Eyal Z. Goren
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Eyal Z. Goren has a total of 10 co-authors. Here are the top co-authors by number of shared publications:
- Ehud de Shalit
Hebrew University of Jerusalem, Israel
7 shared publications - Benjamin Howard
Department of Mathematics, Boston College, Chestnut Hill, USA
3 shared publications - Ehud de Shalit
Hebrew University
3 shared publications - Fabrizio Andreatta
Università degli Studi di Milano, Milan, Italy
3 shared publications - Keerthi Madapusi Pera
Department of Mathematics, Boston College, Chestnut Hill, USA
3 shared publications - Payman L. Kassaei
Department of Mathematics , King’s College London, WC2R 2LS, UK
2 shared publications
Latest Publications
- Open Access
- 11 November 2024
- Published by Canadian Mathematical Society in Canadian Journal of Mathematics
- Vol. 77 (6) , 1938-1965
On elements of prescribed norm in maximal orders of a quaternion algebra
Let $\mathcal {O}$ be a maximal order in the quaternion algebra over $\mathbb Q$ ramified at p and $\infty $ . We prove two theorems that allow us to recover the structure of $\mathcal {O}$ from limited information. The first says that for any infinite set S of integers coprime to p , $\mathcal {O}$ is spanned as a ${\mathbb {Z}}$ -module by elements with norm in S . The second says that $\mathcal {O}$ is determined up to isomorphism by its theta function.- 11 November 2024
- Published by Canadian Mathematical Society in Canadian Journal of Mathematics
- Vol. 77 (6) , 1938-1965
Open AccessOn elements of prescribed norm in maximal orders of a quaternion algebra
Let $\mathcal {O}$ be a maximal order in the quaternion algebra over $\mathbb Q$ ramified at p and $\infty $ . We prove two theorems that allow us to recover the structure of $\mathcal {O}$ from limited information. The first says that for any infinite set S of integers coprime to p , $\mathcal {O}$ is spanned as a ${\mathbb {Z}}$ -module by elements with norm in S . The second says that $\mathcal {O}$ is determined up to isomorphism by its theta function. - 5 September 2023
- Published by American Mathematical Society (AMS) in Journal of Algebraic Geometry
- Vol. 33 (3) , 401-454
Foliations on Shimura varieties in positive characteristic
This paper is a continuation of a paper by de Shalit and Goren from 2018. We study foliations of two types on Shimura varieties S S in characteristic p p . The first, which we call tautological foliations , are defined on Hilbert modular varieties, and lift to characteristic 0 0 . The second, the V V -foliations , are defined on unitary Shimura varieties in characteristic p p only, and generalize the foliations studied by us before, when the CM field in question was quadratic imaginary. We determine when these foliations are p p -closed, and the locus where they are smooth. Where not smooth, we construct a successive blowup of our Shimura variety to which they extend as smooth foliations. We discuss some integral varieties of the foliations. We relate the quotient of S S by the foliation to a purely inseparable map from a certain component of another Shimura variety of the same type, with parahoric level structure at p p , to S . S.
- 5 September 2023
- Published by American Mathematical Society (AMS) in Journal of Algebraic Geometry
- Vol. 33 (3) , 401-454
Foliations on Shimura varieties in positive characteristic
This paper is a continuation of a paper by de Shalit and Goren from 2018. We study foliations of two types on Shimura varieties S S in characteristic p p . The first, which we call tautological foliations , are defined on Hilbert modular varieties, and lift to characteristic 0 0 . The second, the V V -foliations , are defined on unitary Shimura varieties in characteristic p p only, and generalize the foliations studied by us before, when the CM field in question was quadratic imaginary. We determine when these foliations are p p -closed, and the locus where they are smooth. Where not smooth, we construct a successive blowup of our Shimura variety to which they extend as smooth foliations. We discuss some integral varieties of the foliations. We relate the quotient of S S by the foliation to a purely inseparable map from a certain component of another Shimura variety of the same type, with parahoric level structure at p p , to S . S.
- Open Access
Foliations on Shimura varieties in positive characteristic
This paper is a continuation of [G-dS1]. We study foliations of two types on Shimura varieties $S$ in characteristic $p$. The first, which we call "tautological foliations", are defined on Hilbert modular varieties, and lift to characteristic $0$. The second, the "$V$-foliations", are defined on unitary Shimura varieties in characteristic $p$ only, and generalize the foliations studied by us before, when the CM field in question was quadratic imaginary. We determine when these foliations are $p$-closed, and the locus where they are smooth. Where not smooth, we construct a "successive blow up" of our Shimura variety to which they extend as smooth foliations. We discuss some integral varieties of the foliations. We relate the quotient of $S$ by the foliation to a purely inseparable map from a certain component of another Shimura variety of the same type, with parahoric level structure at $p$, to $S.$Open AccessFoliations on Shimura varieties in positive characteristic
This paper is a continuation of [G-dS1]. We study foliations of two types on Shimura varieties $S$ in characteristic $p$. The first, which we call "tautological foliations", are defined on Hilbert modular varieties, and lift to characteristic $0$. The second, the "$V$-foliations", are defined on unitary Shimura varieties in characteristic $p$ only, and generalize the foliations studied by us before, when the CM field in question was quadratic imaginary. We determine when these foliations are $p$-closed, and the locus where they are smooth. Where not smooth, we construct a "successive blow up" of our Shimura variety to which they extend as smooth foliations. We discuss some integral varieties of the foliations. We relate the quotient of $S$ by the foliation to a purely inseparable map from a certain component of another Shimura variety of the same type, with parahoric level structure at $p$, to $S.$ - Open Access
$p$-adic Dynamics of Hecke Operators on Modular Curves
In this paper we study the $p$-adic dynamics of prime-to-$p$ Hecke operators on the set of points of modular curves in both cases of good ordinary and supersingular reduction. We pay special attention to the dynamics on the set of CM points. In the case of ordinary reduction we employ the Serre-Tate coordinates, while in the case of supersingular reduction, we use a parameter on the deformation space of the unique formal group of height $2$ over $\overline{\mathbb{F}}_p$, and take advantage of the Gross-Hopkins period map.Open Access$p$-adic Dynamics of Hecke Operators on Modular Curves
In this paper we study the $p$-adic dynamics of prime-to-$p$ Hecke operators on the set of points of modular curves in both cases of good ordinary and supersingular reduction. We pay special attention to the dynamics on the set of CM points. In the case of ordinary reduction we employ the Serre-Tate coordinates, while in the case of supersingular reduction, we use a parameter on the deformation space of the unique formal group of height $2$ over $\overline{\mathbb{F}}_p$, and take advantage of the Gross-Hopkins period map. - Open Access
- 9 October 2019
- Published by Mathematical Sciences Publishers in Algebra & Number Theory
- Vol. 13 (8) , 1829-1877
Theta operators on unitary Shimura varieties
We define a theta operator on p-adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which p is inert. We study its effect on Fourier-Jacobi expansions and prove that it extends holomorphically beyond the µ-ordinary locus, when applied to scalar-valued forms.
- 9 October 2019
- Published by Mathematical Sciences Publishers in Algebra & Number Theory
- Vol. 13 (8) , 1829-1877
Open AccessTheta operators on unitary Shimura varieties
We define a theta operator on p-adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which p is inert. We study its effect on Fourier-Jacobi expansions and prove that it extends holomorphically beyond the µ-ordinary locus, when applied to scalar-valued forms.
- Open Access
Theta operators on unitary Shimura varieties
We define a theta operator on p-adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which p is inert. We study its effect on Fourier-Jacobi expansions and prove that it extends holomorphically beyond the {\mu}-ordinary locus, when applied to scalar-valued forms.
Open AccessTheta operators on unitary Shimura varieties
We define a theta operator on p-adic vector-valued modular forms on unitary groups of arbitrary signature, over a quadratic imaginary field in which p is inert. We study its effect on Fourier-Jacobi expansions and prove that it extends holomorphically beyond the {\mu}-ordinary locus, when applied to scalar-valued forms.
- 11 October 2018
- Published by Cambridge University Press (CUP) in Compositio Mathematica
- Vol. 154 (11) , 2267-2304
Foliations on unitary Shimura varieties in positive characteristic
When $p$ is inert in the quadratic imaginary field $E$ and $m<n$ , unitary Shimura varieties of signature $(n,m)$ and a hyperspecial level subgroup at $p$ , carry a natural foliation of height 1 and rank $m^{2}$ in the tangent bundle of their special fiber $S$ . We study this foliation and show that it acquires singularities at deep Ekedahl–Oort strata, but that these singularities are resolved if we pass to a natural smooth moduli problem $S^{\sharp }$ , a successive blow-up of $S$ . Over the ( $\unicode[STIX]{x1D707}$ -)ordinary locus we relate the foliation to Moonen’s generalized Serre–Tate coordinates. We study the quotient of $S^{\sharp }$ by the foliation, and identify it as the Zariski closure of the ordinary-étale locus in the special fiber $S_{0}(p)$ of a certain Shimura variety with parahoric level structure at $p$ . As a result, we get that this ‘horizontal component’ of $S_{0}(p)$ , as well as its multiplicative counterpart, are non-singular (formerly they were only known to be normal and Cohen–Macaulay). We study two kinds of integral manifolds of the foliation: unitary Shimura subvarieties of signature $(m,m)$ , and a certain Ekedahl–Oort stratum that we denote $S_{\text{fol}}$ . We conjecture that these are the only integral submanifolds.- 11 October 2018
- Published by Cambridge University Press (CUP) in Compositio Mathematica
- Vol. 154 (11) , 2267-2304
Foliations on unitary Shimura varieties in positive characteristic
When $p$ is inert in the quadratic imaginary field $E$ and $m<n$ , unitary Shimura varieties of signature $(n,m)$ and a hyperspecial level subgroup at $p$ , carry a natural foliation of height 1 and rank $m^{2}$ in the tangent bundle of their special fiber $S$ . We study this foliation and show that it acquires singularities at deep Ekedahl–Oort strata, but that these singularities are resolved if we pass to a natural smooth moduli problem $S^{\sharp }$ , a successive blow-up of $S$ . Over the ( $\unicode[STIX]{x1D707}$ -)ordinary locus we relate the foliation to Moonen’s generalized Serre–Tate coordinates. We study the quotient of $S^{\sharp }$ by the foliation, and identify it as the Zariski closure of the ordinary-étale locus in the special fiber $S_{0}(p)$ of a certain Shimura variety with parahoric level structure at $p$ . As a result, we get that this ‘horizontal component’ of $S_{0}(p)$ , as well as its multiplicative counterpart, are non-singular (formerly they were only known to be normal and Cohen–Macaulay). We study two kinds of integral manifolds of the foliation: unitary Shimura subvarieties of signature $(m,m)$ , and a certain Ekedahl–Oort stratum that we denote $S_{\text{fol}}$ . We conjecture that these are the only integral submanifolds.- 18 September 2018
- Published by Springer Nature
- p. 81-152
On the Bad Reduction of Certain U(2, 1) Shimura Varieties
- 18 September 2018
- Published by Springer Nature
- p. 81-152
On the Bad Reduction of Certain U(2, 1) Shimura Varieties
- Open Access
Foliations on unitary Shimura varieties in positive characteristic
When $p$ is inert in the quadratic imaginary field $E$ and $m<n$, unitary Shimura varieties of signature $(n,m)$ and a hyperspecial level subgroup at $p$, carry a natural foliation of height 1 and rank $m^2$ in the tangent bundle of their special fiber $S$. We study this foliation and show that it acquires singularities at deep Ekedahl-Oort strata, but these singularities are resolved if we pass to a natural smooth moduli problem $S^\sharp$, a successive blow-up of $S$. Over the ($\mu$-)ordinary locus we relate the foliation to Moonen's generalized Serre-Tate coordinates. We study the quotient of $S^\sharp$ by the foliation, and identify it as the Zariski closure of the ordinary-\'etale locus in the special fibre $S_0(p)$ of a certain Shimura variety with parahoric level structure at $p$. As a result we get that this "horizontal component" of $S_0(p)$, as well as its multiplicative counterpart, are non-singular (formerly they were only known to be normal and Cohen-Macaulay). We study two kinds of integral manifolds of the foliation: unitary Shimura subvarieties of signature $(m,m)$, and a certain Ekedahl-Oort stratum that we denote $S_{fol}$. We conjecture that these are the only integral submanifolds.Open AccessFoliations on unitary Shimura varieties in positive characteristic
When $p$ is inert in the quadratic imaginary field $E$ and $m<n$, unitary Shimura varieties of signature $(n,m)$ and a hyperspecial level subgroup at $p$, carry a natural foliation of height 1 and rank $m^2$ in the tangent bundle of their special fiber $S$. We study this foliation and show that it acquires singularities at deep Ekedahl-Oort strata, but these singularities are resolved if we pass to a natural smooth moduli problem $S^\sharp$, a successive blow-up of $S$. Over the ($\mu$-)ordinary locus we relate the foliation to Moonen's generalized Serre-Tate coordinates. We study the quotient of $S^\sharp$ by the foliation, and identify it as the Zariski closure of the ordinary-\'etale locus in the special fibre $S_0(p)$ of a certain Shimura variety with parahoric level structure at $p$. As a result we get that this "horizontal component" of $S_0(p)$, as well as its multiplicative counterpart, are non-singular (formerly they were only known to be normal and Cohen-Macaulay). We study two kinds of integral manifolds of the foliation: unitary Shimura subvarieties of signature $(m,m)$, and a certain Ekedahl-Oort stratum that we denote $S_{fol}$. We conjecture that these are the only integral submanifolds. - Open Access
On the bad reduction of certain U(2, 1) Shimura varieties
Let $E$ be a quadratic imaginary field and let $p$ be a prime which is inert in $E.$ We study three types of Picard modular surfaces in positive characteristic $p$ and the morphisms between them. The first Picard surface, denoted $S$, parametrizes triples $(A,\phi,\iota)$ comprised of an abelian threefold $A$ with an action $\iota$ of the ring of integers $\mathcal{O}_{E}$, and a principal polarization $\phi$. The second surface, $S_{0}(p)$, parametrizes, in addition, a suitably restricted choice of a subgroup $H\subset A[p]$ of rank $p^{2}$. The third Picard surface, $\widetilde{S}$, parametrizes triples $(A,\psi,\iota)$ similar to those parametrized by $S$, but where $\psi$ is a polarization of degree $p^{2}$. We study the components, singularities and naturally defined stratifications of these surfaces, and their behaviour under the morphisms. A particular role is played by a foliation we define on the blow-up of $S$ at its superspecial points.Open AccessOn the bad reduction of certain U(2, 1) Shimura varieties
Let $E$ be a quadratic imaginary field and let $p$ be a prime which is inert in $E.$ We study three types of Picard modular surfaces in positive characteristic $p$ and the morphisms between them. The first Picard surface, denoted $S$, parametrizes triples $(A,\phi,\iota)$ comprised of an abelian threefold $A$ with an action $\iota$ of the ring of integers $\mathcal{O}_{E}$, and a principal polarization $\phi$. The second surface, $S_{0}(p)$, parametrizes, in addition, a suitably restricted choice of a subgroup $H\subset A[p]$ of rank $p^{2}$. The third Picard surface, $\widetilde{S}$, parametrizes triples $(A,\psi,\iota)$ similar to those parametrized by $S$, but where $\psi$ is a polarization of degree $p^{2}$. We study the components, singularities and naturally defined stratifications of these surfaces, and their behaviour under the morphisms. A particular role is played by a foliation we define on the blow-up of $S$ at its superspecial points.