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Equidistribution is an important theme in number theory. The Sato-Tate conjecture, which was established by Richard Taylor et.al. in 2008, asserts that given an elliptic curve over Q without complex m...
Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, ρE : Gal(k/k) → GL2(Z b). For a fixed number field k, we describe the ima...
Let E be an elliptic curve over the rationals that does not have complex multiplication. For each prime `, the action of the absolute Galois group on the `-torsion points of E can be given in terms of...
POSSIBLE INDICES FOR THE GALOIS IMAGE OF ELLIPTIC CURVES OVER.
We nd it interesting that such a natural naive approach as we will describe actually works, and yields a desingularization with these nice properties.
We study the geometry of moduli spaces of genus 0 and 1 curves in Pn with speci ed contact with a hyperplane H. We compute intersection numbers on these spaces that correspond to the number of degre...
with c  0 mod N and d  p mod N. The operators Tp for p6 jN can be simultaneously diagonalised on the space Sk(N) and a simultaneous eigenvector is called an eigenform. If f is an eigenform then the...
Deligne and Rapoport developed the theory of generalized elliptic curves over arbitrary schemes and they proved that various moduli stacks for (ample) “level-N” structures on generalized elliptic cu...
Given an elliptic curve E over Q, we can then consider E over the finite field Fp. If Np is the number of points on the curve over Fp, then we define ap(E) = p+1-Np. We say primes p for which ap(E) = ...
Let $F$ be a number field and let $A$ be an abelian algebraic group defined over $F$. For a prime $\ell$ and a point $\alpha \in A(F)$, we obtain the tower of extensions $F([\ell^n]^{-1}(\alpha))$ by ...
We prove the existence and nonexistence of elliptic curves having good reduction everywhere over certain real quadratic fields Q(m) for m≤200. These results of computations give best-possible data inc...
Given an elliptic curve $E$ and a finite Abelian group $G$, we consider the problem of counting the number of primes $p$ for which the group of points modulo $p$ is isomorphic to $G$. Under a certain ...
Let E be an elliptic curve over Q, and let F be a finite abelian extension of Q. Using Beilinson's theorem on a suitable modular curve, we prove a weak version of Zagier's conjecture for L(E/F,2), whe...
Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$. A famous theorem of Serre states t...

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