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## Publications

Publications (107)

This exposition reviews what exactly Gauss asserted and what did he prove in the last chapter of {\sl Disquisitiones Arithmeticae} about dividing the circle into a given number of equal parts. In other words, what did Gauss claim and actually prove concerning the roots of unity and the construction of a regular polygon with a given number of sides....

This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form p = 8k +1, receives attention.

This exposition reviews what exactly Gauss asserted and what did he prove in the last chapter of {\sl Disquisitiones Arithmeticae} about dividing the circle into a given number of equal parts. In other words, what did Gauss claim and actually prove concerning the roots of unity and the construction of a regular polygon with a given number of sides....

Isomonodromy for the fifth Painlev\'e equation P_5 is studied in detail in the context of certain moduli spaces for connections, monodromy, the Riemann--Hilbert morphism, and Okamoto-Painlev\'e spaces. This involves explicit formulas for Stokes matrices and parabolic structures. The rank 4 Lax pair for P_5, introduced by Noumi--Yamada et al., is sh...

For the hyperelliptic curve Cp with equation y2=x(x−2p)(x−p)(x+p)(x+2p) with p a prime number, we discuss bounds for the rank of its Jacobian over Q, find many cases having 2-torsion in the associated Shafarevich-Tate group, and we present some results on rational points of Cp.

This paper discusses prime numbers that are (resp. are not) congruent numbers. Particularly the only case not fully covered by earlier results, namely primes of the form $p=8k+1$, receives attention.

This note reformulates Mazur’s result on the possible orders of rational torsion points on elliptic curves over $$\mathbb {Q}$$ Q in a way that makes sense for arbitrary genus one curves, regardless whether or not the curve contains a rational point. The main result is that explicit examples are provided of ‘pointless’ genus one curves over $$\math...

For the hyperelliptic curve C_p with equation y^2=x(x-2p)(x-p)(x+p)(x+2p) with p a prime number, we discuss bounds for the rank of its Jacobian over Q, find many cases having 2-torsion in the associated Shafarevich-Tate group, and we present some results on rational points of C_p.

In this paper, we give explicit equations for homogeneous spaces corresponding to a rational isogeny of degree $3$. An explicit set of elliptic curves with elements of order $3$ in their Tate-Shafarevich group is constructed. Combining this gives explicit examples of plane cubics over $\Q$ that have a point everywhere locally, but not globally.

We prove the Hasse-Weil inequality for genus 2 curves given by an equation of the form y^2 = f(x) with f a polynomial of degree 5, using arguments that mimic the elementary proof of the genus 1 case obtained by Yu. I. Manin in 1956.

Given a polynomial f with coefficients in a field of prime characteristic p, it is known that there exists a differential operator that raises 1/f to its pth power. We first discuss a relation between the “level” of this differential operator and the notion of “stratification” in the case of hyperelliptic curves. Next, we extend the notion of level...

For a finite irreducible subgroup H⊂PSL(Cn) and an irreducible, H-invariant curve Z⊂P(Cn) such that C(Z)H=C(t), a standard differential operator Lst∈C(t)[ddt] is constructed. For n=2 this is essentially Klein's work. For n>2 an actual calculation of Lst is done by computing an evaluation of invariants C[X1,…,Xn]H→C(t) and applying a scalar form of...

This note presents explicit equations (up to birational equivalence over $\mathbb{F}_2$) for a complete, smooth, absolutely irreducible curve $X$ over $\mathbb{F}_2$ of genus $50$ satisfying $#X(\mathbb{F}_2)=40$. In his 1985 Harvard lecture notes on curves over finite fields, J-P.~Serre already showed the existence of such a curve: he used class f...

Motivated by some string and plaster models dating back from the late 19th and early 20th century, this note recalls some of the early history of the classification of plane cubic curves over the real numbers. Examples of different classifications are provided, showing their connection with some of the models in the Schilling collection.

We prove the Hasse-Weil inequality for genus 2 curves given by an equation of the form y^2 = f(x) with f a polynomial of degree 5, using arguments that mimic the elementary proof of the genus 1 case obtained by Yu. I. Manin in 1956.

The problem of algebraic dependence of solutions to (non-linear) first order autonomous equations over an algebraically closed field of characteristic zero is given a `complete' answer, obtained independently of model theoretic results on differentially closed fields. Instead, the geometry of curves and generalized Jacobians provides the key ingred...

We study hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs (q,d) such that the hyperelliptic curve C over a finite field F q 2 given by y ² =φ d (x) is maximal over the finite field F q 2 of cardinality q ² . Here φ d (x) denotes the Chebyshev polynomial of degree d. The same question is stu...

Given a polynomial $f$ with coefficients in a field of prime characteristic $p$, it is known that there exists a differential operator that raises $1/f$ to its $p$th power. We first discuss a relation between the `level' of this differential operator and the notion of `stratification' in the case of hyperelliptic curves. Next we extend the notion o...

We study hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs $(q,d)$ such that the hyperelliptic curve $\cC$ over a finite field $\FF_{q^2}$ corresponding to the equation $y^2 = \varphi_{d}(x)$ is maximal over the finite field $\FF_{q^2}$ of cardinality $q^2$. Here $\varphi_{d}(x)$ denotes the...

For a differential operator $L$ of order $n$ over $C(z)$ with a finite (differential) Galois group $G\subset {\rm GL}(C^n)$, there is an algorithm, by M. van Hoeij and J.-A.~Weil, which computes the associated evaluation of the invariants $ev:C[X_1,\dots ,X_n]^G\rightarrow C(z)$. The procedure proposed here does the opposite: it uses a theorem of E...

We construct rational Poncelet configurations, which means finite sets of pairwise distinct [Formula: see text]-rational points [Formula: see text] in the plane such that all [Formula: see text] are on a fixed conic section defined over [Formula: see text], and moreover the lines [Formula: see text] are all tangent to some other fixed conic section...

The Fricke-Macbeath curve is a smooth projective algebraic curve of genus 7 with automorphism group PSL2(F8). We recall two models of it (introduced, respectively, by Maxim Hendriks and by Bradley Brock) defined over ℚ, and we establish an explicit isomorphism defined over ℚ(√−7) between these models. Moreover, we decompose up to isogeny over ℚ the...

Schoof’s classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a g...

This note uses a diophantine problem arising in elementary geometry as a prerequisite to illustrate some theory of elliptic curves. As a typical example, Proposition 2.4 and Theorem 3.1 determine the exact set of rational numbers for which the specialization homomorphism from the torsion free rank 2 group of rational points on some elliptic curve o...

This paper intends to focus on the universal property of this Hesse pencil and of its twists. The main goal is to do this as explicit and elementary as possible, and moreover to do it in such a way that it works in every characteristic different from three.

In this note we extend the theory of twists of elliptic curves as presented in various standard texts for characteristic not equal to two or three to all remaining cases. For this, we make explicit use of the correspondence between the twists and the first Galois cohomology set with values in the automorphism group of the elliptic curve. The result...

The Morales-Ramis theorem can be applied to Painlev\'e equations. As in a paper of Morales-Ruiz we associate an autonomous Hamiltonian $H$ to a Painlev\'e equation $P$. Reducibility of $P$ and complete integrability of $H$ are discussed. The normal variational equation of $H$ is shown to be equivalent to the variational equation of $P$. This leads...

Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a g...

This is a sequel to papers by the last two authors making the Riemann-Hilbert correspondence and isomonodromy explicit. For the degenerate fifth Painleve equation, the moduli spaces for connections and for monodromy are explicitly computed. It is proven that the extended Riemann--Hilbert morphism is an isomorphism. As a consequence these equations...

Een tijdje geleden stond in dit tijdschrift een artikel over zogeheten Littlewood-nulpunten.
Het ging toen vooral om de prachtige plaatjes die deze nulpunten opleveren. In dit artikel
gaan majken roelfszema, Eduardo ruíz duarte en Jaap top, alle drie van de rijksuniversiteit groningen, dieper in op de wiskunde achter deze nulpunten.

The notions of equivalence and strict equivalence for order one
differential equations are introduced. The more explicit notion of
strict equivalence is applied to examples and questions concerning
autonomous equations and equations having the Painleve property. The
order one equation determines an algebraic curve. If this curve has
genus zero or o...

Using elementary techniques, a question named after the famous Russian mathematician I. M. Gelfand is answered. This concerns the leading (i.e., most significant) digit in the decimal expansion of integers 2n , 3n , . . . , 9n . The history of this question, some of which is very recent, is reviewed.

This text discusses triangles with the property that a bisector at one vertex, the median at another, and the altitude at the third vertex are collinear. It turns out that since the 1930s, such triangles appeared in the problem sections of various journals. We recall their well known relation with points on a certain elliptic curve, and we present...

In 1956 Yu.I. Manin published an elementary proof of Helmut Hasse's 1933 result stating that the Riemann hypothesis holds in the case of an elliptic function field over a finite field. We briefly explain how Manin's proof relates to more modern proofs of the same result. This enables us to present an analogous elementary proof for the case of finit...

Stratification for nonlinear differential equations in positive characteristic is introduced. Testing this notion for first order equations is discussed, and related to the Cartier operator on curves. A variant of the Grothendieck–Katz conjecture is formulated, and proved in a special case.

The Riemann-Hilbert approach for the equations PIII (D 6 ) and PIII (D 7 ) is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlevé varieties, the Painlevé property, special solutions and explicit Bäcklund transformations.

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevé equation. One obtains a Riemann–Hilbert correspondence between moduli spaces of rank two connections on 1 and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto–Painlevé varieties and the Painlevé property follows. For an ex...

This note contains an application of the algebraic study by Schütt and Shioda of the elliptic modular surface attached to the commutator subgroup of the modular group. This is used here to provide algebraic descriptions of certain coverings of a j-invariant 0 elliptic curve, unramified except over precisely one point.

This paper applies methods of Van der Put and Van derPut-Saito to the fourth
Painlev\'e equation. One obtains a Riemann--Hilbert correspondence between
moduli spaces of rank two connections on $\mathbb{P}^1$ and moduli spaces for
the monodromy data. The moduli spaces for these connections are identified with
Okamoto--Painlev\'e varieties and the Pa...

The illustration1 shown here appears in the master’s thesis of Mary Emily Sinclair (1878–1955). Her thesis, supervised by Oscar Bolza at the University of Chicago in 1903, deals with quintic polynomials p(t) =t5 + xt3 + yt + z. Every (real) triple (x, y, z) yields such a polynomial, and Sinclair describes the sets

The text shown here is from lectures by Felix Klein (1849–

In this article we recall how to describe the twists of a curve over a finite field and we show how to compute the number of rational points on such a twist by methods of linear algebra. We illustrate this in the case of plane quartic curves with at least 16 automorphisms. In particular we treat the twists of the Dyck–Fermat and Klein quartics. Our...

In this paper, an explicit construction of binary self-dual cyclic codes of length n going to infinity with a minimal distance at least half the square root of n is presented. The same idea is also used to construct more general binary cyclic codes with a large minimal distance. Finally, in the special case of self-dual cyclic codes, a simplified v...

Extending a geometric construction due to Sederberg and to Bajaj, Holt, and Netravali, an algorithm is presented for parameterizing a nonsingular cubic surface by polynomials of degree three. The fact that such a parametrization exists is classical. The present algorithm is, by its purely geometric nature, a very natural one. Moreover, it contains...

This note presents a relatively straightforward proof of the fact that, under certain congruence conditions on a, b, c ∈ Q, the group of rational points over Q¯(t) on the elliptic curve given by y^2 = x^3 + t^3(t^2 + at + b)^2 (t + c)x + t^5(t^2 + at + b)^3 is trivial. This is used to show that a related elliptic curve yields a free abelian group o...

New historical aspects of the classification, by Cayley and Cremona, of ruled quartic surfaces and the relation to string models and plaster models are presented. In a `modern' treatment of the classification of ruled quartic surfaces the classical one is corrected and completed. A conceptual proof is presented of a result of Rohn concerning curves...

Throughout this exposition K will denote a number field and C/K will be an absolutely irreducible (smooth and complete) curve of genus g ≥ 1. In fact more generally one can take for K any field equipped with a product formula as defined in [70, p. 7]. We make the standing assumption that over K, a divisor class of degree 1 on C exists (this can alw...

In this note we classify, up to conjugation, all algebraic subgroups of GL2(ℂ).

The topological classification of smooth real cubic surfaces is recalled and compared to the classification in terms of the number of real lines and of real tritangent planes, as obtained by L. Schläfli in 1858. Using this, explicit examples of surfaces of every possible type are given.

Let E → B be an elliptic surface defined over the algebraic closure of a finite field of characteristic greater than 5. Let W be a resolution of singularities of E ×B E. We show that the l-adic Abel–Jacobi map from the l-power-torsion in the second Chow group of W to H3(W,Zl(2)) ⊗ Ql/Zl is an isomorphism for almost all primes l. A main tool in the...

This paper studies the arithmetic of the extremal elliptic K3 surface with configuration of singular fibres [19,1,1,1,1,1]. We give a model over Q such that the Neron Severi group is generated by divisors over Q, and we describe the local Hasse-Weil zeta-functions in terms of a modular form of weight 3. Furthermore we verify the Tate conjecture for...

We determine conductor exponent, minimal discriminant and fibre type for elliptic curves over discrete valued fields of equal characteristic 3. Along the same lines, partial results are obtained in equal characteristic 2.

A curve over a field k is pointless if it has no k-rational points. We show that there exist pointless genus-3 hyperelliptic curves over a finite field F_q if and only if q < 26, that there exist pointless smooth plane quartics over F_q if and only if either q < 24 or q = 29 or q = 32, and that there exist pointless genus-4 curves over F_q if and o...

In a recent paper Ahlgren, Ono and Penniston described the L-series of K3 surfaces from a certain one-parameter family in
terms of those of a particular family of elliptic curves. The Tate conjecture predicts the existence of a correspondence between
these K3 surfaces and certain Kummer surfaces related to these elliptic curves. A geometric constru...

We present explicit equations of semi-stable elliptic surfaces (i.e., having only type $I_n$ singular fibers) which are associated to the torsion-free genus zero congruence subgroups of the modular group as classified by A. Sebbar.

We present a table containing the maximal number of rational points on a genus 3 curve over a field of cardinality q, for all q<100. Also, some remarks on Frobenius non-classical quartics over finite fields are given.

We show that every elliptic curve over a finite field of odd characteristic whose number of rational points is divisible by 4 is isogenous to an elliptic curve in Legendre form, with the sole exception of a minimal respectively maximal elliptic curve. We also collect some results concerning the supersingular Legendre parameters.

We explain a naive approach towards the problem of finding genus 3 curves C over any given finite field \(
\mathbb{F}_q
\)
of odd characteristic, with a number of rational points close to the Hasse-Weil-Serre upper bound \(
q + 1 + 3\left[ {2\sqrt q } \right]
\)
. The method turns out to be successful at least in characteristic 3.

this paper we list some Hecke eigenvalues of several automorphic forms for congruence subgroups of SL(3; Z). Combining the methods from [AGG] with the Lenstra--Lenstra--Lov'asz algorithm we were able to handle much larger levels than in [AGG] and [vG-T]. Comparing these tables with results from computations of Galois representations, we find furthe...

Using some theory of (rational) elliptic surfaces plus elementary properties of a Mordell-Weil group regarded as module over the endomorphism ring of a (CM) elliptic curve, we present examples of such surfaces with j-invariant zero. In particular, a K3 example is given with the maximal possible rank 18, plus that many explicit independent sections...

We study the surface arising from the diophantine equation $m^3+(m+1)^3+...+(m+k-1)^3=l^2$. It turns out that this is a $K3$ surface with Picard number 20. We stduy its aritmetic properties in detail. We construct elliptic fibrations on it, and we find a parametric solution to the original equation. Also, we determine the Hasse-Weil zeta function o...

this paper some Hecke eigenvalues of several automorphic forms for congruence subgroups of SL(3; Z). By combining the methods from [Ash et al. 1984] with the Lenstra--Lenstra-- Lov'asz algorithm, we were able to handle much larger levels than was the case in [Ash et al. 1984] and [van Geemen and Top 1994]. Comparing these tables with results from c...

A variant of a conjecture of Beilinson and Bloch relates the rank of the Griffiths group of a smooth projective variety over a number field to the order of vanishing of an L-function at the center of the critical strip. Presently, there is little evidence to support the conjecture, especially when the L-function vanishes to order greater than 1. We...

In this paper, Hecke eigenvalues of several automorphic forms for congruence subgroups of SL(3; Z) are listed. To compute such tables, we describe an algorithm which combines techniques developed by Ash, Grayson and Green with the Lenstra--Lenstra--Lov'asz algorithm.

We list here Hecke eigenvalues of several automorphic forms for congruence subgroups of $\SL(3,{}$\funnyZ$)$. To compute such tables, we describe an algorithm that combines techniques developed by Ash, Grayson and Green with the Lenstra--Lenstra--Lovász algorithm. With our implementation of this new algorithm we were able to handle much larger leve...

Some arithmetic of elliptic curves and theory of elliptic surfaces is used to nd all rational solutions (r;s;t) in the function eld Q(m;n) of the pair of equations r(r + 1)=2 = ms(s + 1)=2 r(r + 1)=2 = nt(t + 1)=2: )

This paper gives an expository account of our experiments concerning relations between modular forms for congruence subgroups of SL(3,Z) and three dimensional Galois representations. The main new result presented here is a calculation of the variations of the Hodge structure corresponding to the motives we consider in realizing the Galois represent...

this paper was written at BYU in Provo, and the computercalculations were done on the system of the Math. Dept. of the Univ. of Colorado in Boulder.2 The construction

The Langlands philosophy contemplates the relation between auto-morphic representations and Galois representations. A particularly interesting case is that of the non-selfdual automorphic representations of GL 3. Clozel conjectured that the L-functions of certain of these are equal to L-functions of Galois representations. Here we announce that we...

The theory of Mordell-Weil lattices is applied to a specific example of a rational elliptic surface. This provides a complete description of the sections of this surface, and of the sections which are defined over Q. The surface is related to the diophantine problem of expressing squares as a sum of consecutive squares. Some consequences which our...

This bibliography updates and on some minor spots corrects the one published in B.L. van der Waerden, Zur algebraischen Geometrie (Selected Papers), mit einem Geleitwort van F. Hirze-bruch, Springer Verlag, Berlin etc. 1983. The choice to basically repeat the much longer list of publications before 1983 here again is made for at least two reasons....

In his work on Diophantine equations of the form y2=ax4+bx3+cx2+dx+e, Fermat introduced the notion of primitive solutions. In this expository note we intend to interpret this notion more geometrically, and explain what it means in terms of the arithmetic of elliptic curves. The specific equation y2=x4+4x3+10x2+20x+1 was used extensively by Fermat a...

The aim of this paper is to present a very explicit construction of one parameter families of hyperelliptic curves C of genus ( p −1 )/ 2, for any odd prime number p , with the property that the endomorphism algebra of the jacobian of C contains the real subfield Q (2 cos(2 π / p )) of the cyclotomic field Q ( e 2π i/p ).
Two proofs of the fact tha...

A list is given of elliptic curves over ℚ having additive reduction at exactly one prime. It is also proved that for primes congruent to 5 modulo 12, no such curves having potentially good reduction exist. This enables one to find in a number of cases a complete list of all elliptic curves with bad reduction at only one prime.

A list is given of elliptic curves over Q having additive reduction at exactly one prime. It is also proved that for primes congruent to 5 modulo 12, no such curves having potentially good reduction exist. This enables one to find in a number of cases a complete list of all elliptic curves with bad reduction at only one prime.

Let K be a number field and A an abelian variety over K. The K-rational points of A are known to constitute a finitely generated abelian group (Mordell-Weil theorem). The problem studied in this paper is to find an explicit upper bound for the rank r of its free part in terms of other invariants of A/K. This is achieved by a close inspection of the...

Abstract In this paper families of elliptic curves admitting a rational isogeny of degree 3 are studied. It is known that the 3-torsion in the class group of the,eld dened by the points in the kernel of such an isogeny is related to the rank of the elliptic curve. Families in which almost all the curves have rank at least 3 are constructed. In some...

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