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## Über dieses Buch

Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry, and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas of the field, namely Number Theory, Analysis, and Geometry, representing Lang’s own breadth of interest and impact. A special introduction by John Tate includes a brief and fascinating account of the Serge Lang’s life. This volume's group of 6 editors are also highly prominent mathematicians and were close to Serge Lang, both academically and personally. The volume is suitable to research mathematicians in the areas of Number Theory, Analysis, and Geometry.

## Inhaltsverzeichnis

### Raynaud’s group-scheme and reduction of coverings

Abstract
Let Y K X K be a Galois covering of smooth curves over a field of characteristic 0, with Galois group G. We assume K is the fraction field of a discrete valuation ring R with residue characteristic p. Assuming p 2 G and the p-Sylow subgroup of G is normal, we consider the possible reductions of the covering modulo p. In our main theorem we show the existence, after base change, of a twisted curve $$\mathcal{X} \rightarrow Spec (R)$$, a group scheme $$\mathcal{G}\rightarrow \mathcal{X}$$ and a covering $$Y \rightarrow \mathcal{X}$$ extending Y K X K , with Y a stable curve, such that Y is a $$\mathcal{G}$$-torsor.In case p 2 | G counterexamples to the analogous statement are given; in the appendix a strong counterexample is given, where a non-free effective action of α p 2 on a smooth 1-dimensional formal group is shown to lift to characteristic 0.
Dan Abramovich, Jonathan Lubin

### The modular degree, congruence primes, and multiplicity one

Abstract
The modular degree and congruence number are two fundamental invariants of an elliptic curve over the rational field. Frey and Müller have asked whether these invariants coincide. We find that the question has a negative answer, and show that in the counterexamples, multiplicity one (defined below) does not hold. At the same time, we prove a theorem about the relation between the two invariants: the modular degree divides the congruence number, and the ratio is divisible only by primes whose squares divide the conductor of the elliptic curve. We discuss the ratio even in the case where the square of a prime does divide the conductor, and we study analogues of the two invariants for modular abelian varieties of arbitrary dimension.
Amod Agashe, Kenneth A. Ribet, William A. Stein

### Le théorème de Siegel–Shidlovsky revisité

Abstract
We give a new proof of the Siegel–Shidlovsky theorem, which is based on a new version of Shidlovsky’s lemma and on M. Laurent’s interpolation determinants. We also establish a dual version of the lemma, and yet another proof of the theorem when the monodromy around 0 is trivial (as in the Lindemann–Weierstrass case).
Daniel Bertrand

### Some aspects of harmonic analysis on locally symmetric spaces related to real-form embeddings

Abstract
Let $$G =\mathrm{{ SO}}_{3}(\mathbb{C})$$, Γ = SO3([i]), K = SO(3), and let X be the locally symmetric space ΓGK. In this paper, we present a relationship between the heat kernel on SL3() and SO3(). We write down explicit equations defining a fundamental domain for the action of Γ on GK. The fundamental domain is well adapted for studying the theory of Γ-invariant functions on GK. We write down equations defining a fundamental domain for the subgroup $${\Gamma }_{\mathbb{Z}} =\mathrm{ SO}{(2,1)}_{\mathbb{Z}}$$ of Γ acting on the symmetric space $${G}_{\mathbb{R}}/{K}_{\mathbb{R}}$$, where $${G}_{\mathbb{R}}$$ is the split real form SO(2, 1) of G and $${K}_{\mathbb{R}}$$ is its maximal compact subgroup SO(2). We formulate a simple geometric relation between the fundamental domains of Γ and $${\Gamma }_{\mathbb{Z}}$$ so described. Both the formula for the heat kernel and the fundamental domains are designed to aid in a detailed study of the spectral theory of X and the embedded subspace $${X}_{\mathbb{R}} = {\Gamma }_{\mathbb{Z}}\setminus {G}_{\mathbb{R}}/{K}_{\mathbb{R}}$$.
Eliot Brenner, Andrew Sinton

### Differential characters on curves

Abstract
The δ-characters of an abelian variety [B 95] are arithmetic analogues of the Manin maps [M 63]. Given a smooth projective curve X of genus at least 2 embedded into its Jacobian A, one can consider the restrictions to X of the δ-characters of A; the maps so obtained are referred to as δ-characters of X. It is easy to see that the δ-characters of X have a remarkable symmetry property at the origin. The aim of this paper is to prove that this symmetry property completely characterizes the δ-characters of X.
Alexandru Buium

### Weyl group multiple Dirichlet series of type A 2

Abstract
A Weyl group multiple Dirichlet seriesis a Dirichlet series in several complex variables attached to a root system Φ. The number of variables equals the rank rof the root system, and the series satisfies a group of functional equations isomorphic to the Weyl group Wof Φ. In this paper we construct a Weyl group multiple Dirichlet series over the rational function field using n th order Gauss sums attached to the root system of type A 2. The basic technique is that of [11, 10]; namely, we construct a rational function in rvariables invariant under a certain action of W, and use this to build a “local factor” of the global series.
Gautam Chinta, Paul E. Gunnells

### On the geometry of the diffeomorphism group of the circle

Abstract
We discuss some of the possibilities of endowing the diffeomorphism group of the circle with Riemannian structures arising from right-invariant metrics.

### Harmonic representatives for cuspidal cohomology classes

Abstract
We give a construction of harmonic differentials that uniquely represent cohomology classes of a non-compact Riemann surface of finite topology. We construct these differentials by cutting off all cusps along horocycles and solving a suitable boundary value problem on the truncated surface. We then pass to the limit as the horocycle in each cusp recedes to infinity.
Józef Dodziuk, Jeffrey McGowan, Peter Perry

### About the ABC Conjecture and an alternative

Abstract
After a detailed discussion of the ABC Conjecture, we discuss three alternative conjectures proposed by Baker in 2004. The third alternative is particularly interesting, because there may be a way to prove it using the methods of linear forms in logarithms.
Machiel van Frankenhuijsen

### Unifying themes suggested by Belyi’s Theorem

Abstract
Belyi’s Theorem states that every curve defined over the field of algebraic numbers admits a map to the projective line with at most three branch points. This paper describes a unifying framework, reaching across several different areas of mathematics, inside which Belyi’s Theorem can be understood. The paper explains connections between Belyi’s Theorem and (1) The arithmetic and modularity of elliptic curves, (2) abc-type problems and (3) moduli spaces of pointed curves.
Wushi Goldring

### On the local divisibility of Heegner points

Abstract
We relate the local -divisibility of a Heegner point on an elliptic curve of conductor N, at a prime p which is inert in the imaginary quadratic field, to the first -descent on a related abelian variety of level Np.
Benedict H. Gross, James A. Parson

### Uniform estimates for primitive divisors in elliptic divisibility sequences

Abstract
Let P be a nontorsion rational point on an elliptic curve E, given by a minimal Weierstrass equation, and write the first coordinate of nP as A n D n 2, a fraction in lowest terms. The sequence of values D n is the elliptic divisibility sequence (EDS) associated to P. A prime p is a primitive divisor of D n if p divides D n , and p does not divide any earlier term in the sequence. The Zsigmondy set for P is the set of n such that D n has no primitive divisors. It is known that Z is finite. In the first part of the paper we prove various uniform bounds for the size of the Zsigmondy set, including (1) if the j-invariant of E is integral, then the size of the Zsigmondy set is bounded independently of E and P, and (2) if the abc Conjecture is true, then the size of the Zsigmondy set is bounded independently of E and P for all curves and points. In the second part of the paper, we derive upper bounds for the maximum element in the Zsigmondy set for points on twists of a fixed elliptic curve.
Patrick Ingram, Joseph H. Silverman

### The heat kernel, theta inversion and zetas on Г∖G∕K

Abstract
Direct and precise connections between zeta functions with functional equations and theta functions with inversion formulas can be made using various integral transforms, namely Laplace, Gauss, and Mellin transforms as well as their inversions. In this article, we will describe how one can initiate the process of constructing geometrically defined zeta functions by beginning inversion formulas which come from heat kernels. We state conjectured spectral expansions for the heat kernel, based on the so-called heat Eisenstein series defined in [JoL 04]. We speculate further, in vague terms, the goal of constructing a type of ladder of zeta functions and describe similar features from elsewhere in mathematics.
Jay Jorgenson, Serge Lang

### Applications of heat kernels on abelian groups: ζ(2n), quadratic reciprocity, Bessel integrals

Abstract
The discussion centers around three applications of heat kernel considerations on $$\mathbb{R}$$, $$\mathbb{Z}$$ and their quotients. These are Euler’s formula for ζ(2n), Gauss’ quadratic reciprocity law, and the evaluation of certain integrals of Bessel functions. Some further applications are mentioned, including the functional equation of Riemann’s ζ-function, the reflection formula for the Γ-function, and certain infinite sums of Bessel functions.
Anders Karlsson

### Report on the irreducibility of L-functions

Abstract
In this paper, in honor of the memory of Serge Lang, we apply ideas of Chavdarov and work of Larsen to study the $$\mathbb{Q}$$-irreducibility, or lack thereof, of various orthogonal L-functions, especially L-functions of elliptic curves over function fields in one variable over finite fields. We also discuss two other approaches to these questions, based on work of Matthews, Vaserstein, and Weisfeller, and on work of Zalesskii-Serezkin.
Nicholas M. Katz

### Remark on fundamental groups and effective Diophantine methods for hyperbolic curves

Abstract
In a letter from Grothendieck to Faltings, it was suggested that a positive answer to the section conjecture should imply finiteness of points on hyperbolic curves over number fields. In this paper, we point out instead the analogy between the section conjecture and the finiteness conjecture for the Tate-Shafarevich group of elliptic curves. That is, the section conjecture should provide a terminating algorithm for finding all rational points on a hyperbolic curve equipped with a rational point.
Minhyong Kim

### Ranks of elliptic curves in cubic extensions

Abstract
Let Ebe an elliptic curve defined over the rational field . We examine the rank of the Mordell–Weil group E(K) as Kranges over cubic extensions of.
Hershy Kisilevsky

### On effective equidistribution of expanding translates of certain orbits in the space of lattices

Abstract
We prove an effective version of a result obtained earlier by Kleinbock and Weiss [KW] on equidistribution of expanding translates of orbits of horospherical subgroups in the space of lattices.
D. Y. Kleinbock, G. A. Margulis

### Elliptic Eisenstein series for $${PSL}_{2}(\mathbb{Z})$$

Abstract
Let $$\Gamma\subset \mathrm{{ PSL}}_{2}(\mathbb{R})$$be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane $$\mathbb{H}$$, and let $$\Gamma \setminus \mathbb{H}$$be the associated finite volume hyperbolic Riemann surface. Associated to any cusp of $$\Gamma \setminus \mathbb{H}$$, there is the classically studied non-holomorphic (parabolic) Eisenstein series. In [11], Kudla and Millson studied non-holomorphic (hyperbolic) Eisenstein series associated to any closed geodesic on $$\Gamma \setminus \mathbb{H}$$. Finally, in [9], Jorgenson and the first named author introduced so-called elliptic Eisenstein series associated to any elliptic fixed point of $$\Gamma \setminus \mathbb{H}$$. In this article, we study elliptic Eisenstein series for the full modular group $$\mathrm{{PSL}}_{2}(\mathbb{Z})$$. We explicitly compute the Fourier expansion of the elliptic Eisenstein series and derive from this its meromorphic continuation.
Jürg Kramer, Anna-Maria von Pippich

### Consequences of the Gross–Zagier formulae: Stability of average L-values, subconvexity, and non-vanishing mod p

Abstract
Applying the celebrated results of Gross and Zagier for central values of L-series of holomorphic forms of prime level, we deduce an exact average formula for suitable twists of such L-values, with a relation to the class number of associated imaginary quadratic fieds, thereby strengthening a result of Duke. We also obtain a stability result, as well as subconvexity (in this setting), and certain non-vanishing assertions.
Philippe Michel, Dinakar Ramakrishnan

### A variant of the Lang–Trotter conjecture

Abstract
In 1976, Serge Lang and Hale Trotter formulated general conjectures about the value distribution of traces of Frobenius automorphisms acting on an elliptic curve. In this paper, we study a modular analog. More precisely, we consider the distribution of values of Fourier coefficients of Hecke eigenforms of weight k ≥ 4.
M. Ram Murty, V. Kumar Murty

### Multiplicity estimates, interpolation, and transcendence theory

Abstract
We discuss the problems of interpolation and multiplicity estimates on compactifications of commutative algebraic groups. We consider two extremal cases: one where multiplicity is imposed at a single point and the other where the conditions are imposed on an asymptotically growing set of points. Some conjectures and new results are given in both cases.
Michael Nakamaye

### Sampling spaces and arithmetic dimension

Abstract
This paper introduces the twin concepts of sampling spaces and arithmetic dimension, which together address the question of how to count the number, or measure the size of, families of objects over a number field or global field. It can be seen as an alternative to coarse moduli schemes, with more attention to the arithmetic properties of the ambient base field, and which leads to concrete algorithmic applications and natural height functions. It is compared to the definition of essential dimension.
Catherine O’Neil

### Recovering function fields from their decomposition graphs

Abstract
We develop the global theory of a strategy to tackle a program initiated by Bogomolov in 1990. That program aims at giving a group-theoretical recipe by which one can reconstruct function fields K | k with td(K | k) > 1 and k algebraically closed from the maximal pro- abelian-by-central Galois group Π K c of K, where is any prime number ≠char(k).
Florian Pop

### Irreducible spaces of modular units

Abstract
We give a representation-theoretic decomposition of the group of modular units of prime level. Apart from the formulation, the results obtained are contained in those of Gross [3].
David E. Rohrlich

### Equidistribution and generalized Mahler measures

Abstract
If K is a number field and $$\varphi: {\mathcal{P}}_{K}^{1}\rightarrow {\mathcal{P}}_{K}^{1}$$ is a rational map of degree d > 1, then at each place v of K, one can associate to φ a generalized Mahler measure for polynomials FK[t]. These Mahler measures give rise to a formula for the canonical height h φ(β) of an element $$\beta\in \overline{K}$$; this formula generalizes Mahler’s formula for the usual Weil height h(β). In this paper, we use Diophantine approximation to show that the generalized Mahler measure of a polynomial F at a place v can be computed by averaging log | F | v over the periodic points of φ.
L. Szpiro, T. J. Tucker

Abstract
We give some properties of admissible smooth representations of a reductive p-adic group G over a field of characteristic p, deduced from Pontryagin duality, Ollivier’s equivalence of categories when G = PGL(2, Q p ), and finiteness properties of the Hecke ring of a pro-p-Iwahori-Hecke algebra.
Marie-France Vignéras

### Multiplier ideal sheaves, Nevanlinna theory, and Diophantine approximation

Abstract
This paper states a conjecture for Nevanlinna theory or diophantine approximation, with a sheaf of ideals in place of the normal crossings divisor. This is done by using a correction term involving a multiplier ideal sheaf. This new conjecture trivially implies earlier conjectures in Nevanlinna theory or diophantine approximation, and in fact is equivalent to these conjectures. Although it does not provide anything new, it may be a more convenient formulation for some applications.
Paul Vojta

### Recent advances in Diophantine approximation

Abstract
A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex number, as well as the simultaneous approximation of powers of a real number by rational numbers with the same denominator. Finally we study generalisations of these questions to higher dimensions. Several recent advances have been made by B. Adamczewski, Y. Bugeaud, S. Fischler, M. Laurent, T. Rivoal, D. Roy, and W.M. Schmidt, among others. We review some of these works.
Michel Waldschmidt
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