Algebraic and Arithmetic Geometry

“Algebraic






2021 Joint Mathematics Meetings
Washington, DC
January 6–9, 2021
Walter E. Washington Convention Center










Organizers:

“Marc
Marc Hindry
Institut de Mathématiques de Jussieu-PRG
UFR mathématiques de l'Université Paris 7 Denis Diderot
Bâtiment Sophie Germain
5 rue Thomas Mann
F-75205 Paris CEDEX 13
“tony
Tony Shaska
Department of Mathematics
546 Mathematics Science Center
Oakland University,
Rochester, MI. 48309, USA.

Overview

Diophantine equations are systems of polynomial equations solved over integers or rational numbers. Diophantine geometry is the study of Diophantine equations using ideas and techniques from algebraic geometry. It is one of the oldest subjects of mathematics and the most popular part of number theory connecting it to algebraic geometry.

The goal of this session is to explore recent developments in the theory of arithmetic geometry and with special focus on curves and Jacobian varieties. We intend to bring together mathematicians, working on this area of research, from the USA and from Europe encouraging further cooperation and discussion. We will especially encourage younger mathematicians and graduate students and newcomers in the area.

The session will focus on the following topics, but we will be open and welcoming to talks which do not fall in the list of topics below.

Talks

  1. Title: Local-global principles over semi-global fields.

    Authors: Jean-Louis Colliot-Thelene, David Harbater, Julia Hartmann, Daniel Krashen, R. Parimala, V. Suresh

    Abstract: Local-global principles are central in the study of algebraic objects such as quadratic forms and central simple algebras, over global fields. Such principles can typically be reformulated in terms of the existence of rational points on homogeneous spaces under algebraic groups, especially on principal homogeneous spaces (torsors). This talk will discuss analogous principles over semi-global fields, i.e. function fields of curves over complete discretely valued fields. In a number of cases we show that local-global principles hold in this situation; in other cases we compute the obstruction, which is analogous to the Tate-Shafarevich group.

  2. Title: Dedekind Polylogarithms

    Authors: Ivan Horozov, Pavel Sokolov

    Abstract: In this talk, first I will present some results on multiple Dedekind zeta values (a generalizations of multiple zeta values to number fields) and their relations to algebraic geometry and motives. I will also mention more resent progress (joint work with Pavel Sokolov) on relations between multiple Dedekind zeta values, abelian Artin L-functions and mixed Tate motives via Dedekind polylogarithms.

  3. Title: On codes of Reed-Muller type defined over higher dimensional scrolls.

    Authors: Cicero Carvalho

    Abstract: In 1988 Lachaud introduced the class of projective Reed-Muller codes, defined by evaluating the space of homogeneous polynomials of a fixed degree on the points of a projective space over a finite field. Since then other classes of codes have been obtained by replacing the points of the projective space by the points of a projective variety. In this talk we would like to present results on a class of codes obtained in this way, where the projective variety is a higher dimensional normal scroll. In a joint work with Victor G.L. Neumann, Xavier Ramirez-Mondragon and Horacio Tapia-Recillas we have determined a formula for the dimension of these codes, and the exact value of the minimum distance in a special case.

  4. Title: Eigenvalues of Frobenius endomorphisms of Abelian varieties over finite fields.

    Authors: Yuri G. Zarhin

    Abstract: Let X be a positive-dimensional abelian variety over a finite field of characteristic p, F rX the Frobenius endomorphism of X, and PX[t] the characteristic polynomial of F rX, which is a monic polynomial with integer coefficients. Its roots are eigenvalues of F rX with respect to its action on the `-adic Tate module of X (for all primes ` 6= p). We discuss multiplicative relations between eigenvalues of F rX. As an application we obtain the following result. Theorem. Let g be a positive integer. Then there exists a positive integer N = N(g) that enjoys the following properties. Let X be a g-dimensional abelian variety over a finite field k such that there exist a positive integer n and a prime l 6= char(k) such that the self-product Xn of X carries an exotic l-adic Tate class. Then the self-product XN of X carries an exotic `-adic Tate class for all primes ` 6= char(k). Recall that a Tate class is called exotic if it cannot be presented as a linear combination of products of divisor classes.

  5. Title: Recent progress in algebraic hyperbolicity

    Authors: Izzet Coskun*, Eric Riedl

    Abstract: I will give a survey of recent work with Eric Riedl on algebraic hyperbolicity. I will discuss our proof of the algebraic hyperbolicity of the very general quintic surface. I will explain a classification of 1-clustered families in the Grassmannian and give applications to Lang-type conjectures on hypersurfaces.

  6. Title: Nakajima quiver varieties and irreducible components of Springer fibers.

    Authors: Mee Seong Im*, Chun-Ju Lai, Arik Wilbert

    Abstract: Springer fibers and Nakajima quiver varieties are amongst the most important objects in geometric representation theory. While Springer fibers can be used to geometrically construct and classify irreducible representations of Weyl groups, Nakajima quiver varieties play a key role in the geometric representation theory of Kac–Moody Lie algebras. I will begin by first recalling some background on the objects of interest mentioned above. I will then connect Springer fibers and quiver varieties by realizing the irreducible components of two-row Springer fibers inside a suitable Nakajima quiver variety and describing the resulting subvariety in terms of explicit quiver representations. Next, consider certain fixed-point subvarieties of these quiver varieties, which were studied by Henderson–Licata and Li with the goal of developing the geometric representation theory for certain coideal subalgebras. By applying this machinery, I will give an explicit algebraic description of the irreducible components of all two-row Springer fibers for classical types, thereby generalizing results of Fung and Stroppel–Webster in type A.

  7. Title: An isogeny between certain K3 surfaces of Picard rank 18.

    Authors: Andreas Malmendier, Noah Braeger*

    Abstract: We construct a geometric two-isogeny between two algebraic K3 surfaces obtained as quartic projective hypersurfaces explicitly. One is the Kummer surfaces of the Jacobian of a genus-two curve with elliptic involution, the other is the Shioda-Inose surface associated with the Kummer surface of two non-isogeneous elliptic curves. The construction uses a classical result due to Jacobi and Morrison on Nikulin involutions.

  8. Title: Symplectic automorphisms of order 3 on K3 surfaces: action on cohomology and related "Shioda Inose structures".

    Authors: Alice Garbagnati

    Abstract: The symplectic automorphisms of finite order on a K3 surface induce an essentially unique isometry on the second cohomology group of the K3 surface, i.e. the lattice ΛK3 ' U3 ⊕ E28, as proved by Nikulin at the end of Seventies. In the particular case of the involutions this isometry is very well known and described by Morrison: it switches the two copies of E8 and acts as the identity on the three copies of U. By using this, one is able to state several interesting results: the existence of the Shioda Inose structures; the description of the relations between the Picard groups of K3 surfaces with a symplectic involution and the one of their quotient; the presence of infinite towers of isogenous K3 surfaces. The aim of this talk is to present similar results for symplectic automorphisms of order 3 on K3 surfaces. We will describe explicitly the action of the isometry induced by such an automorphism on the second cohomology group of a K3 surface (as Morrison did for the involutions) by giving a different basis for the ΛK3 and then we will deduce results analogue to the ones mentioned in the case of the involutions; for example we will generalize the Shioda-Inose construction to our case. The talk is based on a joint project with Y. Prieto.

  9. Title: Perfectoid spaces arising from arithmetic jet spaces.

    Authors: Alexandru Buium*, Lance E. Miller

    Abstract: Using arithmetic jet spaces we attach perfectoid spaces to smooth schemes and we attach morphisms of perfectoid spaces to δ-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable δ-morphisms appearing in the theory such as the δ-characters of elliptic curves and the δ-period maps on modular curves.

  10. Title: A Classification of Isogeny-Torsion Graphs over $\mathbb{Q}$.

    Authors: Garen Chiloyan*, \'Alvaro Lozano-Robledo

    Abstract: An isogeny graph is a nice visualization of the isogeny class of an elliptic curve. A theorem 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 classify what torsion subgroups over Q can occur in each vertex of a given isogeny graph of elliptic curves defined over the rationals.

  11. Title: Classifying Jacobian Elliptic Fibrations on a Family of K3 Surfaces of Picard Rank 16.

    Authors: A Clingher A Malmendier Thomas Hill*,

    Abstract: We study a special family of K3 surfaces polarized by the rank-sixteen lattice N = H ⊕ E7(−1) ⊕ E7(−1). Isomorphism classes of Jacobian elliptic fibrations on such a N-polarized K3 surface X are in one-to-one correspondence with primitive lattice embeddings H ,→ N. Our lattice theoretic analysis proves there are exactly four (non isomorphic) primitive lattice embeddings. Therefore, X carries (up to automorphism) four Jacobian elliptic fibrations. Furthermore, we explicitly construct the Weierstrass models for each of these fibrations. We also apply our construction to string theory. Our construction provides a geometric interpretation for the Ftheory/heterotic string duality in eight dimensions with two non-trivial Wilson lines. The moduli space of the family N-polarized K3 surfaces provides a new example where the partial higgsing of the heterotic gauge algebra g = e8 ⊕ e8 or g = so(32) for the associated low energy effective eight-dimensional supergravity theory has inequivalent Coulomb branches and no charged matter fields for the corresponding F-theory model.

  12. Title: Every finite group challenges extending Falting's Theorem.

    Authors: Michael David Fried

    Abstract: Consider finite group G; ` a prime dividing |G| (|G has no Z/` quotient); and C = {C1 . . . Cr} any r ≥ 4 conjugacy classes of order prime to ` elements. Ex: G = A5, C is 4 repetes of the 3-cycle conjugacy class, and ` = 2. For (G, C, `), M0 ∈ I, |I| < ∞ gives a Z` [G] lattice LM0 as kernel of an `-Frattini cover G˜M0 → G =⇒ a moduli space series · · ·→H(G, C, `, L)k → · · ·→ H(. . .)1→ H(G, C, `, L)0 → Jr. Terms are quadi-projective varieties. When r = 4 all are upper half plane quotients; J4 is the classical j-line, minus ∞. Only for G “close to” dihedral (r = 4) are these modular curves. Main Conjecture: Let K be any number field. For k large, projective normalization of H(G, C, `, L)k has general type, and H(G, C, `, L)k has no K points. For r = 4, there are two proofs (myself/Cadoret-Tamagawa). We compare these results and how even this presents an unproven challenge to extending Falting’s Theorem. (

  13. Title: Superelliptic curves with many automorphisms and CM Jacobians.

    Authors: Andrew Obus*, Tony Shaska

    Abstract: Let C be a smooth, projective, genus g ≥ 2 curve, defined over C. Then C has many automorphisms if its corresponding moduli point p ∈ Mg has a neighborhood U in the complex topology, such that all curves corresponding to points in U \ {p} have strictly fewer automorphisms than C. We compute completely the list of superelliptic curves having many automorphisms. For each of these curves, we determine whether its Jacobian has complex multiplication. As a consequence, we prove the converse of Streit’s complex multiplication criterion for these curves.

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Submitting Abstracts

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Algebraic and Arithmetic Geometry

AMS Joint Meeting

For more information on registration, housing, etc, please visit the AMS website, at 2021 Joint Mathematics Meetings