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 construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial groups such that it is not homotopy equivalent toM 0#M 1 unlessM 0 orM 1 is homeomorphic toS 4. LetN be the nucleus of the minimal elliptic Enrique surfaceV 1(2, 2) and putM=N∪ ∂NN. The fundamental group ofM splits as ℤ/2 * ℤ/2. We prove thatM#k(S 2×S2) is diffeomorphic toM 0#M 1 for non-simply connected closed smooth four-manifoldsM 0 andM 1 if and only ifk≥8. On the other hand we show thatM is homeomorphic toM 0#M 1 for closed topological four-manifoldsM 0 andM 1 withπ 1(Mi)=ℤ/2.

/pdf/counterexamples-to-the-kneser-conjecture-in-dimension-four-2706agzktd.pdf

A Course in Arithmetic

Counterexamples to the Kneser conjecture in dimension four

We revisit the classification of rank-1 4d $\mathcal{N}=2$ QFTs in the spirit of Diophantine Geometry, viewing their special geometries as elliptic curves over the chiral ring (a Dedekind domain). The Kodaira-N\'eron model maps the space of non-trivial rank-1 special geometries to the well-known moduli of pairs $(\mathcal{E},F_\infty)$ where $\mathcal{E}$ is a relatively minimal, rational elliptic surface with section, and $F_\infty$ a fiber with additive reduction. Requiring enough Seiberg-Witten differentials yields a condition on $(\mathcal{E},F_\infty)$ equivalent to the "safely irrelevant conjecture". The Mordell-Weil group of $\mathcal{E}$ (with the N\'eron-Tate pairing) contains a canonical root system arising from $(-1)$-curves in special position in the N\'eron-Severi group. This canonical system is identified with the roots of the flavor group $\mathsf{F}$: the allowed flavor groups are then read from the Oguiso-Shioda table of Mordell-Weil groups. Discrete gaugings correspond to base changes. Our results are consistent with previous work by Argyres et al.

/pdf/special-arithmetic-of-flavor-1z7wsupokv.pdf

Special Arithmetic of Flavor

In this survey paper, we explain how weight 2 modular forms on Γ0(N) are related to modular symbols, and how to use this relationship to explicitly compute spaces of modular forms.

/pdf/an-introduction-to-computing-modular-forms-using-modular-1yrgpdpi1y.pdf

An introduction to computing modular forms using modular symbols

Finite groups and Hecke operators

Preface This book is based on notes I created for a one-semester undergraduate course on Algebraic Number Theory, which I taught at Harvard during Spring 2004. The primary sources for the course were chapter 1 of Swinnerton-Dyer's book A Brief Guide to Algebraic Number Theory [SD01] and chapter 2 of Cassels's article Global Fields [Cas67]. I wrote these notes by following closely the above two chapters; in some cases I added substantial text and examples. For example, chapter 1 of [SD01] is 30 pages, whereas my rewrite of it occupies over 100 pages. In contrast, I follow [Cas67] more closely. I have no intent whatever to plagiarize. I acknowledge as such those chapters in this book which are simply a close rewrite of [Cas67]. My goal is to take the useful classical article ([Cas67]) and make it more accessible to students by modernizing the notation, and adding additional explanations and examples. I have no intent to publish this book with a traditional publisher, so it will remain freely available indefinitely. If you have comments, corrections, suggestions for additions, etc., please send them to me! ————————— License: FREE! More precisely, this book my be freely redistributed, copied, or even sold without requiring you to obtain written permission from me. You may even extend or change this book, but this preface page must remain in any derived work, and any derived work must also remain free, including the L A T E X source files. In particular, I have no interest in making any money from this book.

/pdf/a-brief-introduction-to-classical-and-adelic-algebraic-5fqrvkoon4.pdf

A Course in Arithmetic

Citations

Counterexamples to the Kneser conjecture in dimension four

Special Arithmetic of Flavor

An introduction to computing modular forms using modular symbols

Finite groups and Hecke operators

A Brief Introduction to Classical and Adelic Algebraic Number Theory

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