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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...
The aim of this paper is to give a purely rigid-analytic development of the theory of canonical subgroups with arbitrary torsion-level in generalized elliptic curves over rigid spaces over k (using L...
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...
We use arithmetic models of modular curves to establish some properties of Ramanujan’s continued fraction. In particular, we give a new geometric proof that its singular values are algebraic units th...
This lemma is already false for k = p quite generally, and our error in the proof of the lemma occurs in the line beginning “Evidently. . . ” (the formula given for (k, m+jk/p) in that line is wrong...
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...
The moduli spaces of trigonal curves are proven to be rational when the genus is divisible by 4.
We present efficient algorithms for detecting central and mirror symmetry for the case of algebraic curves defined by means of polynomial parametrizations. The algorithms are based on an algebraic rel...
Let M{\subset}\mathbb{R}^{n} be a Riemannian helix submanifold with respect to the unit direction d{\in}\mathbb{R}^{n} and f:M{\to}\mathbb{R} be a eikonal function. We say that M is a f-eikonal helix ...
In this paper we extend the concept of Milnor fiber and Milnor number of a curve singularity allowing the ambient space to be a quotient surface singularity. A generalization of the local {\delta}-inv...
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 ...
We prove uniruledness of some moduli spaces $\bar{M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points using linear systems on nonsingular projective surfaces containing the general curve of...
We prove a general criterion for an algebraic stack to admit a good moduli space. This result may be considered as a weak analog of the Keel-Mori theorem, which guarantees the existence of a coarse mo...

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