# Papers and Preprints

"Expander graphs, gonality, and variation of Galois representations," with C. Hall and E. Kowalski, submitted. We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric (genus and gonality) growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in one-parameter families of abelian varieties.

"Understanding persistent homology and Plex using a networking dataset," with L. Balzano, submitted.

We report on a pilot study of the topological structure of data sets using Plex, a software package for computational homology. Plex assigns to a set of points in R^n a "barcode," which is intended to reveal topological structure in data. In particular, if the data are drawn from a low-dimensional manifold M, the barcode is meant to capture the homology of M. We compared barcodes coming from three sources: bytecount data from the UW-Madison core computer network, Gaussian noise centered at a point, and Gaussian noise convolved with a circle. One of our goals is to understand the result of Plex needs to be in order to distinguish a data set with topological structure from one consisting solely of noise.

"Modeling lambda invariants by p-adic random matrices," with S. Jain and A. Venkatesh, submitted.

We model the variation of lambda-invariants of imaginary quadratic fields by statistics of p-adic random matrices, and test the resulting predictions against numerical data.

"Statistics of number fields and function fields," with A. Venkatesh, to appear, Proceedings of the ICM 2010.

We discuss some problems of arithmetic distribution, including conjectures of Cohen-Lenstra, Malle, and Bhargava; we explain how such conjectures can be heuristically understood for function fields over finite fields, and discuss a general approach to their proof in the function field context based on the topology of Hurwitz spaces. This approach also suggests that the Schur multiplier plays a role in such questions over number fields.

"Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields," with A. Venkatesh and C. Westerland, submitted.

We prove a homological stabilization theorem for Hurwitz spaces: moduli spaces of branched covers of the complex projective line. This topological theorem has the following arithmetic consequence: let l > 2 be prime and A a finite abelian l-group. Then there exists Q = Q(A) such that, for q greater than Q and not congruent to 1 modulo l, a positive fraction of quadratic extensions of Fq(t) have the l-part of their class group isomorphic to A.

"Upper bounds for the growth of Mordell-Weil ranks in pro-p towers of Jacobians," submitted.

We study the variation of Mordell-Weil ranks in the Jacobians of curves in a pro-p tower over a fixed number field. In particular, we show that under mild conditions the Mordell-Weil rank of a Jacobian in the tower is bounded above by a constant multiple of its dimension. In the case of the tower of Fermat curves, we show that the constant can be taken arbitrarily close to 1. The main result is used in the forthcoming paper of Guillermo Mantilla-Soler on the Mordell-Weil rank of the modular Jacobian J(Np^m).

"Linnik's ergodic method and the distribution of integer points on spheres," with P. Michel and A. Venkatesh, submitted.

We discuss Linnik's work on the distribution of integral solutions to x^2+y^2+z^2 =d, as d goes to infinity. We give an exposition of Linnik's ergodic method; indeed, by using large-deviation results for random walks on expander graphs, we establish a refinement of his equidistribution theorem. We discuss the connection of these ideas with modern developments (ergodic theory on homogeneous spaces, L-functions).

"Random p-groups, braid groups, and random tame Galois groups,", with N. Boston, submitted.

Write G_S(p) for the Galois group of the maximal pro-p extension of Q unramified away from a set of primes S not including p. We describe a conjectural probability distribution on the isomorphism class of G_S(p) when S is a random set of g primes. This can be thought of as a non-abelian pro-p version of the Cohen-Lenstra heuristic. We present two heuristic arguments leading to the same conjecture, one in the spirit of Cohen-Lenstra and the other based on analogies between number fields and function fields.

We prove that every algebraic curve X defined over the algebraic closure of the rationals is birational over the complex numbers to a Teichmuller curve.

Let K be a field. A positive motivic measure on the Grothendieck ring K_0(Var_K) is a homomorphism from K_0(Var_K) to the real numbers assigning a nonnegative value to every variety. In this note we show that the only positive motivic measures are the counting measures: measures on K_0(Var_{F_q}) which send a variety to its number of rational points over some fixed finite extension of F_q.

Using the polynomial method of Dvir, we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to familes of curves of bounded degree on algebraic varieties over finite fields. In particular, we prove the Kakeya maximal function conjecture over finite fields, which can be thought of as a generalization of Dvir's proof of the Kakeya set conjecture over finite fields. We also discuss several variants, including maximal operators associated to restriction to k-planes and the case of finite Artinian rings other than fields.

Let A_t be a family of abelian varieties over a number field k parametrized by a rational coordinate t, and suppose the generic fiber of A_t is geometrically simple. For example, we may take A_t to be the Jacobian of the hyperelliptic curve y^2 = f(x)(x-t) for some polynomial f. We give two upper bounds for the number of t \in Q of height at most B such that the fiber A_t is geometrically non-simple. One bound comes from arithmetic geometry, and shows that there are only finitely many such t; but one has very little control over how this finite number varies as f changes. Another bound, from analytic number theory, shows that the number of geometrically non-simple fibers grows quite slowly with B; this bound, by contrast with the arithmetic one, is effective, and is uniform in the coefficients of f. We hope that the paper, besides proving the particular theorems we address, will serve as a good example of the strengths and weaknesses of the two complementary approaches.

In a previous paper, we proved that the equation A^4 + B^2 = C^p has no nontrivial solutions when p is a prime no less than 211. In the present paper, we show that the equations A^4 + B^2 = C^n and A^4 + 2 B^2 = C^n have no nontrivial solutions for n >= 4 apart from 1^4 + 2*11^2 = 3^5. The proof involves a mixture of geometric, analytic, and computational techniques.

In a previous paper with A. Venkatesh, we gave an upper bound on GRH for the order of the p-torsion subgroup of the ideal class group of a number field K, which could be made unconditional in certain cases (for instance, the 3-torsion in the class group of a quadratic field.) In this note, we observe that those bounds could be (slightly) sharpened if one had some control over the number of points on P^1(K) whose height is bounded by some small positive power of the discriminant of K. This problem has apparently not been seriously investigated, apart from some numerical experiments carried out by D. Lecoanet.

Let N(X) be the number of number fields whose discriminant has absolute value at most X, and let N_n(X) be the number of these whose degree over Q is n. In our paper "The number of extensions of a number field with fixed degree and bounded discriminant," we gave an upper bound for N_n(X). In this appendix, we show that the methods of that paper also give a bound for N(X); in particular,

log N(X) < C log X exp(D sqrt(log log X))

for absolute constants C,D.

In Iwasawa theory, one is often faced with a situation in which one wants to compare the rank of a finitely generated Z_p[[G]]-module M with the asymptotic properties of the Z_p-rank of the coinvariant quotient M_{G_n}, where G_n ranges over a descending series of subgroups of G. An old theorem of Harris relates these two in case G_n is a descending series of principal congruence (whence normal) subgroups. In this note, we explain how to extend this theorem to coinvariants by subgroups which are in some sense "far from being normal." As a corollary, we give new examples of torsion modules which are faithful in Venjakob's sense, and an upper bound for the growth of Mordell-Weil ranks of elliptic curves in certain non-Galois towers of field extensions.

We introduce a new method to bound p-torsion in class groups, combining analytic ideas and reflection principles from algebraic number theory. The method gives, in particular, new bounds for the 3-torsion part of class groups in quadratic, cubic and quartic number fields, as well as bounds for arbitrary p for certain families of higher degree fields. (For instance, one can bound the 5-torsion in class groups of quadratic extensions of Q(sqrt(5)).) Conditionally on GRH, we obtain a nontrivial bound for the size of the p-torsion subgroup in the class group of a general number field.

The problem of representing integers by quadratic forms has a long history. More generally, give two quadratic lattices Q' and Q, of ranks m and n, one can ask whether Q' can be embedded into Q. Certainly it is necessary that such an embedding exists everywhere locally. Previously this necessary condition was known to be sufficient when n is at least 2m + 3. We show that in fact a local-to-global theorem holds whenever n >= m+7. The key idea is to relate the problem to a question about actions of p-adic groups, and then apply a version of Ratner's theorem to show, very loosely, that representations of Q' by forms in the genus of Q are equidistributed over the genus of Q. We also include some general discussion about orbits of integral points on homogeneous spaces, and their relation with adelic quotients and cohomology groups.

Warning: The published version of this paper is significantly shorter than the arXiv version linked here, which contains extra material not necessary for the proof of the main theorem -- this consists mostly of philosophical commentary about the approach of the paper and discussion of some related problems. The published version also proves the main theorem when n >= m + 3, subject to some mild conditions. Please cite the published version unless the citation is to material present only in the longer arXiv version.

In a recent paper, Calegari and Dunfield exhibit a sequence of hyperbolic 3-manifolds which have increasing injectivity radius, and which, subject to some conjectures in number theory, are rational homology spheres. We prove unconditionally that these manifolds are rational homology spheres, and give a sufficient condition for a tower of hyperbolic 3-manifolds to have first Betti number 0 at each level. The methods involved are purely pro-p group theoretical.

Published version

In two recent papers, Silverman discusses the problem of bounding the Mordell-Weil ranks of elliptic curves over towers of function fields. We first prove generalizations of the theorems of those two papers, allowing non-abelian Galois groups and removing the dependence on Tate's conjectures. We then prove some theorems about the growth of Mordell-Weil ranks in towers of function fields whose Galois groups are $p$-adic Lie groups; in particular, we give some Galois-theoretic criteria which guarantee that certain curves E/Q(t) have bounded Mordell-Weil rank over C(t^{1/p^n}) as n grows, and show that these criteria are met for elliptic K3 surfaces whose associated Galois representations have sufficiently large image. This contrasts with recent results of Ulmer showing that the Mordell-Weil rank of E/k(t^{1/p^n}) can be unbounded in n when k is a finite field. (Slightly revised after referee report, December 2005)

I am grateful to Brian Conrad for pointing out that a bit more justification is needed in the proof of Lemma 4.7; one needs that E(k^s(C_infty)) has finite torsion subgroup, and what is proven gives only that for each p the subgroup of p-primary torsion is finite. To finish the argument it suffices to show that the mod p Galois representation attached to E(k^s(C)) surjects onto SL_2(F_p) for all sufficiently large p; this follows from the connectedness of modular curves in all characteristics and is for instance shown with explicit bounds in A. Cojocaru, C. Hall, "Uniform results for Serre's theorem for elliptic curves."

A theorem of Heath-Brown shows that a plane curve has at most C H^{2/d} rational points of height at most H, where C is a constant that does not depend on the curve. If the curve has positive genus, the global geometry of its Jacobian can be used to improve this bound; we show that in this case the exponent can be improved to 2/d - delta for some positive delta. If d = 3, we show delta = 1/450 is small enough. (Slightly edited version posted 26 Mar 2005.)

An old conjecture holds that the number N_n(X) of degree n number fields with discriminant less than X is asymptotic to c_n X when X grows and n is fixed. This conjecture was proved by Davenport and Heilbronn for n = 3, and recently for n = 4,5 by Bhargava. For general n, however, the best known upper bound, due to Schmidt, was N_n(X) << X^{(n+2)/4}. We prove the much stronger bound N_n(X) << X^{exp(C sqrt(log n))}. We also show that the number of degree n fields with Galois group S_n and discriminant < X is bounded below by a constant multiple of X^{1/2}, and we give an upper bound on the number of Galois extensions of a number field with fixed degree and bounded discriminant. All our results are proved in the general setting of counting degree-n extensions of an arbitrary number field K.

While the theorem is purely number-theoretic, the proof is almost entirely algebro-geometric; the main idea is to relate the problem of counting number fields to the problem of counting integral points on certain carefully chosen varieties related to Hilbert schemes of points on affine space.

Published version (.pdf)

• "Counting extensions of function fields with specified Galois group and bounded discriminant,", with A. Venkatesh, in Geometric Methods in Algebra and Number Theory, F. Bogomolov and Y. Tschinkel, eds. (2005)

Let G be a subgroup of S_n. Malle has conjectured that the number of number fields which have degree n over Q, Galois group G, and discriminant between -X and X is asymptotic to C X^a (log X)^b where a and b are constants depending on G. We study the problem of counting extensions of F_q(T) with the above properties; this is essentially a problem of counting F_q-rational points on Hurwitz varieties. We show that the heuristic "an irreducible d-fold over F_q has q^d rational points" yields the analogue of Malle's conjecture, with identical values of a and b. Moreover, the function field setting suggests more general heuristics about the distribution of discriminants of number fields. (Remark: in this paper we count only extensions of F_q(T) containing no constant field extension. There are some situations in which the fields with nontrivial constant subextensions actually dominate the ones we count, contrary to the remark on the top of p.154 in our paper.)

"Serre's conjecture over F9," Ann. of Math. 161 (3), 1111-1142 (2005).

We prove that, under certain local conditions at 3,5, and infinity, a representation of the absolute Galois group of Q in GL2(F9) is modular. As a corollary, abelian surfaces over Q with real multiplication and ordinary reduction at 3 and 5 are modular.

Published version (.pdf)

The generic complex K3 surface of degree d has Picard number one. We prove the (somewhat surprisingly, subtler) fact that there exist K3 surfaces of arbitrary degree over Qbar with Picard number one. Note: After this paper was in press, I learned about the paper of T. Terasoma ("Complete intersections with middle Picard number 1 defined over Q", Math. Z. 189 (1985), no.2, 289--296) which treats a related problem and uses a similar method.

A survey paper about the arithmetic of Q-curves, including problems about rational points on modular curves, surjectivity of Galois representations, modularity, and applications to classical Diophantine problems. Accessible open questions are emphasized throughout.

"A sharp diameter bound for unipotent groups of classical type over Z/pZ", with J. Tymoczko, to appear, Forum Math.

We give a sharp bound for the diameter of unipotent groups over Z/pZ, with respect to generators arising from simple roots. This paper generalizes previous unpublished work of the first author pertaining to the unipotent subgroup of GL_n(Z/pZ), as cited, e.g. in P. Diaconis and L. Saloff-Coste, "Moderate growth and random walk on finite groups."

"On the average number of octahedral modular forms," Math. Res. Lett. 10, 269--273 (2003)

In a recent paper, P. Michel and A. Venkatesh, sharpening a result of Duke, show that the number of modular forms of conductor N which are octahedral (associated to Galois representations with projective image S4) is at most N4/5 + e. Any octahedral form gives rise to a Galois representation Phi: Gal(Q) -> S3 by composition with the projection S4 -> S3. Using an amplifier constructed by S. Wong, we show that there are at most N2/3 + e octahedral forms of conductor N associated to a given such Phi. We use this fact, combined with the Davenport-Heilbronn theorem, to show that the average number of octahedral forms of conductor N, as N varies over square-free integers, is at most N2/3 + e.

"On the error term in Duke's estimate for the average special value of L-functions," Canad. Math. Bull. 48, no. 4, 535--546 (2005)

Duke showed, in his 1995 Invent. Math. paper, that the average value of L(f,1), as f ranges over an orthogonal basis for the cusp forms of weight 2 and level N, is 4Pi + O(N^(-1/2) log N). We improve the bound on the error term to O(N^(-1 + e)) for any e > 0. (NOTE: Nathan Ng found an error in an earlier version of this paper that changed the power of log N in the main result. The version dated April 2005 is correct. This change does not affect the results of "Galois representations attached to Q-curves..." below.)

"Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p," Amer. J. Math. 126(4), 763--787 (2004)

We bound the degree of rational isogenies of Q-curves over quadratic number fields, by combining Mazur's formal immersion method with an analytic argument showing that Jacobians of certain twisted modular curves admit quotients with Mordell-Weil rank 0. As a consequence, we show that the generalized Fermat equation A4 + B2 = Cp has no nontrivial primitive solutions for p sufficiently large. In order to make "sufficiently large" not too large, we use a refinement of a result of Duke (see "On the error term..." above.)

Minor revisions, August 2002.

"Galois invariants of dessins d'enfants," in Arithmetic Fundamental Groups and Noncommutative Algebra, M. Fried and Y.Ihara, eds. (2002)

This paper has two goals. First, we discuss some known Galois invariants of dessins d'enfants, such as the cartographic group and the lifting invariants of Fried; we show that these invariants are preserved by the Grothendieck-Teichmuller group. We then define a new invariant which associates to every genus 0 dessin d'enfant a triple of elements in a profinite spherical braid group. We give an example showing two dessins with the same monodromy group and local ramification data which are separated by the braid group invariant.

"Endomorphism algebras of Jacobians," Adv. Math. 162, 243--271 (2001)

Van der Geer and Oort have written:

"...one expects excess intersection of the Torelli locus and the loci corresponding to abelian varieties with very large endomorphism rings; that is, one expects that they intersect much more than their dimensions suggest."

Previous works of Brumer, Mestre, Ekedahl-Serre, and others have justified this expectation by providing examples of families of curves whose Jacobians have large endomorphism rings. We give a general procedure for producing families of branched covers of the line whose Jacobians have extra endomorphisms. We show that many of the examples produced by the above authors are "explained" in this way, and produce some new examples. For instance, we obtain curves whose Jacobians have real multiplication by the index-n subfield of Q(zp), where n is one of 2,4,6,8,10, and p is any prime congruent to 1 mod n. At the end, we discuss some questions about upper bounds for endomorphism algebras of Jacobians.

"On the modularity of Q-curves,",with C. Skinner, Duke Math. J. 109, no. 1, 97--122 (2001)

A Q-curve is an elliptic curve over a number field K which is isogenous to its Galois conjugates. Ribet proved that every quotient of J1(N) is a Q-curve, and conjectured conversely that every Q-curve is a quotient of some J_1(N). We prove this conjecture subject to certain local conditions at 3. The main tools are the deformation theorems of Conrad-Diamond-Taylor and Skinner-Wiles.

"Finite flatness of torsion subschemes of Hilbert-Blumenthal abelian varieties," J. Reine Angew. Math. 532, 1--32 (2001)

Let E be a totally real number field of degree d over Q.  We give a method for constructing a set of Hilbert modular cuspforms f1,...,fd with the following property.  Let K be the fraction field of a complete dvr A, and let X/K be a Hilbert-Blumenthal abelian variety with multiplicative reduction and real multiplication by the ring of integers of E.  Suppose n is an integer such that n divides the minimal valuation of fi(X) for all i.  Then X[n']/K extends to a finite flat group scheme over A, where n' is a divisor of n with n'/n bounded by a constant depending only on f1,..., fd.  When E = Q, the theorem reduces to a well-known property of f1 = D. When E is a quadratic field of discriminant 5 or 8, we produce the desired pairs of Hilbert modular forms explicitly and show how they can be used to compute the group of Néron components of a Hilbert-Blumenthal abelian variety with real multiplication by E.

"Congruence ABC implies ABC," Indag. Math., N.S., 11 (2), 197--200 (2000)

A note proving the following fact:  if the ABC conjecture holds for all A,B,C satisfying a divisibility condition N | ABC, then the full ABC conjecture holds.

"The combinatorics of rewritability in finite groups," with G. Sherman, L. Smithline, C. Sugar, E. Wepsic, in Group Theory (Granville, OH, 1992), 250--261, World Sci. Publishing, River Edge, NJ (1993)

An ordered triple (x1,x2,x3) of elements in a finite group is called rewritable if there exists some nontrivial permutation (xi,xj,xk) of (x1,x2,x3) with xixjxk = x1x2x3. Let r(G) be the proportion of ordered triples of elements of G which are rewriteable. We show that r(G) is at most 17/18 if it is not 1.

This theorem is an analogue to the well-known result that the proportion of ordered pairs of elements in G which commute is at most 5/8 if G is not abelian.

Jordan Ellenberg * ellenber@math.wisc.edu * revised 27 May 2010