Introduction to arithmetic geometry

1 Introduction TO arithmetic geometry (NOTES FROM , FALL 2009)BJORN POONENC ontents1. What is arithmetic geometry ?32. Absolute values on fields33. Thep-adic absolute value onQ44. Ostrowski s classification of absolute values onQ55. Cauchy sequences and completion86. Inverse limits107. DefiningZpas an inverse limit108. Properties ofZp119. The field ofp-adic expansions1311. Solutions to polynomial equations1412. Hensel s lemma1413. Structure ofQ p1514. Squares inQ The case of The casep= analytic functions1816. Algebraic closure1917. Finite fields2018. Inverse limits in general2219. Profinite Topology on a profinite Subgroups2520. Review of field theory2621. Infinite Galois Examples of Galois groups2822. Affine varieties29 Date: December 10, Affine Affine Irreducible Smooth varieties3223. Projective Projective Projective Projective varieties as a union of affine varieties3424.

2 Morphisms and rational maps3625. Quadratic Equivalence of quadratic Numbers represented by quadratic forms3926. Local-global principle for quadratic Proof of the Hasse-Minkowski theorem for quadratic forms in 2 or 3 variables 4127. Rational points on conics4228. Sums of three squares4329. Valuations on the function field of a Closed points4630. Review4631. curves and function fields4732. Degree of a Base Principal Linear equivalence and the Picard group5233. Newton polygons of two-variable polynomials5434. Riemann-Roch theorem5535. Weierstrass equations5736. elliptic curves5837. Group Chord-tangent Torsion points6038. Mordell s theorem6139. The weak Mordell-Weil theorem6240. Height of a rational number66241. Height functions on elliptic curves6742. Descent7043. Faltings is arithmetic geometry ?Algebraic geometry studies the set of solutions of a multivariable polynomial equation(or a system of such equations), usually overRorC.

3 For instance,x2+xy 5y2= 1defines a hyperbola. It uses both commutative algebra (the theory of commutative rings)and geometric geometry is the same except that one is interested instead in the solutionswhere the coordinates lie in other fields that are usually far from being algebraically of special interest areQ(the field of rational numbers) andFp(the finite field ofpelements), and their finite extensions. Also of interest are solutions with coordinates inZ(the ring of integers).Example circlex2+y2= 1 has infinitely many rational points, such as (3/5,4/5).Finding them all is essentially the same as finding all Pythagorean circlex2+y2= 3 has no rational points at all!Example curvex4+y4= 1 has exactly four rational points, namely ( 1,0) and(0, 1). This is the exponent 4 case of Fermat s Last Theorem: this case was proved byFermat ll develop methods for explaining things like values on fieldsOne approach to constructing the fieldQpofp-adic numbers is to copy the constructionofR, but with a twist: the usual absolute value is replaced by an exotic measure of valueon a fieldkis a functionk R 0x7 x such that the following hold forx,y k:(Abs1) x = 0 if and only ifx= 0(Abs2) xy = x y 3(Abs3) x+y x + y ( triangle inequality )Examples: Rwith the usual|| Cwith the usual||(or any subfield of this) any fieldkwith x := 1,ifx6= 00,ifx= is called thetrivial absolute absolute value satisfying(Abs3 ) x+y max( x , y )( nonarchimedean triangle inequality )is said to benonarchimedean.

4 Otherwise it is said to bearchimedean.(Abs3 ) is more restrictive than (Abs3), since max( x , y ) x + y .(Abs3 ) is strange from the point of view of classical analysis: it says that if you add manycopies of a small number, you will never get a large number, no matter how many copiesyou use. This is what givesp-adic analysis its strange the absolute values considered so far, only the trivial absolute value is we will construct others soon. In fact, most absolute values are nonarchimedean! absolute value onQThe fundamental theorem of arithmetic (for integers) implies that every nonzero rationalnumberxcan be factored asx=u ppnp=u2n23n35n5 whereu {1, 1}, andnp Zfor each primep, andnp= 0 for almost allp(so that all butfinitely many factors in the product are 1, making it a finite product).Definition a primep. Thep-adic valuationis the functionvp:Q Zx7 vp(x) :=np,that gives the exponent ofpin the factorization of a nonzero rational numberx. Ifx= 0,then by convention,vp(0) := +.

5 Sometimes the function is called ordpinstead way of saying the definition: Ifxis a nonzero rational number, it can be written inthe formpnrs, whererandsare integersnotdivisible byp, andn Z, and thenvp(x) := havev2(5/24) = 3, since 5/24 = 2 353= 2 33 :(Val1)vp(x) = + if and only ifx= 0(Val2)vp(xy) =vp(x) +vp(y)(Val3)vp(x+y) min(vp(x),vp(y))These hold even whenxoryis 0, as long as one uses reasonable conventions for + ,namely: (+ ) +a= + + a min(+ ,a) =afor anya, includinga= + .Property (Val2) says that if we disregard the input 0, thenvpis a homomorphism fromthe multiplicative groupQ to the additive of(Val3).The cases wherex= 0 ory= 0 orx+y= 0 are easy, so assume thatx,y,andx+yare all nonzero. Writex=pnrs(and)y=pmuvwithr,s,u,vnot divisible byp, sovp(x) =nandvp(y) =m. Without loss of generality,assume thatn m. Thenx+y=pn(rs+pm nuv)= not divisible byp, butNmight be soNmight contribute some extra factors ofp. Thus all we can say is thatvp(x+y) n= min(n,m) = min(vp(x),vp(y)).

6 Definition a primep. Thep-adic absolute valueof a rational numberxis definedby|x|p:=p vp(x).Ifx= 0 ( ,vp(x) = + ), then we interpret this as|0|p:= (Val1), (Val2), (Val3) forvpare equivalent to properties (Abs1), (Abs2), (Abs3 )for||p. In particular,||preally is an absolute value s classification of absolute values onQOnQwe now have absolute values| |2,| |3,| |5, .. , and the usual absolute value| |,which is also denoted|| , for reasons having to do with an analogy with function fields thatwe will not discuss now. Ostrowski s theorem says that these are essentially all of absolute values and on a fieldkare said to beequivalentifthere is a positive real number such that x = x for allx (Ostrowski).Every nontrivial absolute value onQis equivalent to| |pforsomep . be the absolute 1: there exists a positive integerbwith b > the smallest such positiveinteger. Since 1 = 1, it must be thatb >1. Let be the positive real number such that b =b.

7 Any other positive integerncan be written in baseb:n=a0+a1b+ +asbswhere 0 ai< bfor alli, andas6= 0. Then n a0 + a1b + a2b2 + + asbs = a0 + a1 b + a2 b2 + + as bs 1 +b +b2 + +bs (by definition ofb, since 0 ai< b)=(1 +b +b 2 + +b s )bs Cn (sincebs n),whereCis the value of the convergentinfinitegeometric series1 +b +b 2 + .This holds for alln, so for anyN 1 we can substitutenNin place ofnto obtain nN C(nN) ,which implies n N C(n )N n C1/Nn .This holds for allN 1, andC1/N 1 asN , so we obtain n n for eachn next prove the opposite inequality n n for all positive integersn. Givenn,choose an integerssuch thatbs n < bs+1. Then bs+1 n + bs+1 n 6so n bs+1 bs+1 n =b(s+1) bs+1 n (since b =b ) b(s+1) (bs+1 n) (by the previous paragraph) b(s+1) (bs+1 bs) (sincebs n < bs+1)=b(s+1) [1 (1 1b) ]= (bn) [1 (1 1b) ]=cn ,wherecis a positive real number independent ofn. This inequality, n cn holds for allpositive integersn, so as before, we may substituten=nN, takeNthroots, and take thelimit asN to deduce n n.

8 Combining the previous two paragraphs yields n =n for any positive integern. Ifmis another positive integer, then n m/n = m m/n = m / n =m /n = (m/n) .Thus q =q for every positive rational number. Finally, ifqis a positive rational number,then q = 1 q =q =| q| so x =|x| holds for allx Q(including 0).Case 2: b = 1for all positive as in the previous paragraph, the axiomsof absolute values imply that x = 1 for allx Q , contradicting the assumption that is a nontrivial absolute 3: n 1for all positive integersn, and there exists a positive integerbsuch that b < thatbis the smallest such integer. If it were possible to writeb=rsforsome smaller positive integersrands, then r|= 1 and s = 1 by definition ofb, but then b = r s = 1, a contradiction; thusbis a prove (by contradiction) thatpis the only prime satisfying p <1. Suppose thatqwere another such prime. For any positive integerN, the integerspNandqNare relativelyprime, so there exist integersu,vsuch thatupN+vqN= 1,7and then1 = 1 = upN+vqN u p N+ v| q N p N+ q is a contradiction ifNis large enough.

9 So q = 1 for every primeq6= 0< p <1 and 0<|p|p<1, there exists a positive real number such that p =|p| p. Now, for any nonzero rational numberx= primesqincludingpqnqproperty (Abs2) (and 1 = 1) imply x = primesqincludingp q nq= p npsince all the other factors are 1. Since p =|p| p, this becomes x =|p|np p=|x| p. sequences and completionLetkbe a field equipped with an absolute value .Definition sequence (ai) inkconvergesif there exists` ksuch that for every >0, the termsaiare eventually within of`: , for every >0, there exists a positiveintegerNsuch that for alli N, the distance bound ai ` < holds. In this case,`iscalled thelimitof the (ai) converges to`if and only if ai ` 0 asi . The limit is uniqueif it exists: if (ai) converges to both`and` , then ` ` ai ` + ai ` 0 + 0 = 0,so ` ` = 0, so` =`.Definition sequence (ai) inkis aCauchy sequenceif for every >0, the terms areeventually within of each other; , for every >0, there exists a positive integerNsuchthat for alli,j N, the distance bound ai aj < a sequence converges, it is a Cauchy the triangle inequality.

10 Unfortunately, the converse can fieldkiscompletewith respect to if every Cauchy sequence would like every Cauchy sequence to converge, but this might not be the case. Tofix this, for each Cauchy sequence that does not converge, we could formally create a newsymbol that represents the limit and treat it as if it were a new number. But some Cauchysequences look as if they should be converging to the same limit, so we need to identifysome of these symbols. So the new symbols really should correspond to equivalence classesof Cauchy sequences that do not converge. Actually there is no harm in creating symbolsfor Cauchy sequences that converge already, as long as these new symbols are identified withthe pre-existing limits. Finally, we can think of the equivalence classes themselves as beingthe sequences (ai) and (bi) areequivalentif ai bi 0 asi .One can check that this induces an equivalence relation on the set of sequences.