Past Number Theory Seminar

So far, the circle method has been a very useful tool to prove
many cases of Manin's conjecture. Work of B. Birch back in 1961 establishes
this for smooth complete intersections in projective space as soon as the
number of variables is large enough depending on the degree and number of
equations. In this talk we are interested in subvarieties of biprojective
space. There is not much known so far, unless the underlying polynomials are
of bidegree (1,1). In this talk we present recent work which combines the
circle method with the generalised hyperbola method developed by V. Blomer
and J. Bruedern. This allows us to verify Manin's conjecture for certain
smooth hypersurfaces in biprojective space of general bidegree.

Solutions to translation invariant linear forms in dense sets  (for example: k-term arithmetic progressions), have been studied extensively in additive combinatorics and number theory. Finding solutions to translation invariant quadratic forms is a natural generalization and at the same time a simple instance of the hard general problem of solving diophantine equations in unstructured sets. In this talk I will explain how to modify the  classical circle method approach to obtain quantitative results  for quadratic forms with at least 17 variables.

I'll discuss questions about the structure of long sums of

Dirichlet characters --- that is, sums of length comparable to the modulus.

For example: How often do character sums get large? Where do character sums

get large? What do character sums "look like" when then get large? This will

include some combination of theorems and experimental data.

I will discuss conjectures relating cup products of cyclotomic units and modular symbols modulo an Eisenstein ideal. In particular, I wish to explain how these conjectures may be viewed as providing a refinement of the Iwasawa main conjecture. T. Fukaya and K. Kato have proven these conjectures under certain hypotheses, and I will mention a few key ingredients. I hope to briefly mention joint work with Fukaya and Kato on variants.

We will discuss arithmetic restriction phenomena and its relation to Waring's problem, focusing on how recent work of Wooley implies certain restriction bounds.

We prove an analogue of the Tate isogeny conjecture and the
semi-simplicity conjecture for overconvergent crystalline Dieudonne modules
of abelian varieties defined over global function fields of characteristic
p, combining methods of de Jong and Faltings. As a corollary we deduce that
the monodromy groups of such overconvergent crystalline Dieudonne modules
are reductive, and after base change to the field of complex numbers they
are the same as the monodromy groups of Galois representations on the
corresponding l-adic Tate modules, for l different from p.

I will discuss some p-adic (and mod p) criteria ensuring that an elliptic curve over the rationals has algebraic and analytic rank one, as well as some applications.

One of the most subtle aspects of the correspondence between automorphic and Galois representations is the weight part of Serre conjectures, namely describing the weights of modular forms corresponding to mod p congruence class of Galois representations. We propose a direct geometric approach via studying the mod p cohomology groups of certain integral models of modular or Shimura curves, involving Deligne-Lusztig curves with the action of GL(2) over finite fields. This is a joint work with James Newton.

In this talk I will present the best up-to-date bounds for the argument of the Riemann zeta-function on the critical line, assuming the Riemann hypothesis. The method applies to other objects related to the Riemann zeta-function and uses certain special families of functions of exponential type. This is a joint work with Vorrapan Chandee (Montreal) and Micah Milinovich (Mississipi).

Van der Waerden has shown that `almost' all monic integer

polynomials of degree n have the full symmetric group S_n as Galois group.

The strongest quantitative form of this statement known so far is due to

Gallagher, who made use of the Large Sieve.

In this talk we want to explain how one can use recent

advances on bounding the number of integral points on curves and surfaces

instead of the Large Sieve to go beyond Gallagher's result.

Counter-examples to the Hasse principle are known for many families of geometrically rational varieties. We discuss how often such failures arise for Chatelet surfaces and certain higher-dimensional hypersurfaces. This is joint work with Regis de la Breteche.

I will explain the beautiful generalization recently discovered by Y. Tian of Heegner's original proof of the existence of infinitely many primes of the form 8n+5, which are congruent numbers. At the end, I hope to mention some possible generalizations of his work to other elliptic curves defined over the field of rational numbers.

Toby Gee and I have proposed the definition of a "C-group", an extension of Langlands' notion of an L-group, and argue that for an arithmetic version of Langlands' philosophy such a notion is useful for controlling twists properly. I will give an introduction to this business, and some motivation. I'll start at the beginning by explaining what an L-group is a la Langlands, but if anyone is interested in doing some background preparation for the talk, they might want to find out for themselves what an L-group (a Langlands dual group) is e.g. by looking it up on Wikipedia!

A well known theorem of Coleman states that an overconvergent modular eigenform of weight k>1 and slope less than k-1 is classical. This theorem was later reproved and generalized using a geometric method very different from Coleman's cohomological approach. In this talk I will explain how one might go about generalizing the cohomological method to some higher-dimensional Shimura varieties.

In this talk I will explain a notion of p-adic functoriality for inner forms of definite unitary groups. Roughly speaking, this is a morphism between so-called eigenvarieties,  which are certain rigid analytic spaces parameterizing p-adic families  of automorphic forms. We will then study certain properties of classical Langlands functoriality that allow us to prove p-adic functoriality in some "stable" cases.

Let K be a number field and E/K be an elliptic curve. Multiplication by n induces a map from the n^2-Selmer group of E/K to the n-Selmer group. The image of this map contains the image of E(K) in the n-Selmer group and is often smaller. Thus, computing the image of the n^2-Selmer group under multiplication by n can give a tighter bound on the rank of E/K. The Cassels-Tate pairing is a pairing on the n-Selmer group whose kernel is equal to the image of the n^2-Selmer group under multiplication by n. For n=2, Cassels gave an explicit description of the Cassels-Tate pairing as a sum of local pairings and computed the local pairing in terms of the Hilbert symbol. In this talk, I will give a formula for the local Cassels-Tate pairing for n=3 and describe an algorithm to compute it for n an odd prime. This is joint work with Tom Fisher.

Let k be an odd integer and N be a positive integer divisible by 4. Let g be a newform of weight k - 1, level dividing N/2 and trivial character. We give an explicit algorithm for computing the space of cusp forms of weight k/2 that are 'Shimura-equivalent' to g. Applying Waldspurger's theorem to this space allows us to express the critical values of the L-functions of twists of g in terms of the coefficients of modular forms of half-integral weight. Following Tunnell, this often allows us to give a criterion for the n-th twist of an elliptic curve to have positive rank in terms of the number of representations of certain integers by certain ternary quadratic forms.

We will discuss the Littlewood conjecture from Diophantine approximation, and recent variants of the conjecture in which one of the real components is replaced by a p-adic absolute value (or more generally a "pseudo-absolute value''). The Littlewood conjecture has a dynamical formulation in terms of orbits of the action of the diagonal subgroup on SL_3(R)/SL_3(Z). It turns out that the mixed version of the conjecture has a similar formulation in terms of homogeneous dynamics, as well as meaningful connections to several other dynamical systems. This allows us to apply tools from combinatorics and ergodic theory, as well as estimates for linear forms in logarithms, to obtain new results.

In this talk, I will show that Brauer classes of a locally solvable degree 4 del Pezzo surface X are vertical, that is, that every Brauer class is obtained by pullback from an element of Br k(P^1) for some rational map f : X ----> P^1. As a consequence, we see that a Brauer class does not obstruct the existence of a rational point if and only if there exists a fiber of f that is locally solvable. The proof is constructive and gives a simple and practical algorithm, distinct from that in [Bright,Bruin,Flynn,Logan (2007)], for computing all nonconstant classes in the Brauer group of X. This is joint work with Anthony V\'arilly-Alvarado.

Given a variety X over a number field, one is interested in the collection X(F) of rational points on X. Weil defined a variety X' (the restriction of scalars of X) defined over the rational numbers whose set of rational points is naturally equal to X(F). In this talk, I will compare the number of rational points of bounded height on X with those on X'.

I will present an efficient algorithm for computing certain special values of Rankin triple product $p$-adic L-functions and give an application of this to the explicit construction of rational points on elliptic curves.