scholarly journals Big Galois representations and -adic -functions

2014 ◽  
Vol 151 (4) ◽  
pp. 603-664 ◽  
Author(s):  
Haruzo Hida
Keyword(s):  
Power Series ◽  
Irreducible Component ◽  
Principal Ideal ◽  
Congruence Subgroup ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Principal Congruence ◽  
Principal Congruence Subgroup ◽  
Canonical Weight

Let$p\geqslant 5$be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its Galois representation contains, up to finite error, a principal congruence subgroup${\rm\Gamma}(L)$of$\text{SL}_{2}(\mathbb{Z}_{p}[[T]])$for a principal ideal$(L)\neq 0$of$\mathbb{Z}_{p}[[T]]$for the canonical ‘weight’ variable$t=1+T$. If$L\notin {\rm\Lambda}^{\times }$, the power series$L$is proven to be a factor of the Kubota–Leopoldt$p$-adic$L$-function or of the square of the anticyclotomic Katz$p$-adic$L$-function or a power of$(t^{p^{m}}-1)$.

On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves

2005 ◽  
Vol 48 (1) ◽  
pp. 16-31 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Ernst Kani
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Upper Bounds

AbstractLet E be an elliptic curve defined over ℚ, of conductor N and without complex multiplication. For any positive integer l, let ϕl be the Galois representation associated to the l-division points of E. From a celebrated 1972 result of Serre we know that ϕl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which ϕl is not surjective.

ON KAZHDAN CONSTANTS OF FINITE INDEX SUBGROUPS IN SLn(ℤ)

2012 ◽  
Vol 22 (03) ◽  
pp. 1250026
Author(s):  
UZY HADAD
Keyword(s):  
Finite Index ◽  
Congruence Subgroup ◽  
Infinite Family ◽  
Principal Congruence ◽  
The Other ◽  
Index Subgroup ◽  
Generating Set ◽  
Principal Congruence Subgroup ◽  
Finite Index Subgroup ◽  
Finite Index Subgroups

We prove that for any finite index subgroup Γ in SL n(ℤ), there exists k = k(n) ∈ ℕ, ϵ = ϵ(Γ) > 0, and an infinite family of finite index subgroups in Γ with a Kazhdan constant greater than ϵ with respect to a generating set of order k. On the other hand, we prove that for any finite index subgroup Γ of SL n(ℤ), and for any ϵ > 0 and k ∈ ℕ, there exists a finite index subgroup Γ′ ≤ Γ such that the Kazhdan constant of any finite index subgroup in Γ′ is less than ϵ, with respect to any generating set of order k. In addition, we prove that the Kazhdan constant of the principal congruence subgroup Γn(m), with respect to a generating set consisting of elementary matrices (and their conjugates), is greater than [Formula: see text], where c > 0 depends only on n. For a fixed n, this bound is asymptotically best possible.

scholarly journals Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero

2011 ◽  
Vol 203 ◽  
pp. 47-100 ◽  
Author(s):  
Yuichiro Hoshi
Keyword(s):  
Isomorphism Class ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Genus Zero ◽  
Isomorphism Classes ◽  
Tate Module ◽  
Hyperbolic Curve ◽  
Theoretic Characterization

AbstractLet l be a prime number. In this paper, we prove that the isomorphism class of an l-monodromically full hyperbolic curve of genus zero over a finitely generated extension of the field of rational numbers is completely determined by the kernel of the natural pro-l outer Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to Mochizuki which asserts that the isomorphism class of an elliptic curve which does not admit complex multiplication over a number field is completely determined by the kernels of the natural Galois representations on the various finite quotients of its Tate module.

scholarly journals Galois-theoretic characterization of isomorphism classes of monodromically full hyperbolic curves of genus zero

2011 ◽  
Vol 203 ◽  
pp. 47-100 ◽  
Author(s):  
Yuichiro Hoshi
Keyword(s):  
Isomorphism Class ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Genus Zero ◽  
Isomorphism Classes ◽  
Tate Module ◽  
Hyperbolic Curve ◽  
Theoretic Characterization

AbstractLetlbe a prime number. In this paper, we prove that the isomorphism class of anl-monodromically full hyperbolic curve of genus zero over a finitely generated extension of the field of rational numbers is completely determined by the kernel of the natural pro-louter Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to Mochizuki which asserts that the isomorphism class of an elliptic curve which does not admit complex multiplication over a number field is completely determined by the kernels of the natural Galois representations on the various finite quotients of its Tate module.

scholarly journals Modular symmetry by orbifolding magnetized T2 × T2: realization of double cover of ΓN

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Shota Kikuchi ◽  
Tatsuo Kobayashi ◽  
Hajime Otsuka ◽  
Shintaro Takada ◽  
Hikaru Uchida
Keyword(s):  
Modular Forms ◽  
Congruence Subgroup ◽  
Double Cover ◽  
Principal Congruence ◽  
Double Covering ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Zero Modes ◽  
Principal Congruence Subgroup ◽  
Covering Groups

Abstract We study the modular symmetry of zero-modes on $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 and orbifold compactifications with magnetic fluxes, M1, M2, where modulus parameters are identified. This identification breaks the modular symmetry of $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 , SL(2, ℤ)1× SL(2, ℤ)2 to SL(2, ℤ) ≡ Γ. Each of the wavefunctions on $$ {T}_1^2\times {T}_2^2 $$ T 1 2 × T 2 2 and orbifolds behaves as the modular forms of weight 1 for the principal congruence subgroup Γ(N), N being 2 times the least common multiple of M1 and M2. Then, zero-modes transform each other under the modular symmetry as multiplets of double covering groups of ΓN such as the double cover of S4.

scholarly journals On Conjugacy Classes of Congruence Subgroups of PSL(2, R)

2009 ◽  
Vol 12 ◽  
pp. 264-274 ◽  
Author(s):  
C. J. Cummins
Keyword(s):  
Congruence Subgroup ◽  
Principal Congruence ◽  
Conjugacy Classes ◽  
Genus Zero ◽  
Congruence Subgroups ◽  
Principal Congruence Subgroup ◽  
Genus One

AbstractLet G be a subgroup of PSL(2, R) which is commensurable with PSL(2, Z). We say that G is a congruence subgroup of PSL(2, R) if G contains a principal congruence subgroup /overline Γ(N) for some N. An algorithm is given for determining whether two congruence subgroups are conjugate in PSL(2, R). This algorithm is used to determine the PSL(2, R) conjugacy classes of congruence subgroups of genus-zero and genus-one. The results are given in a table.

Free quotients of congruence subgroups of SL2 over a Dedekind ring of arithmetic type contained in a function field

1987 ◽  
Vol 101 (3) ◽  
pp. 421-429 ◽  
Author(s):  
A. W. Mason
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Function Field ◽  
Congruence Subgroup ◽  
Principal Congruence ◽  
Dedekind Ring ◽  
Congruence Subgroups ◽  
Principal Congruence Subgroup

Let R be a commutative ring with identity and let q be an ideal in R. For each n ≽ 2, let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(R, q) be the normal subgroup of En(R) generated by the q-elementary matrices. We put SLn(R, q) = Ker(SLn(R)→SLn(R/q)), the principal congruence subgroup of GLn(R) of level q. (By definition En(R, R) = En(R) and SLn(R, R) = SLn(R).)

A note on cusp forms and representations of SL2(𝔽𝑝)

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Zhe Chen
Keyword(s):  
Finite Field ◽  
Half Plane ◽  
Congruence Subgroup ◽  
Holomorphic Functions ◽  
Principal Congruence ◽  
Cusp Forms ◽  
Cohomology Space ◽  
Principal Congruence Subgroup ◽  
Upper Half Plane ◽  
Space Of Cusp Forms

AbstractCusp forms are certain holomorphic functions defined on the upper half-plane, and the space of cusp forms for the principal congruence subgroup \Gamma(p), 𝑝 a prime, is acted on by \mathrm{SL}_{2}(\mathbb{F}_{p}). Meanwhile, there is a finite field incarnation of the upper half-plane, the Deligne–Lusztig (or Drinfeld) curve, whose cohomology space is also acted on by \mathrm{SL}_{2}(\mathbb{F}_{p}). In this note, we compute the relation between these two spaces in the weight 2 case.

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