Ehud de Shalit
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Ehud de Shalit has a total of 4 co-authors. Here are the top co-authors by number of shared publications:
- Eyal Z. Goren
McGill University
7 shared publications - Charlotte Hardouin
Laboratoire de Mathématiques UMR 8100 CNRS Université de Versailles-St Quentin 45 avenue des États-Unis 78035 Versailles cedex France
2 shared publications - Julien Roques2 shared publications
- Gal Porat
The Hebrew University
1 shared publication
Latest Publications
- Open Access
Hypertranscendence and $q$-difference equations over elliptic functionfields
The differential nature of solutions of linear difference equations over the projective line was recently elucidated. In contrast, little is known about the differential nature of solutions of linear difference equations over elliptic curves. In the present paper, we study power series $f(z)$ with complex coefficients satisfying a linear difference equation over a field of elliptic functions $K$,with respect to the difference operator $φf(z)=f(qz)$, $2\le q\in\mathbb{Z}$,arising from an endomorphism of the elliptic curve. Our main theoremsays that such an $f$ satisfies, in addition, a polynomial differentialequation with coefficients from $K,$ if and only if it belongs tothe ring $S=K[z,z^{-1},ζ(z,Λ)]$ generated over $K$ by$z,z^{-1}$ and the Weierstrass $ζ$-function. This is the first elliptic extension of recent theorems of Adamczewski, Dreyfus and Hardouin concerning the differential transcendence of solutions of difference equations with coefficients in $\mathbb{C}(z),$ in which various difference operators were considered (shifts, $q$-differenceoperators or Mahler operators). While the general approach, of usingparametrized Picard-Vessiot theory, is similar, many features, andin particular the emergence of monodromy considerations and the ring$S$, are unique to the elliptic case and are responsible for non-trivial difficulties. We emphasize that, among the intermediate results, we prove an integrability result for difference-differential systems over ellipticcurves which is a genus one analogue of the integrability results obtained by Sch\''afke and Singer over the projective line.Open AccessHypertranscendence and $q$-difference equations over elliptic functionfields
The differential nature of solutions of linear difference equations over the projective line was recently elucidated. In contrast, little is known about the differential nature of solutions of linear difference equations over elliptic curves. In the present paper, we study power series $f(z)$ with complex coefficients satisfying a linear difference equation over a field of elliptic functions $K$,with respect to the difference operator $φf(z)=f(qz)$, $2\le q\in\mathbb{Z}$,arising from an endomorphism of the elliptic curve. Our main theoremsays that such an $f$ satisfies, in addition, a polynomial differentialequation with coefficients from $K,$ if and only if it belongs tothe ring $S=K[z,z^{-1},ζ(z,Λ)]$ generated over $K$ by$z,z^{-1}$ and the Weierstrass $ζ$-function. This is the first elliptic extension of recent theorems of Adamczewski, Dreyfus and Hardouin concerning the differential transcendence of solutions of difference equations with coefficients in $\mathbb{C}(z),$ in which various difference operators were considered (shifts, $q$-differenceoperators or Mahler operators). While the general approach, of usingparametrized Picard-Vessiot theory, is similar, many features, andin particular the emergence of monodromy considerations and the ring$S$, are unique to the elliptic case and are responsible for non-trivial difficulties. We emphasize that, among the intermediate results, we prove an integrability result for difference-differential systems over ellipticcurves which is a genus one analogue of the integrability results obtained by Sch\''afke and Singer over the projective line. - Open Access
Hypertranscendence and $q$-difference equations over elliptic functionfields
The differential nature of solutions of linear difference equations over the projective line was recently elucidated. In contrast, little is known about the differential nature of solutions of linear difference equations over elliptic curves. In the present paper, we study power series $f(z)$ with complex coefficients satisfying a linear difference equation over a field of elliptic functions $K$,with respect to the difference operator $\phi f(z)=f(qz)$, $2\le q\in\mathbb{Z}$,arising from an endomorphism of the elliptic curve. Our main theoremsays that such an $f$ satisfies, in addition, a polynomial differentialequation with coefficients from $K,$ if and only if it belongs tothe ring $S=K[z,z^{-1},\zeta(z,\Lambda)]$ generated over $K$ by$z,z^{-1}$ and the Weierstrass $\zeta$-function. This is the first elliptic extension of recent theorems of Adamczewski, Dreyfus and Hardouin concerning the differential transcendence of solutions of difference equations with coefficients in $\mathbb{C}(z),$ in which various difference operators were considered (shifts, $q$-differenceoperators or Mahler operators). While the general approach, of usingparametrized Picard-Vessiot theory, is similar, many features, andin particular the emergence of monodromy considerations and the ring$S$, are unique to the elliptic case and are responsible for non-trivial difficulties. We emphasize that, among the intermediate results, we prove an integrability result for difference-differential systems over ellipticcurves which is a genus one analogue of the integrability results obtained by Sch\"afke and Singer over the projective line.Open AccessHypertranscendence and $q$-difference equations over elliptic functionfields
The differential nature of solutions of linear difference equations over the projective line was recently elucidated. In contrast, little is known about the differential nature of solutions of linear difference equations over elliptic curves. In the present paper, we study power series $f(z)$ with complex coefficients satisfying a linear difference equation over a field of elliptic functions $K$,with respect to the difference operator $\phi f(z)=f(qz)$, $2\le q\in\mathbb{Z}$,arising from an endomorphism of the elliptic curve. Our main theoremsays that such an $f$ satisfies, in addition, a polynomial differentialequation with coefficients from $K,$ if and only if it belongs tothe ring $S=K[z,z^{-1},\zeta(z,\Lambda)]$ generated over $K$ by$z,z^{-1}$ and the Weierstrass $\zeta$-function. This is the first elliptic extension of recent theorems of Adamczewski, Dreyfus and Hardouin concerning the differential transcendence of solutions of difference equations with coefficients in $\mathbb{C}(z),$ in which various difference operators were considered (shifts, $q$-differenceoperators or Mahler operators). While the general approach, of usingparametrized Picard-Vessiot theory, is similar, many features, andin particular the emergence of monodromy considerations and the ring$S$, are unique to the elliptic case and are responsible for non-trivial difficulties. We emphasize that, among the intermediate results, we prove an integrability result for difference-differential systems over ellipticcurves which is a genus one analogue of the integrability results obtained by Sch\"afke and Singer over the projective line. - 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 $S.$ - 21 April 2022
- Published by Springer Nature
- p. 245-347
The Fargues-Fontaine Curve and p-Adic Hodge Theory
- 21 April 2022
- Published by Springer Nature
- p. 245-347
The Fargues-Fontaine Curve and p-Adic Hodge Theory
- Open Access
Elliptic (p,q)-difference modules
We study finite dimensional vector spaces over fields of elliptic functions equipped with two commuting aotomorphisms \sigma and \tau induced by isogenies of relatively prime orders. We give a structure theorem for such objects, that reveals a connection to the classification of vector bundles on elliptic curves by Atiyah. As an application we prove an elliptic analogue of a conjecture of Loxton and van der Poorten which has been recently proved by Adamczewski and Bell.
Open AccessElliptic (p,q)-difference modules
We study finite dimensional vector spaces over fields of elliptic functions equipped with two commuting aotomorphisms \sigma and \tau induced by isogenies of relatively prime orders. We give a structure theorem for such objects, that reveals a connection to the classification of vector bundles on elliptic curves by Atiyah. As an application we prove an elliptic analogue of a conjecture of Loxton and van der Poorten which has been recently proved by Adamczewski and Bell.
- 14 August 2020
- Published by Canadian Mathematical Society in Canadian Mathematical Bulletin
- Vol. 64 (3) , 530-540
Criteria for periodicity and an application to elliptic functions
LetPandQbe relatively prime integers greater than 1, and letfbe a real valued discretely supported function on a finite dimensional real vector spaceV. We prove that if $f_{P}(x)=f(Px)-f(x)$ and $f_{Q}(x)=f(Qx)-f(x)$ are both $\Lambda $ -periodic for some lattice $\Lambda \subset V$ , then so isf(up to a modification at $0$ ). This result is used to prove a theorem on the arithmetic of elliptic function fields. In the last section, we discuss the higher rank analogue of this theorem and explain why it fails in rank 2. A full discussion of the higher rank case will appear in a forthcoming work.- 14 August 2020
- Published by Canadian Mathematical Society in Canadian Mathematical Bulletin
- Vol. 64 (3) , 530-540
Criteria for periodicity and an application to elliptic functions
LetPandQbe relatively prime integers greater than 1, and letfbe a real valued discretely supported function on a finite dimensional real vector spaceV. We prove that if $f_{P}(x)=f(Px)-f(x)$ and $f_{Q}(x)=f(Qx)-f(x)$ are both $\Lambda $ -periodic for some lattice $\Lambda \subset V$ , then so isf(up to a modification at $0$ ). This result is used to prove a theorem on the arithmetic of elliptic function fields. In the last section, we discuss the higher rank analogue of this theorem and explain why it fails in rank 2. A full discussion of the higher rank case will appear in a forthcoming work.- Open Access
Criteria for periodicity and an application to elliptic functions
Let P and Q be relatively prime integers greater than 1, and f a real valued discretely supported function on a finite dimensional real vector space V. We prove that if f_{P}(x)=f(Px)-f(x) and f_{Q}(x)=f(Qx)-f(x) are both \Lambda-periodic for some lattice \Lambda\subset V, then so is f (up to a modification at 0). This result is used to prove a theorem on the arithmetic of elliptic function fields. In the last section we discuss the higher rank analogue of this theorem and explain why it fails in rank 2. A full discussion of the higher rank case will appear in a forthcoming work.
Open AccessCriteria for periodicity and an application to elliptic functions
Let P and Q be relatively prime integers greater than 1, and f a real valued discretely supported function on a finite dimensional real vector space V. We prove that if f_{P}(x)=f(Px)-f(x) and f_{Q}(x)=f(Qx)-f(x) are both \Lambda-periodic for some lattice \Lambda\subset V, then so is f (up to a modification at 0). This result is used to prove a theorem on the arithmetic of elliptic function fields. In the last section we discuss the higher rank analogue of this theorem and explain why it fails in rank 2. A full discussion of the higher rank case will appear in a forthcoming work.
- 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.