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ON THE MODULARITY OF ELLIPTIC CURVES OVER Q: WILD 3-ADIC EXERCISES.
Elliptic curve Galois representation modularity
2015/7/6
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...
MODULAR CURVES AND RAMANUJAN’S CONTINUED FRACTION
modular curve Ramanujan continued fraction
2015/7/6
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...
ERRATUM TO MODULAR CURVES AND RAMANUJAN’S CONTINUED FRACTION
q-series modular curve Ramanujan continued fraction
2015/7/6
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...
On the Frequency of Finitely Anomalous Elliptic Curves
Elliptic Curves Galois Representations Number Theory
2014/12/8
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) = ...
GALOIS THEORY OF ITERATED MORPHISMS ON REDUCIBLE ELLIPTIC CURVES AND ABELIAN SURFACES WITH REAL MULTIPLICATION
Kummer map elliptic curves abelian surfaces with real multiplication
2014/12/8
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 ...
On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields
Elliptic Curves over Number Fields Mordell-Weil Group Two-Descent
2013/1/30
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 rationality of the moduli spaces of trigonal curves
trigonal curve rationality SL2 × SL2 bi-transvectant
2012/7/11
The moduli spaces of trigonal curves are proven to be rational when the genus is divisible by 4.
Efficient Detection of Symmetries of Polynomially Parametrized Curves
Efficient Detection Symmetries Polynomially Parametrized Curves Algebraic Geometry
2012/7/11
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...
f-Eikonal helix submanifolds and f-Eikonal helix curves
Helix submanifold Eikonal function Helix line
2012/6/15
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 ...
Local invariants on quotient singularities and a genus formula for weighted plane curves
Local invariants quotient singularities genus formula weighted plane curves Algebraic Geometry
2012/6/30
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...
A Cohen-Lenstra phenomenon for elliptic curves
A Cohen-Lenstra phenomenon elliptic curves Number Theory
2012/6/29
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 ...
Uniruledness of some moduli spaces of stable pointed curves
Pointed curve Moduli space Uniruledness Hyperelliptic curves Complete intersection surfaces
2012/6/27
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...
Existence of good moduli spaces for A_k-stable curves
Existence of good moduli spaces A_k-stable curves Algebraic Geometry
2012/6/25
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...