A family of quintic cyclic fields with even class number parameterized by rational points on an elliptic curve

We give a family of quintic cyclic fields with even class number parametrized by rational points on an elliptic curve associated with Emma Lehmer's quintic polynomial. Further, we use the arithmetic of elliptic curves and the Chebotarev density theorem to show that there are infinitely many such fields.     

The distribution of class groups of function fields

For any sufficiently general family of curves over a finite field F_q and any elementary abelian ?-group H with ? relatively prime to q, we give an explicit formula for the proportion of curves C for which Jac(C)[?](F_q)≅H. In doing so, we prove a conjecture of Friedman and Washington.     

Counting points of fixed degree and bounded height on linear varieties

We count points of fixed degree and bounded height on a linear projective variety defined over a number field k. If the dimension of the variety is large enough compared to the degree we derive asymptotic estimates as the height tends to infinity. This generalizes results of Thunder, Christensen and Gubler and special cases of results of Schmidt and Gao.     

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.     

Central derivatives of <Emphasis Type="Italic">L</Emphasis>-functions in Hida families

We prove a result of the following type: given a Hida family of modular forms, if there exists a weight two form in the family whose L-function vanishes to exact order one at s = 1, then all but finitely many weight two forms in the family enjoy this same property. The analogous result for order of vanishing zero is also true, and is an easy consequence of the existence of the Mazur–Kitagawa two-variable p-adic L-function. This research was supported in part by NSF grant DMS-0556174.     

The extended Bloch groups of biquadratic and dihedral number fields

In this paper, we study the Galois action on the extended Bloch groups of biquadratic and dihedral number fields. We prove that if F is a biquadratic number field, then the index $Q_2(F)$ in Browkin and Gangl's formulas on the Brauer–Kuroda relation can only be 1 or 2. This is exactly what Browkin and Gangl predicted in their paper. Moreover we give the explicit criteria for $Q_2(F)=1$ or 2 in terms of the Tate kernels. We also prove that $Q_2(F)=1$ or p for any dihedral extension $F/\mathbb{Q}$ whose Galois group is the dihedral group of order 2p, where p is an odd prime.     

Higher Heegner points on elliptic curves over function fields

Let E be a modular elliptic curve defined over a rational function field k of odd characteristic. We construct a sequence of Heegner points on E, defined over a -tower of finite extensions of k, and show that these Heegner points generate a group of infinite rank. This is a function field analogue of a result of Cornut and Vatsal.     

The support problem for abelian varieties

Let A be an abelian variety over a number field K. If P and Q are K-rational points of A such that the order of the reduction of Q divides the order of the ) reduction of P for almost all prime ideals , then there exists a K-endomorphism φ of A and a positive integer k such that φ(P)=kQ.     

The Galois representations associated to a Drinfeld module in special characteristic—II: Openness

Let φ be a Drinfeld A-module in special characteristic p₀ over a finitely generated field K. For any finite set P of primes p≠p₀ of A let Γ_P denote the image of Gal(K^sep/K) in its representation on the product of the p-adic Tate modules of φ for all p∈P. We determine Γ_P up to commensurability.     

Image of the group ring of the Galois representation associated to Drinfeld modules

Let φ be a Drinfeld A-module of arbitrary rank and arbitrary characteristic over a finitely generated field K, and set GK=Gal(Ksep/K). Let E=EndK(φ). We show that for almost all primes p of A the image of the group ring A[GK] in EndA(Tp(φ)) is the commutant of E. In the special case E=A it follows that the representation of GK on the p-torsion points φ[p](Ksep) of φ is absolutely irreducible for almost all p.     

The Galois representations associated to a Drinfeld module in special characteristic—III: Image of the group ring

Let K be a finitely generated field of transcendence degree 1 over a finite field, and set G_K?Gal(K^sep/K). Let φ be a Drinfeld A-module over K in special characteristic. Set E?End_K(φ) and let Z be its center. We show that for almost all primes p of A, the image of the group ring A_p[G_K] in End_A(T_p(φ)) is the commutant of E. Thus, for almost all p it is a full matrix ring over Z⊗_AA_p. In the special case E=A it follows that the representation of G_K on the p-torsion points φ[p] is absolutely irreducible for almost all p.     

The Galois representations associated to a Drinfeld module in special characteristic—I: Zariski density

Let ? be a Drinfeld A-module of rank r over a finitely generated field K. Assume that ? has special characteristic p₀ and consider any prime p≠p₀ of A. If End_K^sep(?)=A, we prove that the image of Gal(K^sep/K) in its representation on the p-adic Tate module of ? is Zariski dense in GL_r.     

On the group of modular units and the ideal class group

J.-M. Kim, S. Bae and I.-S. Lee showed that there exists an isomorphism between the p-primary part of the ideal class group and p-primary part of the unit group modulo cyclotomic unit group in Q⁺(ζpn) for all sufficiently large n under some conditions. In the present paper, we shall give an analogue of their result for modular units.     

Adelic openness for Drinfeld modules in generic characteristic

Let φ be a Drinfeld A-module of arbitrary rank and generic characteristic over a finitely generated field K. If the endomorphism ring of φ over an algebraic closure of K is equal to A, we prove that the image of the adelic Galois representation associated to φ is open.     

The vanishing order of certain Hecke L-functions of imaginary quadratic fields

Let −D<−4 denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of exists. Let d be a fundamental discriminant prime to D. Let 2k−1 be an odd natural number prime to the class number of . Let χ be the twist of the (2k−1)th power of a canonical Hecke character of by the Kronecker's symbol . It is proved that the vanishing order of the Hecke L-function L(s,χ) at its central point s=k is determined by its root number when , where the constant implied in the symbol ? depends only on k and ?, and is effective for L-functions with root number −1.     

On Galois representations for abelian varieties with complex and real multiplications

In this paper, we study the image of l-adic representations coming from Tate module of an abelian variety defined over a number field. We treat abelian varieties with complex and real multiplications. We verify the Mumford-Tate conjecture for a new class of abelian varieties with real multiplication.     

Cohomological Mackey functors in number theory

We prove relations between the evaluations of cohomological Mackey functors over complete discrete valuation rings or fields and apply this to Mackey functors that arise naturally in number theory. This provides relations between λ- and μ-invariants in Iwasawa theory, between Mordell-Weil groups, Shafarevich-Tate groups, Selmer groups and zeta functions of elliptic curves, and between ideal class groups and regulators of number fields.     

On the Birch and Swinnerton-Dyer conjecture for abelian varieties attached to Hilbert modular forms

In this paper we prove that if the Birch and Swinnerton-Dyer conjecture holds for abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character, then the Birch and Swinnerton-Dyer conjecture holds for abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character regarded over arbitrary totally real number fields.