The GL2 Main Conjecture for Elliptic Curves without Complex Multiplication

Let G be a compact p-adic Lie group, with no element of order p, and having a closed normal subgroup H such that G/H is isomorphic to Z_p. We prove the existence of a canonical Ore set S^* of non-zero divisors in the Iwasawa algebra Λ(G) of G, which seems to be particularly relevant for arithmetic applications. Using localization with respect to S^*, we are able to define a characteristic element for every finitely generated Λ(G)-module M which has the property that the quotient of M by its p-primary submodule is finitely generated over the Iwasawa algebra of H. We discuss the evaluation of this characteristic element at Artin representations of G, and its relation to the G-Euler characteristics of the twists of M by such representations. Finally, we illustrate the arithmetic applications of these ideas by formulating a precise version of the main conjecture of Iwasawa theory for an elliptic curve E over Q, without complex multiplication, over the field F generated by the coordinates of all its p-power division points; here p is a prime at least 5 where E has good ordinary reduction, and G is the Galois group of F over Q.     

Heegner Point Kolyvagin System and Iwasawa Main Conjecture

We prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points for semi-stable elliptic curves E over a quadratic imaginary field K satisfying a certain generalized Heegner hypothesis, at an ordinary prime p. It states that the square of the index of the anticyclotomic family of Heegner points in E equals the characteristic ideal of the torsion part of its Bloch–Kato Selmer group(see Theorem 1.3 for precise statement). As a byproduct we also prove the equality in the Greenberg–Iwasawa main conjecture for certain Rankin–Selberg product(Theorem 1.7) under some local conditions, and an improvement of Skinner's result on a converse of Gross–Zagier and Kolyvagin theorem(Corollary 1.11).     

The subconvexity problem for GL2

Generalizing and unifying prior results, we solve the subconvexity problem for the L-functions of GL ₁ and GL ₂ automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino–Ikeda.     

Bessel functions for GL 2

In classical analytic number theory there are several trace formulas or summation formulas for modular forms that involve integral transformations of test functions against classical Bessel functions. Two prominent such are the Kuznetsov trace formula and the Voronoi summation formula. With the paradigm shift from classical automorphic forms to automorphic representations, one is led to ask whether the Bessel functions that arise in the classical summation formulas have a representation theoretic interpretation. We introduce Bessel functions for representations of GL ₂ over a finite field first to develop their formal properties and introduce the idea that the γ-factor that appears in local functional equations for L-functions should be the Mellin transform of a Bessel function. We then proceed to Bessel functions for representations of GL ₂(?) and explain their occurrence in the Voronoi summation formula from this point of view. We briefly discuss Bessel functions for GL ₂ over a p-adic field and the relation between γ-factors and Bessel functions in that context. We conclude with a brief discussion of Bessel functions for other groups and their application to the question of stability of γ-factors under highly ramified twists.     

三个测度猜想的否定

给出三个非常容易让人误以为真的测度猜想,通过定理与λ-Cantor集及其余集的构造给出三个猜想的否定答案.     

Gras-Type Conjectures Fattitle for Function Fields

Based on results obtained in [15], we construct groups of special S-units for function fields of characteristic p>0, and show that they satisfy Gras-type Conjectures. We use these results in order to give a new proof of Chinburg's ₃-Conjecture on the Galois module structure of the group of S-units, for cyclic extensions of prime degree of function fields.     

Die irreduziblen Darstellungen von GL2(Z p ), insbesondere GL2(Z 2)

Ohne ZusammenfassungUnterstütz durch den Sonderforschungsbereich 40, Bonn     

On harmonic and 2-stein spaces of Iwasawa type

We study geometric properties of solvable metric Lie groups S of Iwasawa type; in particular harmonicity and the 2-stein condition. One restriction we obtain is that harmonic spaces of Iwasawa type have algebraic rank one, that is, the commutator subgroup of S has codimension one.We show that among Carnot solvmanifolds the only harmonic spaces are the Damek–Ricci spaces. Moreover, this rigidity result remains valid if harmonicity is replaced by the weaker 2-stein condition. As an application, we show that a harmonic Lie group of Iwasawa type with nonsingular 2-step nilpotent commutator subgroup is, up to scaling, a Damek–Ricci space.     

The second moment of GL(3) \times GL(2) L-functions at special points

For a fixed $SL(3,\mathbb Z )$ Maass form $\phi $ , we consider the family of $L$ -functions $L(\phi \times u_j, s)$ where $u_j$ runs over the family of Hecke-Maass cusp forms on $SL(2,\mathbb Z )$ . We obtain an estimate for the second moment of this family of $L$ -functions at the special points ${\frac{1}{2}}+ it_j$ consistent with the Lindelöf Hypothesis. We also obtain a similar upper bound on the sixth moment of the family of Hecke-Maass cusp forms at these special points; this is apparently the first occurrence of a Lindelöf-consistent estimate for a sixth power moment of a family of $GL(2) L$ -functions.     

Curran两个猜想的证明

本文证明：（１）Ｃｕｒｒａ的两个猜想；（２）ｐ^７阶非循环群是ＬＡ群．     

The Stokes and Krasovskii Conjectures for the Wave of Greatest Height

The integral equation:
φ_μ(s) = (1/3 π)∫^π ₀((sin φ_μ(t))/(μ ⁻¹+ ∫^t ₀sin φ_μ(u) d u )) (log((sin½( s + t ))/ (sin½( s − t )))d t
was derived by Nekrasov to describe waves of permanent form on the surface of a nonviscous, irrotational, infinitely deep flow, the function φ_μ giving the angle that the wave surface makes with the horizontal. The wave of greatest height is the singular case μ=∞, and it is shown that there exists a solution φ_∞ to the equation in this case and that it can be obtained as the limit (in a specified sense) as μ→∞ of solutions for finite μ. Stokes conjectured that φ_∞( s )→⅙π as s ↓0, so that the wave is sharply crested in the limit case; and Krasovskii conjectured that sup _{s ∈[0,π]}φ_μ( s )≤⅙π for all finite μ. Stokes' conjecture was finally proved by Amick, Fraenkel, and Toland, and the present article shows Krasovskii's conjecture to be false for sufficiently large μ, the angle exceeding ⅙π in what is a boundary layer.     

On the Iwasawa invariants of certain non-cyclotomic ℤ2-extensions

Let k be an imaginary quadratic field in which the prime 2 splits. We consider the Iwasawa invariants of a certain non-cyclotomic ℤ₂-extension of k and give some sufficient conditions for the vanishing of λ- and μ-invariants.     

Central L-Values and Toric Periods for GL(2)

Let be a cusp form on GL(2) over a number field F and let Ebe a quadratic extension of F. Denote by _E the base change of to E and by a unitary character of A^x_E/ E^x. We use the relativetrace formula to give an explicit formula for L(1/2, _E ) interms of period integrals of Gross–Prasad test vectors.We give an application of this formula to equidistribution ofgeodesics on a hyperbolic 3-fold.     

Iwasawa Theory for Elliptic Curves at Unstable Primes

In this paper we examine the Iwasawa theory of modular elliptic curves E defined over Q without semi-stable reduction at p. By constructing p-adic L-functions at primes of additive reduction, we formulate a "Main Conjecture" linking this L-function with a certain Selmer group for E over the Zp-extension. Thus the leading term is expressible in terms of III_E, E(Q)_tors and a p-adic regulator term.