Potential automorphy over CM fields
نویسندگان
چکیده
Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over without any self-duality condition. deduce that all elliptic curves $E$ are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of different sort, we also Ramanujan Conjecture weight zero cuspidal automorphic $\mathrm{GL}_2(\mathbb{A}_F)$.
منابع مشابه
Stark Conjectures for Cm Curves over Number Fields
In [1], Bloch constructs symbols in K2(E) for a CM elliptic curve E defined over Q, corresponding to divisors supported on torsion points of the curve. This construction, and the special properties of such curves, allowed him to prove the Beilinson conjecture for such curves. In [2], Deninger extends Bloch’s results, for certain elliptic curves ‘of Shimura type ’or ‘type (S) ’. For simplicity a...
متن کاملEfficient CM-constructions of elliptic curves over finite fields
We present an algorithm that, on input of an integer N ≥ 1 together with its prime factorization, constructs a finite field F and an elliptic curve E over F for which E(F) has order N . Although it is unproved that this can be done for all N , a heuristic analysis shows that the algorithm has an expected run time that is polynomial in 2ω(N) logN , where ω(N) is the number of distinct prime fact...
متن کاملOn Cm Abelian Varieties over Imaginary Quadratic Fields
In this paper, we associate canonically to every imaginary quadratic field K = Q(√−D) one or two isogenous classes of CM (complex multiplication) abelian varieties over K, depending on whether D is odd or even (D 6= 4). These abelian varieties are characterized as of smallest dimension and smallest conductor, and such that the abelian varieties themselves descend to Q. When D is odd or divisibl...
متن کاملMordell-Weil growth for GL2-type abelian varieties over Hilbert class fields of CM fields
Let A be a modular abelian variety of GL2-type over a totally real field F of class number one. Under some mild assumptions, we show that the Mordell-Weil rank of A grows polynomially over Hilbert class fields of CM extensions of F .
متن کاملRigid Local Systems and Potential Automorphy: the G 2 -case
For s ∈ P(Q) \ {0, 1}, we study the compatible system of Galois representations (ρ l (3) : Gal(Q̄/Q) → G2(Ql))l introduced in [4], where G2(Ql) ≤ GL7(Ql) is the simple group of type G2. We prove that, under some mild condition on s, the image of the Tate twisted Galois representation ρ l (3) coincides with G2(Zl) for all l up to a set of primes having density zero, and we show that ρ l is potent...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Annals of Mathematics
سال: 2023
ISSN: ['1939-8980', '0003-486X']
DOI: https://doi.org/10.4007/annals.2023.197.3.2