Algebraic and Arithmetic Geometry
2021 Joint Mathematics Meetings
Washington, DC
January 6–9, 2021
Walter E. Washington Convention Center
Organizers:


Marc Hindry
Institut de Mathématiques de JussieuPRG
UFR mathématiques de l'Université Paris 7 Denis Diderot
Bâtiment Sophie Germain
5 rue Thomas Mann
F75205 Paris CEDEX 13

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
 Title: Localglobal principles over semiglobal fields.
Authors: JeanLouis ColliotThelene, David Harbater, Julia Hartmann, Daniel Krashen, R. Parimala, V. Suresh
Abstract: Localglobal 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 semiglobal fields, i.e. function fields of curves over complete discretely valued fields. In a number of cases
we show that localglobal principles hold in this situation; in other cases we compute the obstruction, which is analogous
to the TateShafarevich group.
 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 Lfunctions and mixed Tate
motives via Dedekind polylogarithms.
 Title: On codes of ReedMuller type defined over higher dimensional scrolls.
Authors: Cicero Carvalho
Abstract: In 1988 Lachaud introduced the class of projective ReedMuller 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 RamirezMondragon and Horacio TapiaRecillas we
have determined a formula for the dimension of these codes, and the exact value of the minimum distance in a special
case.
 Title: Eigenvalues of Frobenius endomorphisms of Abelian varieties over finite fields.
Authors: Yuri G. Zarhin
Abstract: Let X be a positivedimensional 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 gdimensional 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 selfproduct Xn of X carries an exotic ladic Tate class.
Then the selfproduct 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.
 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 1clustered families in the Grassmannian
and give applications to Langtype conjectures on hypersurfaces.
 Title: Nakajima quiver varieties and irreducible components of Springer fibers.
Authors: Mee Seong Im*, ChunJu 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 tworow Springer fibers inside a suitable Nakajima
quiver variety and describing the resulting subvariety in terms of explicit quiver representations.
Next, consider certain fixedpoint 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 tworow Springer fibers for
classical types, thereby generalizing results of Fung and Stroppel–Webster in type A.
 Title: An isogeny between certain K3 surfaces of Picard rank 18.
Authors: Andreas Malmendier, Noah Braeger*
Abstract: We construct a geometric twoisogeny between two algebraic K3 surfaces obtained as quartic projective hypersurfaces
explicitly. One is the Kummer surfaces of the Jacobian of a genustwo curve with elliptic involution, the other is the
ShiodaInose surface associated with the Kummer surface of two nonisogeneous elliptic curves. The construction uses a
classical result due to Jacobi and Morrison on Nikulin involutions.
 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 ShiodaInose construction to our
case. The talk is based on a joint project with Y. Prieto.
 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 δmorphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined
by some of the remarkable δmorphisms appearing in the theory such as the δcharacters of elliptic curves and the δperiod
maps on modular curves.
 Title: A Classification of IsogenyTorsion Graphs over $\mathbb{Q}$.
Authors: Garen Chiloyan*, \'Alvaro LozanoRobledo
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.
 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 ranksixteen lattice N = H ⊕ E7(−1) ⊕ E7(−1). Isomorphism
classes of Jacobian elliptic fibrations on such a Npolarized K3 surface X are in onetoone 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 nontrivial Wilson lines. The moduli space of the family
Npolarized 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 eightdimensional supergravity theory has inequivalent Coulomb
branches and no charged matter fields for the corresponding Ftheory model.
 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 3cycle 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 quadiprojective varieties. When r = 4 all are upper half plane quotients; J4 is the classical jline, 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/CadoretTamagawa). We compare these results and how even this presents
an unproven challenge to extending Falting’s Theorem. (
 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
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2021 Joint Mathematics Meetings