Prime analogues of the Erd?s-Kac theorem for elliptic curves

Let E/Q be an elliptic curve. For a prime p of good reduction, let E(Fp) be the set of rational points defined over the finite field Fp. We denote by ω(#E(Fp)), the number of distinct prime divisors of #E(Fp). We prove that the quantity (assuming the GRH if E is non-CM)

     

On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.     

Height difference bounds for elliptic curves over number fields

Let E be an elliptic curve over a number field K. Let h be the logarithmic (or Weil) height on E and be the canonical height on E. Bounds for the difference are of tremendous theoretical and practical importance. It is possible to decompose as a weighted sum of continuous bounded functions Ψ_υ:E(K_υ)→R over the set of places υ of K. A standard method for bounding , (due to Lang, and previously employed by Silverman) is to bound each function Ψ_υ and sum these local ‘contributions’.In this paper, we give simple formulae for the extreme values of Ψ_υ for non-archimedean υ in terms of the Tamagawa index and Kodaira symbol of the curve at υ.For real archimedean υ a method for sharply bounding Ψ_υ was previously given by Siksek [Rocky Mountain J. Math. 25(4) (1990) 1501]. We complement this by giving two methods for sharply bounding Ψ_υ for complex archimedean υ.     

Points entiers et modèles des courbes algébriques

LetC: F(X, Y)=0 be an algebraic curve of genus 1, over a number fieldK. In this work we construct a modelG(Z,W)=0 of the curveC, over a fixed number fieldL with , having the following property: ifx, y are algebraic integers ofK withF(x, y)=0, thenz=Z(x, y), w=W(x, y) are algebraic integers ofL withG(z, w)=0. Also, the total degree and the height of the polynomialG are bounded. As an application of this result, we give a reduction of the problem to determine effectively the integer points on a curve of genus 2, over a number field, to the problem to determine effectively the integer solutions of an equation of degree 4, over a number field. Also we consider a family of curvesF(X, Y)=0, defined over a number fieldK, which are cyclic coverings ofP ¹ and we calculate, using our previous results, an explicit upper bound for the height of the integer points ofF(X, Y)=0 overK.

     

The 2-primary torsion on elliptic curves in the Z p -extensions of Q

Let E be an elliptic curve over Q and p a prime number. Denote by Q_p,∞ the Z_p-extension of Q. In this paper, we show that if p≠3, then where E(Q_p,∞)₍₂₎ is the 2-primary part of the group E(Q_p,∞) of Q_p,∞-rational points on E. More precisely, in case p=2, we completely classify E(Q_2,∞)₍₂₎ in terms of E(Q)₍₂₎; in case p≥5 (or in case p=3 and E(Q)₍₂₎≠{O}), we show that E(Q_p,∞)₍₂₎=E(Q)₍₂₎.     

Higher relative class number formulae

Let E be a totally complex abelian number field with maximal real subfield F, and let denote the non-trivial character of . Similar to the classical case n=1 the value of the Artin L-function at for odd is given by a relative class number formula of the form Here is a higher Q-index, which is equal to 1 or 2 and is a higher relative class number. Here for any number field L the higher class number is the order of the finite group closely related to the order of the higher K-theory group of the ring of integers in L. Received: 4 June 1999 / Revised version: 27 September 2001 / Published online: 26 April 2002     

Euler–Poincaré formula in equal characteristic¶under ordinariness assumptions

Let X be an irreducible smooth projective curve over an algebraically closed field of characteristic p>0. Let ? be either a finite field of characteristic p or a local field of residue characteristic p. Let F be a constructible étale sheaf of $\BF$-vector spaces on X. Suppose that there exists a finite Galois covering π:Y→X such that the generic monodromy of π^* F is pro-p and Y is ordinary. Under these assumptions we derive an explicit formula for the Euler–Poincaré characteristic χ(X,F) in terms of easy local and global numerical invariants, much like the formula of Grothendieck–Ogg–Shafarevich in the case of different characteristic. Although the ordinariness assumption imposes severe restrictions on the local ramification of the covering π, it is satisfied in interesting cases such as Drinfeld modular curves. Received: 7 December 1999 / Revised version: 28 January 2000     

Companion forms over totally real fields

We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2 < k < p, which is ordinary at all primes dividing p and has tamely ramified Galois representation at all primes dividing p, then there is a “companion form” of parallel weight k′ := p + 1 − k. This work generalises results of Gross and Coleman–Voloch for modular forms over Q.     

Application of Greenberg's conjecture in the capitulation problem

Let r be a positive integer. Assume Greenberg's conjecture for some totally real number fields, we show that there exists an infinite family of imaginary cyclic number fields F over the field of rational number field , with an elementary 2‐class group of rank equal to r that capitulates in an unramified quadratic extension over F. Also, we give necessary and sufficient conditions for the Galois group of the unramified maximal 2‐extension over F to be abelian.     

Determination of conductors from Galois module structure

Let E denote an unramified extension of , and set for an odd prime p and . We determine the conductors of the Kummer extensions of F by those elements such that is Galois. This follows from a comparison of the Galois module structure of with the unit filtration of F. Received: 28 August 2000; in final form: 11 October 2001 / Published online: 4 April 2002     

Newton stratification for polynomials: The open stratum

In this paper we consider the Newton polygons of L-functions coming from additive exponential sums associated to a polynomial over a finite field Fq. These polygons define a stratification of the space of polynomials of fixed degree. We determine the open stratum: we give the generic Newton polygon for polynomials of degree d?2 when the characteristic p?3d, and the Hasse polynomial over Fp, i.e. the equation defining the hypersurface complementary to the open stratum.     

Q-curves of degree 5 and jacobian surfaces of GL2-type

We construct a parametric family {E ^(±)(s,t,u)} of minimal Q-curves of degree 5 over the quadratic fields Q , and the family {C(s,t,u)} of genus two curves over Q covering E ^{(+)(s,t,u) whose jacobians are abelian surfaces of GL₂-type. We also discuss the modularity for them and the sign change between E ^{(+)(s,t,u) and its twist E ⁽⁻⁾(s,t,u), which correspond by modularity to cusp forms of trivial and non-trivial Neben type characters, respectively. We find in {C(s,t,u)} concrete equations of curves over Q whose jacobians are isogenous over cyclic quartic fields to Shimura's abelian surfaces A _f attached to cusp forms of Neben type character of level N= 29, 229, 349, 461, and 509. Received: 23 September 1997 / Revised version: 26 May 1998     

On the independence of Heegner points in the function field case

Let ∞ be a fixed place of a global function field k. Let E be an elliptic curve defined over k which has split multiplicative reduction at ∞ and fix a modular parametrization ΦE:X₀(N)→E. Let be Heegner points associated to the rings of integers of distinct quadratic “imaginary” fields K₁,…,Kr over (k,∞). We prove that if the “prime-to-2p” part of the ideal class numbers of ring of integers of K₁,…,Kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in . Moreover, when k is rational, we show that there are infinitely many imaginary quadratic fields for which the prime-to-2p part of the class numbers are larger than C.     

The generalized Rédei-matrix

Let F be a number field with odd class number, and let E be a quadratic extension of F. Our main aim is to prove that the 4-rank of the class group C(E) of E is equal to m − 1 − rank R _E/F , where m is the number of primes of F ramifying in E, R _E/F is the generalized Rédei-matrix of local Hilbert symbols with coefficients in and the rank is the rank over . We determine the generalized Rédei-matrices R _E/F explicitly for biquadratic number fields E. The research is partly supported by NNSF of China (No. 10371054, No. 10771100) and the Morningside Center of Mathematics in Beijing (MCM).     

Legendre polynomials and complex multiplication, I

The factorization of the Legendre polynomial of degree (p−e)/4, where p is an odd prime, is studied over the finite field Fp. It is shown that this factorization encodes information about the supersingular elliptic curves in Legendre normal form which admit the endomorphism , by proving an analogue of Deuring's theorem on supersingular curves with multiplier . This is used to count the number of irreducible binomial quadratic factors of P(p−e)/4(x) over Fp in terms of the class number h(−2p).     

A lower bound for the canonical height on elliptic curves over abelian extensions

Let E/K be an elliptic curve defined over a number field, let ? be the canonical height on E, and let K^ab/K be the maximal abelian extension of K. Extending work of M. Baker (IMRN 29 (2003) 1571-1582), we prove that there is a constant C(E/K)>0 so that every nontorsion point P∈E(K^ab) satisfies .     

On the vanishing of Selmer groups for elliptic curves over ring class fields

Let E/Q be an elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let Hc be the ring class field of K of conductor c prime to ND with Galois group Gc over K. Fix a complex character χ of Gc. Our main result is that if LK(E,χ,1)≠0 then Selp(E/Hc)χ_⊗W=0 for all but finitely many primes p, where Selp(E/Hc) is the p-Selmer group of E over Hc and W is a suitable finite extension of Zp containing the values of χ. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a χ-twisted version of the Birch and Swinnerton-Dyer conjecture for E over Hc (Bertolini and Darmon) and of the vanishing of Selp(E/K) for almost all p (Kolyvagin) in the case of analytic rank zero.     

Abelian functional equations, planar web geometry and polylogarithms

We study some abelian functional equations (Afe). They are equations in the F_i’s of the form F₁(U₁) + ... + F_N(U_N) = 0 where the U_i’s are real rational functions in two variables. First we prove that the local measurable solutions are actually analytic and we characterize their components as solutions of linear differential equations constructed from the U_i’s. Then we propose two methods for solving Afe. Next we apply these methods to the explicit resolution of generalized versions of classical (inhomogeneous) Afe satisfied by low order polylogarithms. Interpreted in the framework of web geometry, these results give us new nonlinearizable maximal rank planar webs. Then we observe that there is a relation between these webs and certain configurations of points in which leads us to define the notion of web associated to a configuration: these webs seem of high rank and could provide numerous new exceptional webs. Finally, we use the preceding results to show that, under weak regularity assumptions, the trilogarithm is the only function which satisfies the Spence–Kummer equation.     

On ramification in the compositum of function fields

The aim of this paper is twofold: Firstly, we generalize well-known formulas for ramification and different exponents in cyclic extensions of function fields over a field K (due to H. Hasse) to extensions E = F(y), where y satisfies an equation of the form f(y) = u · g(y) with polynomials f(y), g(y) ∈ K[y] and u ∈ F. This result depends essentially on Abhyankar’s Lemma which gives information about ramification in a compositum E = E ₁ E ₂ of finite extensions E ₁, E ₂ over a function field F. Abhyankar’s Lemma does not hold if both extensions E ₁/F and E ₂/F are wildly ramified. Our second objective is a generalization of Abhyankar’s Lemma if E ₁/F and E ₂/F are cyclic extensions of degree p = char(K). This result may be useful for the study of wild towers of function fields over finite fields.