Prüfer Sheaves and Generic Sheaves Over the Weighted Projective Lines and Elliptic Curves

In the present paper, we introduce the concepts of Prüfer sheaves and adic sheaves over a weighted projective line of genus one or an elliptic curve, show that Prüfer sheaves and adic sheaves can characterize the category of coherent sheaves. Moreover, we describe the relationship between Prüfer sheaves and generic sheaves, and provide two methods to construct generic sheaves by using coherent sheaves and Prüfer sheaves.     

Hashing to Elliptic Curves of $$j=0$$ and Mordell–Weil Groups

Mathematical Notes -     

Higher Gauss maps of Veronese varieties—A generalization of Boole’s formula and degree bounds for higher Gauss map images

The image of the higher Gauss map for a projective variety is discussed. The notion of higher Gauss maps here was introduced by Fyodor L. Zak as a generalization of both ordinary Gauss maps and conormal maps. The main result is a closed formula for the degree of those images of Veronese varieties. This yields a generalization of a classical formula by George Boole on the degree of the dual varieties of Veronese varieties in 1844. As an application of our formula, degree bounds for higher Gauss map images of Veronese varieties are given.     

A Necessary and Sufficient Condition for Solvability of the Matrix Equation AXB＋CYD＝E Over an Arbitrary Skew Field

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Existence and uniqueness theorems for A -regular generalized solutions of the first boundary-value problem for (A,674-01674-01674-01) — Elliptic equations

For second-order quasilinear degenerate elliptic equations, having the structure of -elliptic equations in a bounded domain Rⁿ, n2, one establishes theorems of existence and uniqueness for the generalized solutions of the first boundary-value problem, bounded together with their A -derivatives of first order and also of first and second order. The case of linear second-order -elliptic equations are separately considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 83–96, 1982.     

Elliptic Curves of Twin-Primes Over Gauss Field and Diophantine Equations

Letp,qbetwinprimenumberswithq-p=2.ConsidertheellipticcurvesE=EcisalsodenotedasE orE-wheng= 1or-l.HeretheMordell-WeilgroupandtherankoftheellipticcurveEovertheGaussfieldK=q(r1)(andovertherationalfieldQ)willbedeterminedinseveralcases;andresultsofsolutionsforrelatedDiophantineequationsandsimultaneousPellianequationswillbegiven.ThearithmeticconstructsoverQoftheellipticcurveEhavebeenstudiedin[ll,theSelmergroupsaredetermined,resultsonMordell-Weilgroup,rank,Shafarevich-Tategroup,andtorsionsubg…     

Elliptic and Hyperelliptic Curves Over Supersimple Fields

It is proved that if F is an infinite field with characteristicdifferent from 2, whose theory is supersimple, and C is an ellipticor hyperelliptic curve over F with generic ‘modulus’,then C has a generic F-rational point. The notion of generityhere is in the sense of the supersimple field F.     

On the Torsion of Elliptic Curves Over Quartic Number Fields

We determine which groups Z/MZZ/NZ occur infinitely often astorsion groups E(K)_tors when K varies over all quartic numberfields and E varies over all elliptic curves over K.     

Subtleties in the Distribution of the Numbers of Points on Elliptic Curves Over a Finite Prime Field

Three questions concerning the distribution of the numbers ofpoints on elliptic curves over a finite prime field are considered.First, the previously published bounds for the distributionare tightened slightly. Within these bounds, there are wildfluctuations in the distribution, and some heuristics are discussed(supported by numerical evidence) which suggest that numbersof points with no large prime divisors are unusually prevalent.Finally, allowing the prime field to vary while fixing the fieldof fractions of the endomorphism ring of the curve, the orderof magnitude of the average order of the number of divisorsof the number of points is determined, subject to assumptionsabout primes in quadratic progressions. There are implications for factoring integers by Lenstra's ellipticcurve method. The heuristics suggest that (i) the subtletiesin the distribution actually favour the elliptic curve method,and (ii) this gain is transient, dying away as the factors tobe found tend to infinity.     

Explicit Determination of Torsion Subgroups of Elliptic Curves Over Multi-Quadratic Fields

LotKbeanalgebraicnUmberfield,EanellipticcurveoverK.Thewell-knownMordelLWeiltheOemassertsthattheK-rationalpointsE(K)ofEisafinitelygeneratedabeliangroup(MordelgrouP):E(K)2E(K)torsxZ,whereE(K)torsisthetorsionsubgroup,ristherank(anonnegativeinteger),Zistheadditivegroupofrationalintegers.TheaboveisomorphismsometimesisaIsowrittellasequality:E(K)=E(K)torsxZ.TherearemanyliteraturesstudyingthetorsionsubgrouPE(K)tors'EspeciallyL.Merelrecenilyprovedtheuniformb0undednessconjecture:F0ranyp…     

Diophantine Equations and Bernoulli Polynomials

Given m, n 2, we prove that, for sufficiently large y, the sum 1 ⁿ +···+ y ⁿ is not a product of m consecutive integers. We also prove that for m n we have 1 ^m +···+ x ^m 1 ⁿ +···+ y ⁿ , provided x, y are sufficiently large. Among other auxiliary facts, we show that Bernoulli polynomials of odd index are indecomposable, and those of even index are almost indecomposable, a result of independent interest.     

与Gauss测度有关的椭圆方程比较结果中等号成立的条件

In this paper, we deal with a Dirichlet problem for linear elliptic equations related to Gauss measure. For this problem, we study the converse of some inequalities proved by other authors, in the sense that we study the case of equalities and show that equalities are achieved only in the ＂symmetrized＂ situations. In addition, under other assumptions, we give a different form of comparison results and discuss the corresponding case of equalities.