Arithmetic geometry is a branch of algebraic geometry studying schemes (usually of finite type) over the spectrum $Spec(\mathbb{Z})$ of the commutative ring of integers. More generally often algebraic geometry over non-algebraically closed fields or fields of positive characteristic is also referred to as “arithmetic algebraic geometry”.
An archetypical application of arithmetic geometry is the study of elliptic curves over the integers and the rational numbers.
For number theoretic purposes, i.e. in actual arithmetic; usually one complements this with some data “at the prime at infinity” leading to a more modern notion of an arithmetic scheme (cf. Arakelov geometry).
The refinement to higher geometry is E-infinity geometry (spectral geometry).
analytic geometry (complex, rigid, global)
geometry+analysis/analytic number theory
analytic space, analytic variety, Berkovich space
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
There is a noticeable analogy between phenomena (theorems) in the theory of number fields and those in the theory of function fields over finite fields (Weil 39, Iwasawa 69, Mazur-Wiles 83), hence between the theories of the two kinds of global fields. When regarding number theory dually as arithmetic geometry, then one may see that this analogy extends further to include complex analytic geometry, the theory of complex curves (e.g. Frenkel 05).
At a very basic level the analogy may be plausible from the fact that both the integers $\mathbb{Z}$ as well as well as the polynomial rings $\mathbb{F}_q[x]$ (over finite fields $\mathbb{F}_q$) are principal ideal domains with finite group of units, which already implies that a lot of arithmetic over these rings is similar. Since number fields are the finite dimensional field extensions of the field of fractions of $\mmathbb{Z}$, namely the rational numbers $\mathbb{Q}$, and since function fields are just the finite-dimensional field extensions of the fields of fractions $\mathbb{F}_q(x)$ of $\mathbb{F}_q[x]$, this similarity plausibly extends to these extensions. (Also the entire holomorphic functions on the complex plane are, while not quite an principal ideal domain still a Bézout domain. )
But the analogy ranges much deeper than this similarity alone might suggest. For instance (Weil 39) defined an invariant of a number field – the genus of a number field– which is analogous to the genus of the algebraic curve on which a given function field is the rational functions. This is such as to make the statement of the Riemann-Roch theorem for algebraic curves extend to arithmetic geometry (Neukirch 92, chapter II, prop.(3.6)).
Another notable part of the analogy is the fact that there are natural analogs of Riemann zeta functions? in all three columns of the analogy. This aspect has found attention notably through the lens of regarding number fields as rational functions on “arithmetic curves over the would-be field with one element $\mathbb{F}_1$”.
It is also the function field analogy which induces the conjecture of the geometric Langlands correspondence by analogy from the the number-theoretic Langlands correspondence. Here one finds that the moduli stack of bundles over a complex curve is analogous in absolute arithmetic geometry to the coset space of the general linear group with coefficients in the ring of adeles of a number field, on which unramified automorphic representations are functions. Under this analogy the Weil conjecture on Tamagawa numbers may be regarded as giving the groupoid cardinality of the moduli stack of bundles in arithmetic geometry.
In summary then the analogy says that the theory of number fieldsand of function fields both looks much like a global analytic geometry-version of the theory complex curves,
To date the function field analogy remains just that, an analogy, though various research programs may be thought of as trying to provide a context in which the analogy would become a consequence of a systematic theory (see e.g. the introduction of v.d. Geer et al 05). This includes
geometry “over F1”;
number fields (“function fields of curves over F1”) | function fields of curves over finite fields $\mathbb{F}_q$ (arithmetic curves) | Riemann surfaces/complex curves | |
---|---|---|---|
affine and projective line | |||
$\mathbb{Z}$ (integers) | $\mathbb{F}_q[t]$ (polynomials, function algebra on affine line $\mathbb{A}^1_{\mathbb{F}_q}$) | $\mathcal{O}_{\mathbb{C}}$ (holomorphic functions on complex plane) | |
$\mathbb{Q}$ (rational numbers) | $\mathbb{F}_q(t)$ (rational functions) | meromorphic functions on complex plane | |
$p$ (prime number/non-archimedean place) | $x \in \mathbb{F}_p$ | $x \in \mathbb{C}$ | |
$\infty$ (place at infinity) | $\infty$ | ||
$Spec(\mathbb{Z})$ | $\mathbb{A}^1_{\mathbb{F}_q}$ (affine line) | complex plane | |
$Spec(\mathbb{Z}) \cup place_{\infty}$ | $\mathbb{P}_{\mathbb{F}_q}$ (projective line) | Riemann sphere | |
genus of the rational numbers = 0 | genus of the Riemann sphere = 0 | ||
formal neighbourhoods | |||
$\mathbb{Z}_p$ (p-adic integers) | $\mathbb{F}_q[ [ t -x ] ]$ (power series around $x$) | $\mathbb{C}[ [t-x] ]$ (holomorphic functions on formal disk around $x$) | |
$\mathbb{Q}_p$ (p-adic numbers) | $\mathbb{F}_q((t-x))$ (Laurent series around $x$) | $\mathbb{C}((t-x))$ (holomorphic functions on punctured formal disk around $x$) | |
$\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p$ (ring of adeles) | $\mathbb{A}_{\mathbb{F}_q((t))}$ ( adeles of function field ) | $\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((t-x))$ (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks) | |
$\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})$ (group of ideles) | $\mathbb{I}_{\mathbb{F}_q((t))}$ ( ideles of function field ) | $\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((t-x)))$ | |
branched covering curves | |||
$K$ a number field ($\mathbb{Q} \hookrightarrow K$ a possibly ramified finite dimensional field extension) | $K$ a function field of an algebraic curve $\Sigma$ over $\mathbb{F}_p$ | $K_\Sigma$ (sheaf of rational functions on complex curve $\Sigma$) | |
$\mathcal{O}_K$ (ring of integers) | $\mathcal{O}_{\Sigma}$ (structure sheaf) | ||
$Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})$ (spectrum with archimedean places) | $\Sigma$ (arithmetic curve) | $\Sigma \to \mathbb{C}P^1$ (complex curve being branched cover of Riemann sphere) | |
genus of a number field | genus of an algebraic curve | genus of a surface | |
formal neighbourhoods | |||
$v$ prime ideal in ring of integers $\mathcal{O}_K$ | $x \in \Sigma$ | $x \in \Sigma$ | |
$K_v$ (formal completion at $v$) | $\mathbb{C}((t_x))$ (function algebra on punctured formal disk around $x$) | ||
$\mathcal{O}_{K_v}$ (ring of integers of formal completion) | $\mathbb{C}[ [ t_x ] ]$ (function algebra on formal disk around $x$) | ||
$\mathbb{A}_K$ (ring of adeles) | $\prod^\prime_{x\in \Sigma} \mathbb{C}((t_x))$ (restricted product of function rings on all punctured formal disks around all points in $\Sigma$) | ||
$\mathcal{O}$ | $\prod_{x\in \Sigma} \mathbb{C}[ [t_x] ]$ (function ring on all formal disks around all points in $\Sigma$) | ||
$\mathbb{I}_K = GL_1(\mathbb{A}_K)$ (group of ideles) | $\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((t_x)))$ | ||
automorphy and bundles | |||
$GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})$ (constant sheaves on this stack form unramified automorphic representations) | $Bun_{GL_n(\mathbb{C})}(\Sigma)$ (moduli stack of bundles on the curve $\Sigma$) | ||
number field Langlands correspondence | function field Langlands correspondence | geometric Langlands correspondence | |
Tamawa-Weil for number fields | Tamagawa-Weil for function fields | ||
zeta functions | |||
Dedekind zeta function (Riemann zeta function for $K = \mathbb{Q}$) | Goss zeta function | zeta function of a Riemann surface |
Original articles includes
André Weil, Sur l’analogie entre les corps de nombres algébrique et les corps de fonctions algébrique, Revue Scient. 77, 104-106, 1939
Kenkichi Iwasawa, Analogies between number fields and function fields, in Some Recent Advances in the Basic Sciences, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965-1966), Belfer Graduate School of Science, Yeshiva Univ., New York, pp. 203–208, MR 0255510
for more on this see: Wikipedia, Main conjecture of Iwasawa theory
Barry Mazur, Andrew Wiles, Analogies between function fields and number fields, American Journal of Mathematics Vol. 105, No. 2 (Apr., 1983), pp. 507-521 (JStor)
Textbook accounts include
Jürgen Neukirch, Algebraische Zahlentheorie (1992), English translation Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften 322, 1999 (pdf)
Michael Roosen, Number theory in function fields, Graduate texts in mathematics, 2002
Reviews and lecture notes include
Edward Frenkel, section 2 of Lectures on the Langlands Program and Conformal Field Theory (arXiv:hep-th/0512172).
Bjorn Poonen, section 2.6 of Lectures on rational points on curves, 2006 (pdf)
A collection of more recent developments is in
Textbook accounts include
Lecture notes include
Andrew Sutherland, Introduction to Arithmetic Geometry, 2013 (web)
C. Soulé, D. Abramovich, J. F. Burnol, J. K. Kramer, Lectures on Arakelov Geometry, Cambridge Studies in Advanced Mathematics 33, 188 pp.
Further resources include
Wikipedia: glossary of arithmetic and Diophantine geometry, Arakelov geometry
Arakelov geometry preprint arxiv, list of links
conferences in arithmetic geometry, at Kiran Kedlaya’s wiki