Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on arithmetic progressions modular forms.

We use Zagier's method to compute the critical values of the symmetric square L-functions of six cuspidal eigenforms of level one with rational coefficients. According to the Bloch-Kato conjecture, certain large primes dividing these critical values must be the orders of elements in generalised Shafarevich-Tate groups. We give some conditional constructions of these elements. One uses Heegner cycles and Ramanujan-style congruences. The other uses Kurokawa's congruences for Siegel modular forms of degree two. The first construction also applies to the tensor product L-function attached to a pair of eigenforms of level one. Here the critical values can be both calculated and analysed theoretically using a formula of Shimura.

background

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A Course in Arithmetic

Symmetric Square L-Functions and Shafarevich-Tate Groups

In this paper we present our results on the homology cobordism group � 3 of the oriented integral homology 3-spheres. We specially emphasize the role played in the subject by the gauge theory including Floer homology and invariants by Donaldson and Seiberg - Witten. A closed oriented 3-manifoldis said to be an integral homology sphere if it has the same integral homology as the 3-sphere S 3 . Two homology spheres �0 and �1 are homology cobordant, if there is a smooth compact oriented 4-manifold W with @W = �0 ∪−�1 such that H∗(W,�0; Z) = H∗(W,�1; Z) = 0. The set of all homology cobordism classes forms an abelian group � 3 with the group operation defined by a connected sum. Here, the zero element is homology cobordism class of 3-sphere S 3 , and the additive inverse is obtained by a reverse of orientation. The Rochlin invariant µ is an epimorphism µ : � 3 → Z2, defined by the formula

On the homology cobordism group of homology 3-spheres

Quadratische Formen und Monodromiegruppen von Singularitäten

In this article we derive analytic and Fourier aspects of a Kronecker limit
					formula for second-order Eisenstein series. Let $\Gamma$ be any Fuchsian group
					of the first kind which acts on the hyperbolic upper half-space $\mathbf{H}$ such that
					the quotient $\Gamma \backslash \mathbf{H}$ has finite volume yet is non-compact.
					Associated to each cusp of $\Gamma \backslash \mathbf{H}$, there is a classically
					studied {\it first-order\/} non-holomorphic Eisenstein series $E(s, z)$ which is
					defined by a generalized Dirichlet series that converges for $\Re(s) > 1$.
					The Eisenstein series $E(s, z)$ admits a meromorphic continuation with a simple
					pole at $s = 1$. Classically, Kronecker's limit formula is the study of the
					constant term $\mathcal{K}_{1}(z)$ in the Laurent expansion of $E(s, z)$ at $s = 1$. A
					number of authors recently have studied what is known as the {\it
					second-order\/} Eisenstein series $E^{\ast}(s, z)$, which is formed by twisting
					the Dirichlet series that defines the series $E(s, z)$ by periods of a given
					cusp form $f$. In the work we present here, we study an analogue of Kronecker's
					limit formula in the setting of the second-order Eisenstein series $E^{\ast}(s,
					z)$, meaning we determine the constant term $\mathcal{K}_{2}(z)$ in the Laurent expansion
					of $E^{\ast}(s, z)$ at its first pole, which is also at $s = 1$. To begin our
					investigation, we prove a bound for the Fourier coefficients associated to the
					first-order Kronecker limit function $\mathcal{K}_{1}$. We then define two families of
					convolution Dirichlet series, denoted by $L^{+}_{m}$ and $L^{-}_{m}$ with $m \in
					\mathbb{N}$, which are formed by using the Fourier coefficients of $\mathcal{K}_{1}$ and the
					weight two cusp form $f$. We prove that for all $m$, $L^{+}_{m}$ and $L^{-}_{m}$
					admit a meromorphic continuation and are holomorphic at $s = 1$. Turning our
					attention to the second-order Kronecker limit function $\mathcal{K}_{2}$, we first
					express $\mathcal{K}_{2}$ as a solution to various differential equations. Then we obtain
					its complete Fourier expansion in terms of the cusp form $f$, the Fourier
					coefficients of the first-order Kronecker limit function $\mathcal{K}_{1}$, and special
					values $L^{+}_{m}(1)$ and $L^{-}_{m}(1)$ of the convolution Dirichlet series.
					Finally, we prove a bound for the special values $L^{+}_{m}(1)$ and
					$L^{-}_{m}(1)$ which then implies a bound for the Fourier coefficients of
					$\mathcal{K}_{2}$. Our analysis leads to certain natural questions concerning the
					holomorphic projection operator, and we conclude this paper by examining certain
					numerical examples and posing questions for future study.

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Convolution Dirichlet series and a Kronecker limit formula for second-order Eisenstein series

Index Theory with Applications to Mathematics and Physics describes, explains, and explores the Index Theorem of Atiyah and Singer, one of the truly great accomplishments of twentieth-century mathematics whose influence continues to grow, fifty years after its discovery. The Index Theorem has given birth to many mathematical research areas and exposed profound connections between analysis, geometry, topology, algebra, and mathematical physics. Hardly any topic of modern mathematics stands independent of its influence. In this ambitious new work, authors David Bleecker and Bernhelm Booss-Bavnbek give two proofs of the Atiyah-Singer Index Theorem in impressive detail: one based on K-theory and the other on the heat kernel approach. As a preparation for this, the authors explain all the background information on such diverse topics as Fredholm operators, pseudo-differential operators, analysis on manifolds, principal bundles and curvature, and K-theory--carefully and with concern for the reader. Many applications of the theorem are given, as well as an account of some of the most important recent developments in the subject, with emphasis on gauge theoretic physical models and low-dimensional topology. The 18 chapters and two appendices of the book introduce different topics and aspects, often beginning from scratch without presuming full knowledge of all the preceding chapters. Learning paths, through a restricted selection of topics and sections, are suggested and facilitated. The chapters are written for students of mathematics and physics: some for the upper-undergraduate level, some for the graduate level, and some as an inspiration and support for researchers. Index Theory with Applications to Mathematics and Physics is a textbook, a reference book, a survey, and much more. Written in a lively fashion, it contains a wealth of basic examples and exercises. The authors have included many discussion sections that are both entertaining and informative, which illuminate the thinking behind the more general theory. A detailed bibliography and index facilitate the orientation.

A Course in Arithmetic

Citations

Symmetric Square L-Functions and Shafarevich-Tate Groups

On the homology cobordism group of homology 3-spheres

Quadratische Formen und Monodromiegruppen von Singularitäten

Convolution Dirichlet series and a Kronecker limit formula for second-order Eisenstein series

Index Theory with Applications to Mathematics and Physics

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