14 citations · 67 across the 20 of their papers we have counts for
22 papers
S-integrality for families of ordinary K3 surfaces and algebraicity theorems
Ruofan Jiang, Ananth N. Shankar
We prove an -integrality theorem for special divisors on GSpin Shimura varieties in positive characteristic. Let be a generically ordinary curve in such a Shimura variety no…
Powers of abelian varieties over not isogenous to a Jacobian
Olivier de Gaay Fortman, Ananth N. Shankar
We prove the existence of abelian varieties over with no power isogenous to a Jacobian. Moreover, given a positive integer , we prove the existence of…
Finiteness of function field-valued points on exceptional Shimura varieties
Benjamin Bakker, Ananth N. Shankar, Jacob Tsimerman
Let be a smooth curve over a finite field of characteristic . We prove that there are finitely many principally polarized abelian schemes of given dimension over …
-adic hyperbolicity for Shimura varieties and period images
Benjamin Bakker, Abhishek Oswal, Ananth N. Shankar +1
We prove that Shimura varieties and geometric period images satisfy a -adic extension property for large enough primes . More precisely, let $\mathsf{D}^{\times}\subset \math…
A characteristic p analogue of the André--Pink--Zannier conjecture
Yeuk Hay Joshua Lam, Ananth N. Shankar
We investigate the analogue of the André--Pink--Zannier conjecture in characteristic . Precisely, we prove it for ordinary function field-valued points with big monodromy, in Sh…
Integral canonical models of exceptional Shimura varieties
Benjamin Bakker, Ananth N Shankar, Jacob Tsimerman
We prove that Shimura varieties admit integral canonical models for sufficiently large primes. In the case of abelian-type Shimura varieties, this recovers work of Kisin-Kottwitz f…
-adic hyperbolicity for moduli spaces of abelian motives
Abhishek Oswal, Ananth N. Shankar, Xinwen Zhu +1
We prove that Shimura varieties of abelian type satisfy a -adic Borel-extension property over discretely valued fields. More precisely, let denote the rigid-analyti…
Just-likely intersections on Hilbert modular surfaces
Asvin G., Qiao He, Ananth N. Shankar
In this paper, we prove an intersection-theoretic result pertaining to curves in certain Hilbert modular surfaces in positive characteristic. Specifically, we show that given two a…
Finiteness of reductions of Hecke orbits
Mark Kisin, Yeuk Hay Joshua Lam, Ananth N. Shankar +1
We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive m…
Canonical Heights on Shimura Varieties and the André-Oort Conjecture
Jonathan Pila, Ananth N. Shankar, Jacob Tsimerman +2
The main purpose of this work is to prove the André-Oort conjecture in full generality.
Abelian varieties not isogenous to Jacobians over global fields
Ananth N. Shankar, Jacob Tsimerman
We prove the existence of abelian varieties not isogenous to Jacobians over characterstic function fields. Our methods involve studying the action of degree Hecke operators…
Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture
Davesh Maulik, Ananth N. Shankar, Yunqing Tang
Let be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve in characteristic . We prove that the geometric…
Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields
Ananth N. Shankar, Arul Shankar, Yunqing Tang +1
Given a K3 surface over a number field with potentially good reduction everywhere, we prove that the set of primes of where the geometric Picard rank jumps is infinite.…
Families of elliptic curves ordered by conductor
Ananth N. Shankar, Arul Shankar, Xiaoheng Wang
In this article, we study the family of elliptic curves , having good reduction at and , and whose -invariants are small. Within this set of elliptic curves…
Almost ordinary abelian varieties over finite fields
Abhishek Oswal, Ananth N. Shankar
We provide a characterization of almost ordinary abelian varieties over finite fields, and use this characterization to provide lower bounds for the sizes of some almost ordinary i…
Reductions of abelian surfaces over global function fields
Davesh Maulik, Ananth N. Shankar, Yunqing Tang
Let be a non-isotrivial ordinary abelian surface over a global function field with good reduction everywhere. Suppose that does not have real multiplication by any real qua…
The rank two -curvature conjecture on generic curves
Anand Patel, Ananth N. Shankar, Junho Peter Whang
We prove the -curvature conjecture for rank two vector bundles with connection on generic curves, by combining deformation techniques for families of varieties and topological a…
Exceptional splitting of reductions of abelian surfaces
Ananth N. Shankar, Yunqing Tang
Heuristics based on the Sato--Tate conjecture suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for ab…
Serre-Tate theory for Shimura varieties of Hodge type
Ananth N. Shankar, Rong Zhou
We study the formal neighbourhood of a point in -ordinary locus of an integral model of a Hodge type Shimura variety. We show that this formal neighbourhood has a structure of a…
2-Selmer groups of hyperelliptic curves with two marked points
Ananth N. Shankar
We consider the family of hyperelliptic curves over $\Q$ of fixed genus along with a marked rational Weierstrass point and a marked rational non-Weierstrass point. When these curve…
The -curvature conjecture and monodromy about simple closed loops
Ananth N. Shankar
The Grothendieck-Katz -curvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a vari…
Unlikely Intersections in Finite Characteristic
Ananth N Shankar, Jacob Tsimerman
We present a heuristic argument based on Honda-Tate theory against many conjectures in `unlikely intersections' over the algebraic closure of a finite field; notably, we conjecture…