
Michael ScheinBar Ilan University | BIU · Department of Mathematics
Michael Schein
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27
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Introduction
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Publications
Publications (27)
Let be a nilpotent algebra of class two over a compact discrete valuation ring of characteristic zero or of sufficiently large positive characteristic. Let be the residue cardinality of . The ideal zeta function of is a Dirichlet series enumerating finite‐index ideals of . We prove that there is a rational function in , , , and giving the ideal zet...
We construct infinite families of irreducible supersingular mod p representations of GL2(F) with GL2(OF)-socle compatible with Serre’s modularity conjecture, where F/ℚp is any finite extension with residue field Fp2 and ramification degree e ≤ (p − 1)/2. These are the first such examples for ramified F/ℚp.
We compute the local pro-isomorphic zeta functions at all but finitely many primes for a certain family of class-two-nilpotent Lie lattices of even rank, parametrized by irreducible non-linear polynomials $f(x) \in \mathbb{Z} [x]$, that corresponds to a family of groups introduced by Grunewald and Segal. The result is expressed in terms of a combin...
Let $L$ be a nilpotent algebra of class two over a compact discrete valuation ring $A$ of characteristic zero or of sufficiently large positive characteristic. Let $q$ be the residue cardinality of $A$. The ideal zeta function of $L$ is a Dirichlet series enumerating finite-index ideals of $L$. We prove that there is a rational function in $q$, $q^...
We present a method for computing upper bounds on the systolic length of certain Riemann surfaces uniformized by congruence subgroups of hyperbolic triangle groups, admitting congruence Hurwitz curves as a special case. The uniformizing group is realized as a Fuchsian group and a convenient finite generating set is computed. The upper bound is deri...
We construct infinite families of irreducible supersingular mod $p$ representations of $\mathrm{GL}_2(F)$ with $\mathrm{GL}_2(\mathcal{O}_F)$-socle compatible with Serre's modularity conjecture, where $F / \mathbb{Q}_p$ is any finite extension with residue field $\mathbb{F}_{p^2}$ and ramification degree $e \leq (p-1)/2$.
We present a method for computing upper bounds on the systolic length of certain Riemann surfaces uniformized by congruence subgroups of hyperbolic triangle groups, admitting congruence Hurwitz curves as a special case. The uniformizing group is realized as a Fuchsian group and a convenient finite generating set is computed. The upper bound is deri...
We consider pro-isomorphic zeta functions of the groups $\Gamma(\mathcal{O}_K)$, where $\Gamma$ is a nilpotent group scheme defined over $\mathbb{Z}$ and $K$ varies over all number fields. Under certain conditions, we show that these functions have a fine Euler decomposition with factors indexed by primes of $K$ and depending only on the structure...
We introduce a new class of combinatorially defined rational functions and apply them to deduce explicit formulae for local ideal zeta functions associated to the members of a large class of nilpotent Lie rings which contains the free class-2-nilpotent Lie rings and is stable under direct products. Our computations unify and generalize a substantia...
Given a quadratic number field $k=\mathbb{Q}(\sqrt{d})$ with narrow class number $h_d^+$ and discriminant $\Delta_k$, let $\underline{\textbf{O}}_d$ be the orthogonal $\mathbb{Z}$-group of the associated norm form $q_d$. In this paper we describe the structure of the pointed set $H^1_{\mathrm{fl}}(\mathbb{Z},\underline{\textbf{O}}_d)$, which classi...
Let $\mathcal{O}$ be an order of index $m$ in the maximal order of a quadratic number field $k=\mathbb{Q}(\sqrt{d})$. Let $\underline{\mathbf{O}}_{d,m}$ be the orthogonal $\mathbb{Z}$-group of the associated norm form $q_{d,m}$. We describe the structure of the pointed set $H^1_{\mathrm{fl}}(\mathbb{Z},\underline{\mathbf{O}}_{d,m})$, which classifi...
The irreducible supersingular mod p representations of G=GL2(F)G=GL2(F), where F is a finite extension of QpQp, are the building blocks of the mod p representation theory of G . They all arise as irreducible quotients of certain universal supersingular representations. We investigate the structure of these universal modules in the case when F/QpF/Q...
We compute explicitly the normal zeta functions of the Heisenberg groups
$H(R)$, where $R$ is a compact discrete valuation ring of characteristic zero.
These zeta functions occur as Euler factors of normal zeta functions of
Heisenberg groups of the form $H(\mathcal{O}_K)$, where $\mathcal{O}_K$ is the
ring of integers of an arbitrary number field~$...
We give a detailed description of the arithmetic Fuchsian group of the Bolza
surface and the associated quaternion order. This description enables us to
show that the corresponding principal congruence covers satisfy the bound
sys(X) > 4/3 log g(X) on the systole, where g is the genus. We also exhibit the
Bolza group as a congruence subgroup, and c...
These are the lecture notes from a five-hour mini-course given at the Winter
School on Galois Theory held at the University of Luxembourg in February 2012.
Their aim is to give an overview of Serre's modularity conjecture and of its
proof by Khare, Wintenberger, and Kisin, as well as of the results of other
mathematicians that played an important r...
To determine an unknown function belonging to a known n-dimensional space, it suffices to evaluate the function at n generic points. We apply the character theory of finite groups towards finding optimal designs of such points.
Let $K$ be a number field with ring of integers $\mathcal{O}_K$. We compute
explicitly the local factors of the normal zeta functions of the Heisenberg
groups $H(\mathcal{O}_K)$ that are indexed by rational primes which are
unramified in $K$. We show that these local zeta functions satisfy functional
equations upon the inversion of the prime.
Let F be a totally ramified extension of Qp. We consider supersingular representations of GL2(F) whose socles as GL2(OF)-modules are of a certain form that is expected to appear in the mod p local Langlands correspondence and establish a condition under which they are irreducible.
Let 𝒪 be the ring of integers of a p -adic field and 𝔭 its maximal ideal. This paper computes the Jordan–Hölder decomposition of the reduction modulo p of the cuspidal representations of GL 2 (𝒪/𝔭 e ) for e ≥ 1. An alternative formulation of Serre's conjecture for Hilbert modular forms is then provided.
Let ρ : Gal (over(Q, -) / Q) → GLn (over(F, -)p) be an n-dimensional mod p Galois representation. If ρ is modular for a weight in a certain class, called p-minute, then we restrict the Fontaine-Laffaille numbers of ρ; in other words, we specify the possibilities for the restriction of ρ to inertia at p. Our result agrees with the Serre-type conject...
Let F be a totally real field, p ≥ 3 a rational prime unramified in F , and p a place of F over p. Let ρ : Gal(F /F) → GL2(Fp) be a two-dimensional mod p Galois representation which is assumed to be modular of some weight and whose restriction to a decomposition subgroup at p is irreducible. We specify a set of weights, determined by the restrictio...
Let F be a totally real field and p ≥ 3 a prime. If ρ :
is continuous, semisimple, totally odd, and tamely ramified at all places of F dividing p, then we formulate a conjecture specifying the weights for which ρ is modular. This extends the conjecture of Diamond, Buzzard, and Jarvis, which required p to be unramified in F. We also prove a theorem...
Let F be a totally real field and ρ: Gal(F /F) → GL2(Fp) a Galois representation whose restriction to a decomposition group at some place dividing p is irreducible. Suppose that ρ is modular of some weight σ. We specify a set of weights, not containing σ, such that ρ is modular for at least one weight in this set. Let F be a totally real field and...
Lesson study is a professional-development process in which teachers collaboratively plan, execute, observe, and discuss lessons in the classroom. Lesson study can be implemented in a college environment to develop and improve the teaching skills of both new and experienced faculty. Focusing on a single lesson allows the lesson study participants t...
Let F be a totally real field and p an odd prime. If r is a continuous, semisimple, totally odd mod p representation of the absolute Galois group of F which is tamely ramified at all places of F dividing p, then we formulate a conjecture specifying the weights for which r is modular. This extends the conjecture of Diamond, Buzzard, and Jarvis, whic...
In this mostly expository article we give a survey of some of the generalizations of Serre's conjecture and results towards them that have been obtained in recent years. We also discuss recent progress towards a mod p local Langlands correspondence for p-adic fields and its connections with Serre's conjecture. A theorem describing the structure of...