Dihedral Galois Representations and Katz Modular Forms
نویسندگان
چکیده
We show that any two-dimensional odd dihedral representation ρ over a finite field of characteristic p > 0 of the absolute Galois group of the rational numbers can be obtained from a Katz modular form of level N , character 2 and weight k, where N is the conductor, 2 is the prime-to-p part of the determinant and k is the so-called minimal weight of ρ. In particular, k = 1 if and only if ρ is unramified at p. Direct arguments are used in the exceptional cases, where general results on weight and level lowering are not available. 2000 Mathematics Subject Classification: 11F11, 11F80, 14G35
منابع مشابه
On Modular Forms and the Inverse Galois Problem
In this article new cases of the inverse Galois problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL2(Fpn ) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are cons...
متن کاملOn Galois Representations of Weight One
A two-dimensional Galois representation into the Hecke algebra of Katz modular forms of weight one over a finite field of characteristic p is constructed and is shown to be unramified at p in most cases. 2010 Mathematics Subject Classification: Primary 11F80M Secondary 11F33, 11F25
متن کاملOn the average number of octahedral modular forms
λf (p) = Tr(ρ(Frobp)) χ(p) = det(ρ(Frobp)) for all p not dividing N . Following [4], we define SArtin 1/4,k (N,χ) to be the finite set of primitive weight k cupsidal eigenforms which admit an associated Galois representation. If ρ : Gal(Q̄/Q)→ GL2(C) is a Galois representation, we define Pρ to be the composition of ρ with the natural projection GL2(C)→ PGL2(C). Two-dimensional complex Galois rep...
متن کاملMock-modular Forms of Weight One
The object of this paper is to initiate a study of the Fourier coefficients of a weight one mock-modular form and relate them to the complex Galois representations associated to the form’s shadow, which is a weight one newform. In this paper, our focus will be on weight one dihedral newforms of prime level p ≡ 3 (mod 4). In this case we give properties of the Fourier coefficients themselves tha...
متن کاملOn the computation of Galois representations associated to level one modular forms
In this paper we explicitly compute mod-l Galois representations associated to modular forms. To be precise, we look at cases with l ≤ 23 and the modular forms considered will be cusp forms of level 1 and weight up to 22. We present the result in terms of polynomials associated to the projectivised representations. As an application, we will improve a known result on Lehmer’s non-vanishing conj...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2004