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Beyond two criteria for supersingularity: coefficients of division polynomials

Christophe Debry (2014)

Journal de Théorie des Nombres de Bordeaux

Let f ( x ) be a cubic, monic and separable polynomial over a field of characteristic p ≥ 3 and let E be the elliptic curve given by y 2 = f ( x ) . In this paper we prove that the coefficient at x 1 2 p ( p - 1 ) in the p –th division polynomial of E equals the coefficient at x p - 1 in f ( x ) 1 2 ( p - 1 ) . For elliptic curves over a finite field of characteristic p , the first coefficient is zero if and only if E is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the zero loci...

BG-ranks and 2-cores.

Chen, William Y.C., Ji, Kathy Q., Wilf, Herbert S. (2006)

The Electronic Journal of Combinatorics [electronic only]

Bicyclotomic polynomials and impossible intersections

David Masser, Umberto Zannier (2013)

Journal de Théorie des Nombres de Bordeaux

In a recent paper we proved that there are at most finitely many complex numbers t ≠ 0 , 1 such that the points ( 2 , 2 ( 2 - t ) ) and ( 3 , 6 ( 3 - t ) ) are both torsion on the Legendre elliptic curve defined by y 2 = x ( x - 1 ) ( x - t ) . In a sequel we gave a generalization to any two points with coordinates algebraic over the field Q ( t ) and even over C ( t ) . Here we reconsider the special case ( u , u ( u - 1 ) ( u - t ) ) and ( v , v ( v - 1 ) ( v - t ) ) with complex numbers u and v .

Bielliptic and hyperelliptic modular curves X(N) and the group Aut(X(N))

Francesc Bars, Aristides Kontogeorgis, Xavier Xarles (2013)

Acta Arithmetica

We determine all modular curves X(N) (with N ≥ 7) that are hyperelliptic or bielliptic. We also give a proof that the automorphism group of X(N) is PSL₂(ℤ/Nℤ), whence it coincides with the normalizer of Γ(N) in PSL₂(ℝ) modulo ±Γ(N).

Bihomogeneous forms in many variables

Damaris Schindler (2014)

Journal de Théorie des Nombres de Bordeaux

We count integer points on varieties given by bihomogeneous equations using the Hardy-Littlewood method. The main novelty lies in using the structure of bihomogeneous equations to obtain asymptotics in generically fewer variables than would be necessary in using the standard approach for homogeneous varieties. Also, we consider counting functions where not all the variables have to lie in intervals of the same size, which arises as a natural question in the setting of bihomogeneous varieties.

Bilinear forms for SL(2,q), An and similar groups.

Alexandre Turull (1992)

Publicacions Matemàtiques

The set of invariant symmetric bilinear forms on irreducible modules over fields of characteristic zero for certain groups is studied. Results are obtained under the presence in a finite group of elements of order four whose square is central. In particular, we find that the relevant modules for the groups mentioned in the title always accept an invariant symmetric bilinear form under which the module admits an orthonormal basis.

Billiard complexity in the hypercube

Nicolas Bedaride, Pascal Hubert (2007)

Annales de l’institut Fourier

We consider the billiard map in the hypercube of ℝ d . We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that n 3 d - 3 is the order of magnitude of the complexity.

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