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- Riemann surface (3)
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- Compact Riemann surface (1)
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- Group action (1)
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- Hecke algebra (1)
- Isomorphism of curves (1)
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- Quasi-platonic surface (1)
- Ramification (1)
- Real elliptic curve (1)
- Root of polynomials (1)
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Articles 1 - 9 of 9
Full-Text Articles in Algebraic Geometry
Probability Distributions For Elliptic Curves In The Cgl Hash Function, Dhruv Bhatia, Kara Fagerstrom, Max Watson
Probability Distributions For Elliptic Curves In The Cgl Hash Function, Dhruv Bhatia, Kara Fagerstrom, Max Watson
Mathematical Sciences Technical Reports (MSTR)
Hash functions map data of arbitrary length to data of predetermined length. Good hash functions are hard to predict, making them useful in cryptography. We are interested in the elliptic curve CGL hash function, which maps a bitstring to an elliptic curve by traversing an inputdetermined path through an isogeny graph. The nodes of an isogeny graph are elliptic curves, and the edges are special maps betwixt elliptic curves called isogenies. Knowing which hash values are most likely informs us of potential security weaknesses in the hash function. We use stochastic matrices to compute the expected probability distributions of the …
Topological And H^Q Equivalence Of Cyclic N-Gonal Actions On Riemann Surfaces - Part Ii, Sean A. Broughton
Topological And H^Q Equivalence Of Cyclic N-Gonal Actions On Riemann Surfaces - Part Ii, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
We consider conformal actions of the finite group G on a closed Riemann surface S, as well as algebraic actions of G on smooth, complete, algebraic curves over an arbitrary, algebraically closed field. There are several notions of equivalence of actions, the most studied of which is topological equivalence, because of its close relationship to the branch locus of moduli space. A second important equivalence relation is that induced by representation of G on spaces of holomorphic q-differentials. The notion of topological equivalence does not work well in positive characteristic. We shall discuss an alternative to topological equivalence, …
Branching Matrices For The Automorphism Group Lattice Of A Riemann Surface, Sean A. Broughton
Branching Matrices For The Automorphism Group Lattice Of A Riemann Surface, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
Let S be a Riemann surface and G a large subgroup of Aut(S) (Aut(S) may be unknown). We are particularly interested in regular n-gonal surfaces, i.e., the quotient surface S/G (and hence S/Aut(S)) has genus zero. For various H the ramification information of the branched coverings S/K -> S/H may be captured in a matrix. The ramification information, in particular strong branching, may be then be used in analyzing the structure of Aut(S). The ramification information is conjugation invariant so the matrix's rows and columns may be indexed by conjugacy classes of subgroups. The only required …
Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces (Corrected), Sean A. Broughton
Topological And Hq Equivalence Of Prime Cyclic P-Gonal Actions On Riemann Surfaces (Corrected), Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
Two Riemann surfaces S1 and S2 with conformal G-actions have topologically equivalent actions if there is a homeomorphism h : S1 -> S2 which intertwines the actions. A weaker equivalence may be defined by comparing the representations of G on the spaces of holomorphic q-differentials Hq(S1) and Hq(S2). In this note we study the differences between topological equivalence and Hq equivalence of prime cyclic actions, where S1/G and S2/G have genus zero.
Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, Sean A. Broughton
Quasi-Platonic Psl2(Q)-Actions On Closed Riemann Surfaces, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
This paper is the first of two papers whose combined goal is to explore the dessins d'enfant and symmetries of quasi-platonic actions of PSL2(q). A quasi-platonic action of a group G on a closed Riemann S surface is a conformal action for which S/G is a sphere and S->S/G is branched over {0, 1,infinity}. The unit interval in S/G may be lifted to a dessin d'enfant D, an embedded bipartite graph in S. The dessin forms the edges and vertices of a tiling on S by dihedrally symmetric polygons, generalizing the idea of a …
Continuous Dependence Of Solutions Of Equations On Parameters, Sean A. Broughton
Continuous Dependence Of Solutions Of Equations On Parameters, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
It is shown under very general conditions that the solutions of equations depend continuously on the coefficients or parameters of the equations. The standard examples are solutions of monic polynomial equations and the eigenvalues of a matrix. However, the proof methods apply to any finite map T : Cn -> Cn.
Isomorphisms Of Elliptic Curves Over Extensions Of Finite Fields, Mathew Niemerg
Isomorphisms Of Elliptic Curves Over Extensions Of Finite Fields, Mathew Niemerg
Mathematical Sciences Technical Reports (MSTR)
Our main interest lies in exploring isomorphisms of elliptic curves. In particular, we focus on two curves defined over a base field and look at which extension fields the curves are isomorphic over. Elliptic curves have a fascinating structure behind them. This structure allows for much to be explored and studied.
The Birational Isomorphism Types Of Smooth Real Elliptic Curves, Sean A. Broughton
The Birational Isomorphism Types Of Smooth Real Elliptic Curves, Sean A. Broughton
Mathematical Sciences Technical Reports (MSTR)
In this note we determine all birational isomorphism types of real elliptic curves and show that it is the same as the orbit space of smooth cubic real curves in real projective space under linear projective equivalence. There are two families, each depending polynomially on a real parameter in a open subinterval of R. We further show that the complexification of a real elliptic curve has exactly two real forms. Thus the real elliptic curves come in pairs which are isomorphic over C. Finally, the map taking a real elliptic curve to its j-invariant maps the two families …
The Galois Correspondence For Branched Covering Spaces And Its Relationship To Hecke Algebras, Matthew Ong
The Galois Correspondence For Branched Covering Spaces And Its Relationship To Hecke Algebras, Matthew Ong
Mathematical Sciences Technical Reports (MSTR)
There is a very beautiful correspondence between branched covers of the Riemann sphere P1 and subgroups of the fundamental group π1(P1 − {branch points}), exactly analogous to the correspondence between subfields of an algebraic extension E/F and subgroups of the Galois group Gal(E/F). This paper explores the concept of a Hecke algebra, which in this context is a generalization of the Galois group to the case of non- Galois covers S/P1. Specifically, we show that the isomorphism type of a Hecke algebra C[H\G/H] is completely determined by the decomposition of …