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# ArithmeTexas 2005

## Texas A&M University Department of Mathematics April 2-3, 2005

All talks will take place in Blocker 158. See campus map.

## Saturday, April 2

 8:15-9:00 Coffee/Tea/Pastries

 9:00-9:50 Alf van der Poorten Brown University & CeNTRe, Sydney Somos sequences
 Abstract: In the early nineties Michael Somos asked for the inner meaning of the fact that the quadratic recursions C_{h-2}C_{h+2}=C_{h-1}C_{h+1}+C_{h}^2 and B_{h-2}B_{h+3}=B_{h-1} B_{h+2} + B_{h}B_{h+1} are respectively satisfied by the bi-directional sequences ..., 2, 1, 1, 1, 1, 2, 3, 7, 23, 59, ..., and ..., 3, 2, 1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, ..., surprisingly consisting entirely of integers [which incidentally, increase at a hefty clip: the log of their h-th term is O(h^2) as h -> infinity]. In the meantime I had found quite independent reason for studying the continued fraction expansion of square roots of polynomials. Nonetheless, I found that those expansions inter alia produce sequences of integers satisfying quadratic recursions of Somos type. For instance, the sequence (C_h) [called 4-Somos by some of its friends] reports the denominators of a sequence of points on the quartic curve Y^2=(X^2-3)^2+4(X-2), equivalently of the points M+hS, with M=(-1,1), S=(0,0) on the cubic curve V^2-V=U^3+3U^2+2U. Indeed all Somos 4 and Somos 5 sequences are `elliptic sequences' in such a sense. I will tell the story from very first principles and will mention generalisations. Thus, again for instance, the sequence ..., 2, 1, 1, 1, 1, 1, 1, 2, 3, 4, 8, 17, 50, ..., given by the recursion B_{h-3}B_{h+3}=B_{h-2}B_{h+2}+B_{h}^2 arises from adding multiples of the class of the divisor at infinity on the Jacobian of the curve Y^2=(X^3-4X+1)^2+4(X-2) of genus 2.

 10:00-10:20 Jeff Vaaler University of Texas Counting algebraic points of low height and fixed degree
 Abstract: Joint work with David Masser.

 10:25-10:45 Ling Long Iowa State University A short proof of Milne's formulae for sums of integer squares
 Abstract: Representing a natural number $n$ as the sums of integer squares is a long standing and difficult problem. We give a short proof of Milne's formulae for sums of 4n^2 and 4n^2+4n integer squares using the theory of modular forms. Joint work with Yifan Yang.

 10:50-11:10 Jayce Getz University of Wisconsin Intersection numbers of Hecke correspondences
 Abstract: We prove a formula relating the intersection numbers of certain correspondences on the product of two Hilbert modular varieties to traces of Hecke operators. As an application, we show that the generating function for these intersection numbers is a Hilbert modular form.

 11:15-11:35 Ze-Li Dou Texas Christian University Period relations of automorphic forms
 Abstract: Periods of automorphic forms are investigated in many different guises, as integrals, as special values of L-functions, as Fourier coefficients of automorphic forms, and so on. There are many challenges in understanding their mutual relations; a comprehensive network of conjectures has been proposed by Shimura. I shall begin by introducing the Shimura period conjectures, and some of the methods and known results. I shall then report on some recent progress obtained via theta correspondence with respect to a quadratic base change.

 11:35-1:00 Lunch

 1:00-1:50 Doug Hensley Texas A&M University Revisiting the Hurwitz complex continued fraction
 Abstract: The Hurwitz complex continued fraction algorithm computes a sequence p_n/q_n = a_0 + 1/(a_1 + 1/(a_2 + ... + 1/a_n)) of Gaussian rational approximations to a complex \emph{target}. It is the natural analog in a complex setting to the centered variant of the classical continued fraction. We establish a Gauss-Kuzmin type theorem for the algorithm, announce related conjectures concerning the average runtime of the algorithm in its role as a gcd algorithm for Gaussian integers, and show that for a class of algebraic targets which have degree 4 over Q(i), the successive remainders, the frequency of a_j values, and the approximation properties, are neither periodic nor in line with the average behavior that holds by virtue of our Gauss-Kuzmin theorem for almost all targets. We indicate some similarities to recent results concerning simultaneous diophantine approximation in R^n to targets of the form (alpha, ..., alpha degree[\alpha] - 1)$.  2:00-2:20 William Banks University of Missouri On values taken by the largest prime factor of shifted primes  Abstract: Let Pi denote the set of prime numbers, and let$P(n)$denote the largest prime factor of an integer$n>1$. We show that, for any real number$25/13\eta<(4+3\sqrt{2})/4$and every integer$a\ne 0$, the set { p in Pi : p = P(q-a) for some prime q with p^eta < q < 12.5p^eta } has relative asymptotic density one in the set of all prime numbers. Moreover, in the range$2 <= eta < (4+3*sqrt{2})/4\$, one can replace 12.5 by any absolute constant c > 1. In particular, for every real number 0.486 <= theta <= 0.520, the relation P(q-a) is asymptotic to q^theta holds for infinitely many primes q. We use this result to derive a lower bound for the value of the Carmichael function taken on a product of shifted primes. This is joint work with Igor Shparlinski.

 2:25-2:45 Chris Rasmussen Rice University On the torsion of Jacobian varieties of X(p^n)
 Abstract: In this talk, we study the fixed field of the kernel of a particular representation of the absolute Galois group, into the outer automorphisms of the (pro-p) fundamental group of the projective line minus three points. Although well studied, many properties of this representation are still unknown, such as the size of the field in question. We will present new work, following the techniques of Anderson and Ihara, demonstrating fields of p-power torsion of the Jacobian varieties of modular curves of level p^N are rational over this field, in the case p=3. The result rests on both the arithmetic and geometry of X(p^N), when viewed as a cover of the projective line minus three points. This work is joint with Matt Papanikolas.

 2:50-3:10 Paul Jenkins University of Wisconsin Borcherds products and p-adic properties of singular moduli
 Abstract: We give a new proof of some identities of Zagier relating traces of singular moduli to the coefficients of certain half integral weight modular forms. These results imply a new proof of the infinite product isomorphism announced by Borcherds in his 1994 ICM lecture. In addition, we derive a simple expression for writing twisted traces as an infinite series, and discuss p-adic properties of traces and the congruences that follow.

 3:10-3:50 Coffee/Tea

 3:50-4:10 Bogdan Petrenko Texas A&M University On pairs of matrices that generate matrix rings
 Abstract: I will discuss some algebraic and topological properties of the pairs of matrices that generate matrix rings. Joint work with Said Sidki.

 4:15-4:35 Sungkon Chang University of Georgia On the arithmetic of twists of Jacobian varieties
 Abstract: Let l be a prime number, and let lambda := 1 - zeta_l. Let K be a global field of characteristic not equal to l containing zeta_l, a primitive l-th root of unity. Let C/K be a superelliptic curve given by y^l = f (x), and for D in K^* let C_D/K be the l-th power twist of C given by Dy^l = f (x). Let J/K and J_D/K denote the Jacobian varieties of C and C_D, respectively. Suppose that K is the l-th cyclotomic extension of Q with class number not divisible by l, and f(x) in Z[x] is irreducible over K and has prime degree p not equal to l. Then, for a large positive integer X, I find a lower bound on the number of l-power free positive integers D < X such that the Mordell-Weil rank over Q, denoted by rank J_D(Q), is bounded by the dimension of the l-Selmer group of J/K. This result has applications to showing that there are infinitely many l-th power twists C_D such that #C_D(Q) is bounded. An analogous statement is true for K being a function field F_q(t), q not l, and it proves for some cases that there are infinitely many l-th power twists C_D with rank J_D(K) = 0. I shall also show examples of superelliptic curves C/K (for each prime number l such that the l-Selmer group of J_D/K can be arbitrarily large.

 4:40-5:00 Lenny Fukshansky Texas A&M University Effective decompositions of quadratic spaces
 Abstract: A classical theorem of Witt states that a bilinear space can be decomposed into an orthogonal sum of hyperbolic planes, singular, and anisotropic components. I will discuss the existence of such a decomposition of bounded height for a symmetric bilinear space over a number field. I will also talk about an effective version of Cartan-Dieudonne theorem on representation of an isometry of a regular symmetric bilinear space as a product of reflections. All bounds on height are explicit and polynomial in the main parameters.

## Sunday, April 3

 8:15-9:00 Coffee/Tea/Pastries

 9:00-9:50 Pierre Debes Université de Lille Inverse Galois theory and profinite arithmetic geometry
 Abstract: The Hurwitz moduli space theory provides an arithmetic-geometric approach to the regular inverse Galois problem: the question is reduced to finding rational points on varieties. Patching techniques can be used over p-adic fields, and extend to the realization of profinite groups. In this profinite context, the moduli aspect leads to towers of varieties, for which there are new specific arithmetic issues. There are positive results over p-adic fields, negative results over number fields, and over other fields like large fields, the situation is unclear.

 10:00-10:20 Gergely Harcos University of Texas A Burgess-like subconvex bound for twisted L-functions
 Abstract: This talk presents joint work with Valentin Blomer and Philippe Michel. I will discuss how the Kuznetsov trace formula and the Deshouillers-Iwaniec spectral large sieve can be used to improve on existing upper bounds for L-functions associated with character twists of a cuspidal newform. The method applies equally to holomorphic and Maass forms of arbitrary level and nebentypus. In particular, in the conductor aspect the trivial exponent 1/2 can be replaced by 1/2-1/8 assuming the Ramanujan-Petersson conjecture for Maass forms. This is analogous to Burgess' 40-year old unsurpassed bound for Dirichlet L-functions which replaces the trivial exponent 1/4 by 1/4-1/16.

 10:25-10:45 Igor Shparlinski Macquarie University Pseudoprimes in nonlinear sequences
 Abstract: We show that if a >1 is any fixed integer, then for a sufficiently large x > 1, the n-th Fibonacci number F_n is a base a pseudoprime only for at most O(x/\log x) of positive integers n <= x. The same result holds for Mersenne-like sequences such as 2^n - 1, and for one more general class of Lucas sequences. A slight modification of our method also leads to similar results for polynomial sequences f(n), where f is in Z[X]. Finally, we use a different technique to get a much sharper upper bound on the counting function of the positive integers n such that phi(n) is a base a pseudoprime, where phi is the Euler function. Joint work with Florian Luca.

 10:50-11:10 Marius Somodi University of Northern Iowa Wild sets for rational self-equivalences
 Abstract: To any Hilbert symbol equivalence between two number fields one associates a set of prime ideals, called the wild set of the Hilbert symbol equivalence. We will focus on the Hilbert symbol self-equivalences of the field of rational numbers (called rational self-equivalences), and present a characterization of the finite sets of primes that are wild sets for rational self-equivalences.

 11:15-11:35 Ian Connell McGill University Apecs8
 Abstract: Apecs is a computer program written in Maple to do calculations related to elliptic curves. Apecs8 is the latest version and is considerably expanded from earlier versions. In particular, for elliptic curves defined over number fields it can carry through Tate's algorithm at the bad primes and use Silverman's algorithm to compute canonical heights, and so calculate the regulator of a list of points of infinte order (or find an explicit relation mod torsion). Algorithms of Cohen et al for number fields are implemented in apecs to "minimize" the polynomials defining the fields, find integral bases, etc. Positive charcteristics are not neglected: for example, Tate's algorithm is implemented for the function fields GF(p^n)(T). As many of these capabilities of apecs as time permits are illustrated by examples.

 11:35-1:00 Lunch

 1:00-1:50 Joseph Silverman Brown University Divisibility sequences and algebraic groups
 Abstract: A divisibility sequence is a sequence of integers d_n with the property that m|n ==> d_m|d_n. Examples of divisibility sequences include the sequence a^n-1, the Fibonacci sequence F_n, and the sequence D_n obtained by writing x(nP)=A_n/D_n^2 for the multiples of a rational point P of infinite order on an elliptic curve E/Q. These examples are special cases of the divisibility sequences attached to points of infinite order on algebraic groups. More precisely, let G/Z be a group scheme and let g in G(Z) be an element of infinite order. Associated to g is a divisibility sequence d_n defined by the property that d_n is the largest integer such that g^n = 1 (mod d_n). We will discuss how Vojta's conjectures applied to G_Q blown up at the origin leads to a strong bound on the growth rate of such divisibility sequences and will describe recent work of Ailon, Bugeaud, Corvaja, Rudnick, Zannier, and the speaker proving such bounds for G_m^d and for other tori over Q.

 2:00-2:20 Kathrin Bringmann University of Wisconsin Traces of singular moduli on Hilbert modular surfaces
 Abstract: Suppose that p = 1 (mod 4) is a prime, and that O_p is the ring of integers of K := Q(sqrt{p}). A classical result of Hirzebruch and Zagier asserts that certain generating functions for the intersection numbers of Hirzebruch-Zagier divisors on the Hilbert modular surface (H x H)/\SL_2(O_p) are weight 2 holomorphic modular forms. Using recent work of Bruinier and Funke, we show that the generating functions of traces of singular moduli over these intersection points are often weakly holomorphic weight 2 modular forms. For the singular moduli of J_1(z) = j(z) - 744, we explicitly determine these generating functions using classical Weber functions, and we factorize their "norms" as products of Hilbert class polynomials.

 2:25-2:45 Florian Luca Universidad Nacional Autónoma de México Diophantine m-tuples
 Abstract: Let n be a nonzero integer. A set with the property D(n) is a set of nonzero integers A = { a_1, ..., a_m } such that a_i*a_j + n is a square for all i not j. What is of interest in general is to find upper bounds on m, the size of a set with the property D(n). In my talk, I will survey various known results about this problem and report on a few new ones. For example, one of the new results is that if n is a prime, then m < 3*2^144. Joint work with Andrej Dujella.

 2:50-3:10 Ahmad El-Guindy Texas A&M University Galois actions on Weierstrass points of X_0(p)