME:20210512T153000Z DTEND;VALUE=DATE-TIME:20210512T163000Z DTSTAMP;VALUE=DATE-TIME:20240328T084015Z UID:DioGal2021/9 DESCRIPTION:Title: The Shafarevich conjecture for hypersurfaces in abelian varieties\n by Will Sawin (Columbia University) as part of Rational Points and Galois Representations\n\n\nAbstract\nFaltings proved the statement\, previously conjectured by \nShafarevich\, that there are finitely many abelian variet ies of \ndimension $n$\, defined over a fixed number field\, with good red uction \noutside a fixed finite set of primes\, up to isomorphism. In join t work \nwith Brian Lawrence\, we prove an analogous finiteness statement for \nhypersurfaces in a fixed abelian variety with good reduction outside a \nfinite set of primes. I will give an introduction to some of the idea s \nin the proof\, which builds on $p$-adic Hodge theory techniques from w ork \nof Lawrence and Venkatesh as well as a little-known area of algebrai c \ngeometry\n LOCATION:https://researchseminars.org/talk/DioGal2021/9/ END:VEVENT BEGIN:VEVENT SUMMARY:David Zureick-Brown (moderator) (Emory University) DTSTART;VALUE=DATE-TIME:20210512T170000Z DTEND;VALUE=DATE-TIME:20210512T180000Z DTSTAMP;VALUE=DATE-TIME:20240328T084015Z UID:DioGal2021/10 DESCRIPTION:Title: Problem discussion session\nby David Zureick-Brown (moderator) (Em ory University) as part of Rational Points and Galois Representations\n\n\ nAbstract\nThis problem discussion session features advance contrib utions (pdf) from \n