Brian Conrad

On the Modularity of Elliptic Curves Over Q: Wild 3-Adic Exercises

In this paper, building on work of Wiles [Wi] and of Wiles and one of us (R.T.) [TW], we will prove the following two theorems (see §2.2). Theorem A. If E/Q is an elliptic curve, then E is modular. Theorem B. If ρ : Gal(Q/Q) → GL2(F5) is an irreducible continuous representation with cyclotomic determinant, then ρ is modular. We will first remind the reader of the content of these results and then briefly outline the method of proof. If N is a positive integer then we let Γ1(N) denote the subgroup of SL2(Z) consisting of matrices that modulo N are of the form ( 1 ∗ 0 1 ) .

ARITHMETIC MODULI OF GENERALIZED ELLIPTIC CURVES

1.1. Motivation. In [DR], Deligne and Rapoport developed the theory of generalized elliptic curves over arbitrary schemes and they proved that various moduli stacks for (ample) “level-N” structures on generalized elliptic curves over Z[1/N ]-schemes are Deligne–Mumford stacks over Z[1/N ]. These stacks were proved to be Z[1/N ]-proper, and also finite flat over the Z[1/N ]-localization of the proper Z-smooth moduli stack M 1 = M 1,1 of stable marked curves of genus 1 with one marked point. Hence, by normalization over M 1 one gets proper normal flat stacks over Z but the method gives no moduli interpretation in “bad” characteristics. In [KM], Katz and Mazur developed the theory of Drinfeld level structures on elliptic curves over arbitrary schemes, thereby removing the etaleness restriction on the level when working away from the cusps. When there is “enough” level (to remove non-trivial isotropy groups), the work in [KM] constructs affine moduli schemes over Z for Drinfeld level structures on elliptic curves. These schemes were proved to be normal and finite flat over the “j-line” AZ, so they extend to proper flat Z-schemes by normalization over P 1 Z. If there is “enough” level then the Z-proper constructions of Deligne–Rapoport and Katz–Mazur coincide. The approach in [KM] does not give a moduli interpretation at the cusps (in the sense of Deligne and Rapoport), and [DR] uses methods in deformation theory that often do not work at the cusps in bad characteristics. One expects that the theory of Drinfeld structures on generalized elliptic curves should provide a moduli-theoretic explanation for the proper Z-structures made in [DR] and [KM]. In unpublished work, Edixhoven [Ed] carried out an analysis of the situation for level structures of the types Γ(N), Γ1(N), and Γ0(n) for squarefree n, as well as some mixtures of these level structures. He proved that the moduli stacks in all of these cases are proper and flat Deligne–Mumford stacks over Z that are moreover regular. Edixhoven used a method resting on considerations with an etale cover by a scheme, and he proved that the moduli stacks are isomorphic to normalizations that were constructed in [DR]. In view of the prominent role of the modular curve X0(n) in the study of elliptic curves over Q, it is natural to ask if the restriction to squarefree n is really necessary in order that the moduli stack for Γ0(n)structures be a reasonable algebro-geometric object over Z. Let us recall why non-squarefree n may seem to present a difficulty. Suppose n has a prime factor p with ordp(n) ≥ 2. Choose any d|n and let Cd denote the standard d-gon over an algebraically closed field of characteristic p, considered as a generalized elliptic curve in the usual manner. Assume p|d, so Aut(Cd) = μd o 〈inv〉 contains μp (with “inv” denoting the unique involution of Cd that restricts to inversion on the smooth locus C d = Gm × (Z/dZ)). If moreover p|(n/d) then there exist cyclic subgroup schemes G in the smooth locus C d such that G is ample on Cd and has order n with p-part that is non-etale and disconnected. Such a subgroup contains the p-torsion μp in the identity component of C d , and so the infinitesimal subgroup μp in the automorphism scheme of Cd preserves G. In particular, the finite automorphism scheme of the Γ0(n)-structure (Cd, G) contains μp and thus is not etale. The moduli stack for Γ0(n)-structures therefore cannot be a Deligne–Mumford stack in characteristic p if p|n. However, failure of automorphism groups to be etale does not prevent the possibility that such stacks can be Artin stacks.

Gross--Zagier Revisited

ly isomorphic as rings are conjugate in B. 146 BRIAN CONRAD (APPENDIX BY W. R. MANN) Remark A.10. The order ( A mn A A ) in (2) is the intersection of the maximal orders M2(A) and γnM2(A)γ−1 n = ( A mn m−n A ) , where γn = ( 0 πn 1 0 ) . This example is called a standard Eichler order. Proof. For the first part, it suffices to show that the set R of elements of B integral over R is an order. That is, we must show R is finite as an A-module and is a subring of B. Note that R is stable under the involution b 7→ b. The key to the subring property is that if b ∈ B has N(b) ∈ A, then Tr(b) ∈ A. Indeed, F [b] is a field on which the reduced norm and trace agree with the usual norm and trace (relative to F ), and by completeness of A we know that the valuation ring of F [b] is characterized by having integral norm. Thus, to show that R is stable under multiplication we just need that if x, y ∈ R then N(xy) ∈ A. But N(xy) = N(x)N(y). Meanwhile, for addition (an issue because noncommutativity does not make it evident that a sum of integral elements is integral), we note that if x, y ∈ R then N(x + y) = (x + y)(x + y) = N(x) + N(y) + Tr(xy). But this final reduced trace term lies in A because xy ∈ R. Hence, R is a subring of B. In particular, R is an A-submodule since A ⊆ R. To show that R is A-finite, we may pick a model for B as in Example A.2, and may assume i = e, j = f ∈ A. Thus, i, j ∈ R, so ij ∈ R. For any x ∈ R, we have x, xi, xj, xij ∈ R. Taking reduced traces of all four of these elements and writing x = c + c1i + c2j + c3ij for c, c1, c2, c3 ∈ F , we get x ∈ 1 2 A + 1

Power laws for monkeys typing randomly: the case of unequal probabilities

An early result in the history of power laws, due to Miller, concerned the following experiment. A monkey types randomly on a keyboard with N letters (N>1) and a space bar, where a space separates words. A space is hit with probability p; all other letters are hit with equal probability (1-p)/N. Miller proved that in this experiment, the rank-frequency distribution of words follows a power law. The case where letters are hit with unequal probability has been the subject of recent confusion, with some suggesting that in this case the rank-frequency distribution follows a lognormal distribution. We prove that the rank-frequency distribution follows a power law for assignments of probabilities that have rational log-ratios for any pair of keys, and we present an argument of Montgomery that settles the remaining cases, also yielding a power law. The key to both arguments is the use of complex analysis. The method of proof produces simple explicit formulas for the coefficient in the power law in cases with rational log-ratios for the assigned probabilities of keys. Our formula in these cases suggests an exact asymptotic formula in the cases with an irrational log-ratio, and this formula is exactly what was proved by Montgomery.

Complex Multiplication and Lifting Problems

Introduction Algebraic theory of complex multiplication CM lifting over a discrete valuation ring CM lifting of

Nagata compactification for algebraic spaces

p

Finiteness theorems for algebraic groups over function fields

-divisible groups CM lifting of abelian varieties up to isogeny Some arithmetic results for abelian varieties CM lifting via

Non-archimedean analytification of algebraic spaces

p

Modular curves and Ramanujan's continued fraction

-adic Hodge theory Notes on quotes Glossary of notations Bibliography Index

Prime specialization in higher genus I

We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes.