Serre Lie algebras of generalized Jacobians

In this work we construct an example of a generalized Jacobian of an elliptic curve defined over a field of algebraic numbers k such that the Serre Lie algebra p-adic representation of the Galois group of the algebraic closure of the field k in its Tate module is irreducible.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 571–576, April, 1976.In conclusion the authors thank O. N. Vvedenskii for guidance in this work.     

Tate-Shafarevich products in elliptic curves over pseudolocal fields with residue fields of characteristic 3

Let k be a general local field with pseudolocal residue field x, char x=3, and A an elliptic curve defined over k. It is proved that the Tate-Shafarevich product H¹(k, A)×A_k Q/ of the group H¹(k, A) of principal homogeneous spaces of the curve A over k and the group A_k of its k-rational points is left nondegenerate.Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1157–1165, September, 1992.     

Elliptic curves of rank 1 satisfying the 3-part of the Birch and Swinnerton–Dyer conjecture

Let E be an elliptic curve over Q of conductor N and K be an imaginary quadratic field, where all prime divisors of N split. If the analytic rank of E over K is equal to 1, then the Gross and Zagier formula for the value of the derivative of the L-function of E over K, when combined with the Birch and Swinnerton–Dyer conjecture, gives a conjectural formula for the order of the Shafarevich–Tate group of E over K. In this paper, we show that there are infinitely many elliptic curves E such that for a positive proportion of imaginary quadratic fields K, the 3-part of the conjectural formula is true.     

Ebene algebraische Kurven vom Typ p,q

Let p,q be relatively prime integers with 2pr ^p,q be the numerical semigroup generated by p,q,{(p–1) (q–1)–1–(ip+jq)¦i+jr–2}. Then there exists a smooth projective curve X and a point x on X, such that H _r ^p,q is the set of orders of poles of the rational functions on X, which are regular on X\{x}; in other words: H _r ^p,q is a Weierstraß semigroup.     

Elliptic Tate curves over local Γ-extensions

The group A(K)/N is computed, where A(K) is the group of points of a Tate curve over a local field while N is the group of universal norms from the group of points over a -extension. As an application, the Mazurl-modulus of modular elliptic curves is computed for values ofl dividing the denominator of the absolute invariant.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 531–539, April, 1973.In conclusion, I wish to thank Yu. I. Manin for having guided this work.     

Local–global invariants of finite and infinite groups: Around Burnside from another side

This expository essay is focused on the Shafarevich–Tate set of a group G$G$. Since its introduction for a finite group by Burnside, it has been rediscovered and redefined more than once. We discuss its various incarnations and properties as well as relationships (some of them conjectural) with other local–global invariants of groups.     

A Remark on an Algebraic Riemann—Roch Formula for Flat Bundles

Let f:X S be a smooth projective morphism over an algebraically closed field, with X and S regular. When E, ) is a flat bundle over X, then its Gauss–Manin bundles on S have a flat connection and one may ask for a Riemann–Roch formula relating the algebraic Chern–Simons and Cheeger–Simons invariants. We give an answer for X = Y × S, f = projection. The method of proof is inspired by the work of Hitchin and Simpson.     

A Duality Theorem for Étale p-Torsion Sheaves on Complete Varieties over a Finite Field

Let X be an arbitrary variety over a finite field k and p=char k,n N. We will construct a complex of étale sheaves on X together with trace isomorphism from the highest étale cohomology group of this complex onto Z/pⁿZ such that for every constructible Z/pⁿZ-sheaf on X the Yoneda pairing is a nondegenerate pairing of finite groups. If X is smooth, this complex is the Gersten resolution of the logarithmic de Rham–Witt sheaf introduced by Gros and Suwa. The proof is based on the special case proven by Milne when the sheaf is constant and X is smooth, as well as on a purity theorem which in turn follows from a theorem about the cohomological dimension of C_i-fields due to Kato and Kuzumaki. If the existence of the Lichtenbaum complex is proven, the theorem will be the p-part of a general duality theorem for varieties over finite fields.     

Lubin–Tate Formal Groups over the Ring of Integers of a Multidimensional Local Field

Some isomorphism criteria for Lubin–Tate formal groups over the ring of integers of a multidimensional local field are given. Bibliography: 4 titles.     

Sylow 2-subgroups of the group U(q2)

We obtain a classification of the Sylow 2-subgroups of the unitary group U(q²) over the finite field GF(q²) for an odd number q.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1091–1093, July–August, 1991.     

Modular Form Congruences and Selmer Groups

Motivated by Cremona and Mazur's notion of visibility of elementsin Shafarevich–Tate groups [6, 27], there have been anumber of recent works which test its compatibility with theBirch and Swinnerton–Dyer conjecture and the Bloch–Katoconjecture. These conjectures provide formulas for the ordersof Shafarevich–Tate groups in terms of values of L-functions.For example, one may see recent work of Agashe, Dummigan, Steinand Watkins [1, 2, 10, 11]. In their examples, they find thatthe presence of visible elements agrees with the expected divisibilityproperties of the relevant L-values.     

Two-dimensional exponential model of the Euclidean field theory

It is proved that the two-dimensional exponential model of the field theory is trivial for ² > 8.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 469–477, April, 1990.     

Endomorphisms of abelian varieties and points of finite order in characteristic p

An analog of the Tate hypothesis on homomorphisms of Abelian varieties is proved, in which points of sufficiently large prime order figure in place of the Tate modules. As is the case with the Tate hypothesis, this assertion follows formally from a finiteness hypothesis for isogenies of Abelian varieties, which is proved in characteristic p > 2 and for finite fields. The same methods are used to prove the finiteness of the set of Abelian varieties of a given dimension over a finite field.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 737–744, June, 1977.     

A theorem of Suslin

Suppose that f is the field of algebraic functions of one variable over a field of constants k, v is a point of the field f, and that A_v is the ring of functions which do not have poles outside {inv}. It is proved that if E0₄(A_v) is a normal divisor in 0₄(A_v) then degV=1,Fk(X), and, consequently A_vK[X].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 119–121, 1983.     

特征≠2的有限域上对称矩阵方程解的计数公式及其q超几何级数表达

设Ｆｑ是特征≠２的有限域．本文利用Ｆｑ上奇异正交几何的理论，给出当Ａ，Ｂ分别是Ｆｑ上阶ｎ秩２ν＋δ和阶ｍ秩２ｓ＋γ的对称矩阵时，Ｆｑ上适合方程ＸＡＸ′＝Ｂ的秩ｋ的解Ｘ的个数和解Ｘ的个数的明显公式，并且用ｑ超几何级数简化表达解数公式．     

The group F2K(X) as a birational invariant of a surface

The birational invariance is proved of the group F²K(X), where X is a smooth projective surface over a field.Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 15–20, January, 1972.The author expresses his gratitude to Yu. I. Manin for his interest.     

One theorem of Cohn

Let F be the field of algebraic functions of one variable over the field of constants k, v be a point of field F/k, and A_v be the ring of functions not having poles outside point v. It is proved that A_v is a GE₂-ring if and only if it coincides with the ring k[X] of polynomials of one variable over field k.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 127–130, 1976.     

On the conjugacy classes in maximal unipotent subgroups of simple algebraic groups

Let G be a simple algebraic group over the algebraically closed field k of characteristic p ≥ 0. Assume p is zero or good for G. Let B be a Borel subgroup of G; we write U for the unipotent radical of B and u for the Lie algebra of U. Using relative Springer isomorphisms} we analyze the adjoint orbits of U in u. In particular, we show that an adjoint orbit of U in u contains a unique so-called minimal representative. In case p > 0, assume G is defined and split over the finite field of p elements F_p. Let q be a power of p and let G(q) be the finite group of F_q-rational points of G. Let F be the Frobenius morphism such that G(q) = G^F. Assume B is F-stable, so that U is also F-stable and U(q) is a Sylow p-subgroup of G(q). We show that the conjugacy classes of U(q) are in correspondence with the F-stable adjoint orbits of U in u. This allows us to deduce results about the conjugacy classes of U(q).     

On the cardinality of blocking sets in PG(2,q)

Denote by PG(2,q) the finite desarguesian projective plane of order q, where q=p^h, p a prime, q>2. We define the function m(q) as follows: m(q)=q, if q is a square; m(q)=(q+1)/2, if q is a prime; m(q)=p^h–d, if q=p^h with h an odd integer, where d denotes the greatest divisor of h different from h. The following theorem is proved: For any integer k with q+m(q)+1 k q²–m(q), there exists a blocking set in PG(2,q) having exactly k elements.To Professor Adriano Barlotti on his 60th birthday.Research partially supported by G.N.S.A.G.A. (CNR)     

Beispiele dreidimensionaler Hilbertscher Modulmannigfaltigkeiten von allgemeinem Typ

Let k be a totally real algebraic number field of degree n with ring of integers o. By studying certain invariants r_m of the isolated singularities of the usual compact ification X(k) of the quotient ⁿ / Sl (2,0) we construct some example of totally real cubic number fields k such that the field of meromorphic functions of X(k) is not rational.