Past Number Theory Seminar

Let $G=SL(2,\R)\ltimes R^2$ and $\Gamma=SL(2,Z)\ltimes Z^2$. Building on recent work of Strombergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of $\Gamma\G$, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of $\sqrt n$ mod 1.

We investigate certain (hopefully new) arithmetic aspects of abelian varieties defined over function fields of curves over finitely generated fields. One of the key ingredients in our investigation is a new specialisation theorem a la N\'eron for the first Galois cohomology group with values in the Tate module, which generalises N\'eron specialisation theorem for rational points. Also, among other things, we introduce a discrete version of Selmer groups, which are finitely generated abelian groups. We also discuss an application of our investigation to anabelian geometry (joint work with Akio Tamagawa).

For Logic Seminar: Note change of time and place.

For Logic Seminar: Note change of time and location!

A distinct covering system of congruences is a collection

\[

(a_i \bmod m_i), \qquad 1\ \textless\ m_1\ \textless\ m_2\ \textless\ \ldots\ \textless\ m_k

\]

whose union is the integers. Erd\"os asked whether there are covering systems for which $m_1$ is arbitrarily large. I will describe my negative answer to this problem, which involves the Lov\'{a}sz Local Lemma and the theory of smooth numbers.

Note: Change of time and (for Logic) place! Joint with Number Theory (double header)

The goal of this lecture is describing recent joint work with Henri Darmon, in which we construct an Euler system of twisted Gross-Kudla diagonal cycles that allows us to prove, among other results, the following statement (under a mild non-vanishing hypothesis that we shall make explicit):

Let $E/\mathbb{Q}$ be an elliptic curve and $K=\mathbb{Q}(\sqrt{D})$ be a real quadratic field. Let $\psi: \mathrm{Gal}(H/K) \rightarrow \mathbb{C}^\times$ be an anticyclotomic character. If $L(E/K,\psi,1)\ne 0$ then the $\psi$-isotypic component of the Mordell-Weil group $E(H)$ is trivial.

Such a result was known to be a consequence of the conjectures on Stark-Heegner points that Darmon formulated at the turn of the century. While these conjectures still remain highly open, our proof is unconditional and makes no use of this theory.

After a short survey of the notion of level of distribution for
arithmetic functions, and its importance in analytic number theory, we
will explain how our recent studies of twists of Fourier coefficients of
modular forms (and especially Eisenstein series) by "trace functions"
lead to an improvement of the results of Friedlander-Iwaniec and
Heath-Brown for the ternary divisor function in arithmetic progressions
to prime moduli.

This is joint work with É. Fouvry and Ph. Michel.

We combine recent breakthroughs in modularity lifting with a
3-5-7 modularity switching argument to deduce modularity of elliptic curves over real
quadratic fields. We
discuss the implications for the Fermat equation. In particular we
show that if d is congruent
to 3 modulo 8, or congruent to 6 or 10 modulo 16, and $K=Q(\sqrt{d})$
then there is an
effectively computable constant B depending on K, such that if p>B is prime,
and $a^p+b^p+c^p=0$ with a,b,c in K, then abc=0.   This is based on joint work with Nuno Freitas (Bayreuth) and Bao Le Hung (Harvard).

 This lecture will cover two recent works on the mock modular
forms of Ramanujan.

I. Solution of Ramanujan's original conjectures about these functions.
(Joint work with Folsom and Rhoades)

II. A new theorem that mock modular forms are "generating functions" for
central L-values and derivatives of quadratic twist L-functions.
(Joint work with Alfes, Griffin, Rolen).

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.

The $p$-adic Gross-Zagier formula for diagonal cycles and the $p$-adic Beilinson formulae described in the lectures of Rotger and Bertolini respectively suggest a connection between certain  {\em $p$-adic iterated integrals} attached to modular forms and rational points on elliptic curves. I will describe an ongoing project (in collaboration with Alan Lauder and Victor Rotger) whose goal is to explore these relationships numerically, with the goal of better understanding the notion of {\em Stark-Heegner points}. It is hoped that these experiments might suggest new perspectives on Stark-Heegner points based on suitable {\em $p$-adic deformations} of the global objects--diagonal cycles, Beilinson-Kato and Beilinson-Flach elements-- described in the lectures of Rotger, Bertolini, Dasgupta, and Loeffler, following  the influential approach to $p$-adic $L$-functions pioneered by Coates-Wiles, Kato, and Perrin-Riou.

We define an integral version of Sczech cocycle on ${\rm GL}_n(\mathbf{Z})$ by raising the level at a prime $\ell$.As a result, we obtain a new construction of the $p$-adic L-functions of Barsky/Cassou-Nogu\`es/Deligne-Ribet. This cohomological construction further allows for a study of the leading term of these L-functions at $s=0$:\\1) we obtain a new proof that the order of vanishing is at least the oneconjectured by Gross. This was already known as result of Wiles.\\2) we deduce an analog of the modular symbol algorithm for ${\rm GL}_n$ from the cocyclerelation and LLL. It enables for the efficient computation of the special values of these $p$-adic L-functions.\\When combined  with a refinement of the Gross-Stark conjecture, we obtain some examples of numerical construction of $\mathfrak p$-units in class fields of totally real (cubic) fields.This is joint work with Samit Dasgupta.

I will describe a construction of special cohomology classes over the cyclotomic tower for the product of the Galois representations attached to two modular forms, which $p$-adically interpolate the "Beilinson--Flach elements" of Bertolini, Darmon and Rotger. I will also describe some applications to the Iwasawa theory of modular forms over imaginary quadratic fields.

We show that the $p$-adic L-function associated to the tensor square of a $p$-ordinary eigenform factors as the product of the symmetric square $p$-adic L-function of the form with a Kubota-Leopoldt $p$-adic L-function.  Our method of proof follows that of Gross, who proved a factorization for Katz's $ p$-adic L-function for a character arising as the restriction of a Dirichlet character.  We prove certain special value formulae for classical and $p$-adicRankin L-series at non-critical points.  The formula of Bertolini, Darmon, and Rotger in the $p$-adic setting is a key element of our proof.  As demonstrated by Citro, we obtain as a corollary of our main result a proof of the exceptional zero conjecture of Greenberg for the symmetric square.

I will report on $p$-adic Beilinson's formulas, relating the values of certain Rankin $p$-adic L-functions outside their range of classical interpolation, to $p$-adic syntomic regulators of Beilinson-Kato and Beilinson-Flach elements. Applications to the theory of Euler systems and to the Birch and Swinnerton-Dyer conjecture will also be discussed. This is joint work with Henri Darmon and Victor Rotger.

In this lecture I shall introduce certain generalised Gross-Kudla-Schoen diagonal cycles in the product of three Kuga-Sato varieties and a $p$-adic formula of Gross-Zagier type which relates the images of these diagonal cycles under the $p$-adic Abel-Jacobi map to special values of the $p$-adic L-function attached to the Garrett triple convolution of three  Hida families of  modular forms. This formula has applications to the Birch--Swinnerton-Dyer conjecture and the theory of Stark-Heegner points. (Joint work with Henri Darmon.)

The main result of the talk is that two curves over a finite field are isomorphic, up to automorphisms of the ground field, if and only if there is an isomorphism of groups of Dirichlet characters such that the corresponding L-series are all equal. This can be shown by combining Uchida's proof of the anabelian theorem for global function fields with methods from (noncommutative) dynamical systems. I will also discuss how to turn this theorem into a theoretical algorithm that, given a listing of L-functions, determines an equation for the corresponding curve(s).

I will present an overview of a series of joint works with Akio Tamagawa about l-adic representations of etale fundamental group of curves (to simplify, over finitely generated fields of characteristic 0).
More precisely, when the generic representation is GLP (geometrically Lie perfect) i.e. the Lie algebra of the geometric etale fundamental group is perfect, we show that the associated local $\ell$-adic Galois representations satisfies a strong uniform open image theorem (ouside a `small' exceptional locus). Representations on l-adic cohohomology provide an important example of GLP representations. In that case, one can even provethat the exceptional loci that appear in the statement of our stronguniform open image theorem are independent of $\ell$, which was predicted by motivic conjectures.
Without the GLP assumption, we prove that the  associated local l-adic Galois representations still satisfy remarkable rigidity properties: the codimension of the image at the special fibre in the image at the generic fibre is at most 2 (outside a 'small' exceptional locus) and its Lie algebra is controlled by the first terms of the derived series of the Lie algebra of the image at the generic fibre.
I will state the results precisely, mention a few applications/open questions and draw a general picture of the proof in the GLP case (which,in particular, intertwins via the formalism of Galois categories, arithmetico-geometric properties of curves and $\ell$-adic geometry). If time allows, I will also give a few hints about the $\ell$-independency of the exceptional loci or the non GLP case.