B. Sury

Elliptic Curves over Finite Fields

Jacobi was the first person to suggest (in 1835) using the group law on a cubic curve E. The chord-tangent method does give rise to a group law if a point is fixed as the zero element. This can be done over any field over which there is a rational point.

On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

Abstract Let g ( y ) ϵ Q [ Y ] be an irreducible polynomial of degree n ≥ 3. We prove that there are only finitely many rational numbers x, y with bounded denominator and an integer m ≥ 3 satisfying the equation x ( x + 1) ( x + 2)…( x + ( m − 1) ) = g ( y ). We also obtain certain finiteness results when g(y) is not an irreducible polynomial.

Bernoulli numbers and the riemann zeta function

In this article, we shall discuss several methods of evaluat ing the above sum. For instance, Marikkannan and Ravichandran have wri t ten about a method of evaluat ion using integration. Apart from Bernoulli s m e t h o d which we shall recall, we give a me thod akin to using integration, and one using differentiation. These methods are often useful in evaluating more general sums too as we shall indicate. Finally, we discuss the connections with the R iemann Zeta function.

Arithmetic groups and Salem numbers

We show that the existence of a sequence of elements from cocompact torsion-free arithmetic subgroups ofSL(2,R) converging to the identity is equivalent to the density of Salem numbers in [1,∞).

Cyclotomy and cyclotomic polynomials

As these n points on the circle are also the corners of a regular n-gon, the problem of cyclotomy is equivalent to the problem of constructing the regular n-gon using only a ruler and a compass. Euclids school constructed the equilateral triangle, the square, the regular pentagon and the regular hexagon. For more than 2000 years mathematicians had been unanimous in their view that for no prime p bigger than 5 can the p-gon be constructed by ruler and compasses. The teenager Carl Friedrich Gauss proved a month before he was 19 that the regular 17-gon is constructible. He did not stop there but went ahead to completely characterise all those n for which the regular n-gon is constructible! This achievement of Gauss is one of the most surprising discoveries in mathematics. This feat was responsible for Gauss dedicating his life to the study of mathematics instead of philology 1 in which too he was equally proficient.

The Chevalley-Warning theorem and a combinatorial question on finite groups

Recently, W. D. Gao (1996) proved the following theorem: For a cyclic group G of prime order, and any element a in it, and an arbitrary sequence 91i... 92p-1 of 2p 1 elements from G, the number of ways of writing a as a sum of exactly p of the gj s is 1 or 0 modulo p according as a is zero or not. The dual purpose of this note is (i) to give an entirely different type of proof of this theorem; and (ii) to solve a conjecture of J. E. Olson (1976) by answering an analogous question affirmatively for solvable groups.

Generators for all principal congruence subgroups of SL(n, Z) with n ≥ 3

We show that there is a uniform bound for the numbers of generators for all principal congruence subgroups of SL(n, Z) for n ≥ 3. On the other hand, we show that the numbers are unbounded if we work with all arithmetic subgroups of SL(n, Z).

A pretty binomial identity

Elementary proofs abound: the first identity results from choosing x = y = 1 in the binomial expansion of (x+y). The second one may be obtained by comparing the coefficient of x in the identity (1 + x)(1 + x) = (1 + x). The reader is surely aware of many other proofs, including some combinatorial in nature. At the end of the previous century, the evaluation of these sums was trivialized by the work of H. Wilf and D. Zeilberger [8]. In the preface to the charming book [8], the authors begin with the phrase You’ve been up all night working on your new theory, you found the answer, and it is in the form that involves factorials, binomial coefficients, and so on, ... and then proceed to introduce the method of creative telescoping discussed in Section 3. This technique provides an automatic tool for the verification of these type of identities. The points of view presented in [3] and [10] provide an entertaining comparison of what is admissible as a proof. In this short note we present a variety of proofs of the identity

A note on the special unitary group of a division algebra

If D is a division algebra with its center a number field K and with an involution of the second kind, it is unknown if the group SU(1, D)/[U(1, d), U(1, D)] is trivial. We show that, by contrast, if K is a function field in one variable over a number field, and if D is an algebra with center K and with an involution of the second kind, the group SU(1, D)/[U(1, d), U(1, D)] can be infinite in general. We give an infinite class of examples.

Congruence subgroup problem for anisotropic groups over semilocal rings

In Chapter I, a theorem of Margulis which gives the structure of normal subgroups ofSL(1,D) for a quaternion division algebraD over a global fieldK of characteristic not 2, is generalized to semi-local ringsR inK. Using this, we obtain in Chapter II, a description of normal subgroups ofG(R) forK-anisotropic algebraic groupsG of typesA3, Bn, Cn,1Dn,2Dn and some forms of2An. As a Corollary, a proof of the Platonov-Margulis conjecture is obtained for the above groups.