Lehmer's conjecture. I

One formulates a conjecture which contains Lehmer's conjecture as a special case and one derives an identity which is equivalent to the negation of the conjecture.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 91, pp. 52–80, 1979.     

A reduction theorem for the Alperin weight conjecture

The Alperin Weight Conjecture was stated by J. Alperin in 1986 and constitutes one of the main problems in the representation theory of finite groups. In this paper, we provide a reduction to simple groups and as an application we prove the conjecture in several cases.     

On the Ramanujan-Petersson conjecture for modular forms of half-integral weight

It is shown that a “character twist” of a growth estimate for certain weighted infinite sums of Kloosterman sums which is equivalent to the Ramanujan-Petersson conjecture for modular forms of half-integral weight, can easily be proved using Deligne’s theorem (previously the Ramanujan-Petersson conjecture for modular forms of integral weight).     

Proof of an exceptional zero conjecture for elliptic curves over function fields

Based on the analogy between number fields and function fields of one variable over finite fields, we formulate and prove an analogue of the exceptional zero conjecture of Mazur, Tate and Teitelbaum for elliptic curves defined over function fields. The proof uses modular parametrization by Drinfeld modular curves and the theory of non-archimedean integration. As an application we prove a refinement of the Birch-Swinnerton-Dyer conjecture if the analytic rank of the elliptic curve is zero.     

Index four orientable embeddings and case zero of the Heawood conjecture

By imposing a special symmetry, we are able to construct index four triangular embeddings of graphs in compact orientable 2-manifolds. Because of the complexity of the current graphs required, such embeddings have heretofore been unattainable, but the imposed symmetry reduces the problem to constructing a special kind of index two current graph. We illustrate the method with a solution for case zero of the Heawood conjecture, using an abelian group, thus completing a constructive proof of the Heawood map color theorem, and eliminating the need for Galois field theory and nonabelian groups in its solution. The method has also been used in the determination of the genus of K_n,n,n,n.     

The standard identity in characteristicp a conjecture of I.B. Volichenko

In this paper we present a full proof of the result which was announced at the International Congress of Mathematicians in Kyoto (August, 1990): Any associative PI-algebra over a field of characteristicp satisfies the standard identity. The final version of this paper was written while the author was visiting Bar-Ilan University.     

Elliptic curves of rank zero satisfying the p-part of the Birch and Swinnerton-Dyer conjecture

Let ${p \in \{3,5,7\}}$ and ${E/\mathbb{Q}}$ an elliptic curve with a rational point P of order p. Let D be a square-free integer and E _D the D-quadratic twist of E. Vatsal (Duke Math J 98:397–419, 1999) found some conditions such that E _D has (analytic) rank zero and Frey (Can J Math 40:649–665, 1988) found some conditions such that the p-Selmer group of E _D is trivial. In this paper, we will consider a family of E _D satisfying both of the conditions of Vatsal and Frey and show that the p-part of the Birch and Swinnerton-Dyer conjecture is true for these elliptic curves E _D . As a corollary we will show that there are infinitely many elliptic curves ${E/\mathbb{Q}}$ such that for a positive portion of D, E _D has rank zero and satisfies the 3-part of the Birch and Swinnerton-Dyer conjecture. Previously only a finite number of such curves were known, due to James (J Number Theory 15:199–202, 1982).     

The Conley conjecture for the cotangent bundle

We prove the Conley conjecture for cotangent bundles of oriented, closed manifolds, and Hamiltonians which are quadratic at infinity, i.e., we show that such Hamiltonians have infinitely many periodic orbits. For the conservative systems, similar results have been proven by Lu and Mazzucchelli using convex Hamiltonians and Lagrangian methods. Our proof uses Floer homological methods from Ginzburg’s proof of the Conley conjecture for closed symplectically aspherical manifolds.     

The Lalonde-McDuff conjecture for nilmanifolds

We prove that any Hamiltonian bundle whose fiber is a nilmanifold c-splits.     

Algebraic classification of physical structures with zero. I

An algebraic version is considered of the axiomatization of physical structures. An arbitrary set R with a distinguished element O (zero) is taken as a set of measurements. Under an additional condition, understood to be an analog of the requirement that a physical structure is one-metric, the structure of a topological skew field with zero O is introduced on R; and on the object sets $\mathcal{M}$ and $\mathcal{N}$ , the structure of finite-dimensional vector spaces over the skew field is introduced. This leads to a complete classification of the corresponding physical structures. The classification theorem can be considered also as a variant of the axiomatics connected with a bilinear form on a pair of finite-dimensional vector spaces over a skew field; i.e., the variant which uses, as axioms, only the combinatorial properties of a bilinear form as a map $\mathcal{M} \times \mathcal{N} \to R$ (i.e., without the axioms of addition and multiplication).     

On the Petersson Norm of Certain Siegel Modular Forms

We shall prove a rationality result for a quotient of scalar products involving the Ikeda lift of an elliptic cusp form.     

The falsity of the reconstruction conjecture for tournaments

The conjecture that for all sufficiently large p any tournament of order p is uniquely reconstructable from its point-deleted subtournaments is shown to be false. Counterexamples are presented for all orders of the form 2ⁿ + 1 and 2ⁿ + 2. The largest previously known counterexamples were of order 8.