Past Number Theory Seminar

 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.

In this talk, I will describe a construction of a $p$-adic L-function attached to a triple of $p$-adic Coleman families of cusp forms.  This function interpolates algebraic parts of special values of Garrett triple product L-functions at balanced triples of weights.  Our construction is complementary to that of Harris and Tilouine which treats the case of unbalanced weights.

Given a $p$-adic weight and a finite slope we describe a Hecke and Galois equivariant geometric map relating elliptic overconvergent modular symbols and overconvergent modular forms of that slope, appropriate weights and $\mathbf{C}_p$-coefficients. We show that for a fixed slope, with the possible exception of a discrete family of weights, this map is an isomorphism.

Using Faltings' theory of the Hodge-Tate sequence of an abelian scheme we construct certain sheaves $\Omega^\kappa$, where $\kappa$ is a not-necessarily integral weight, over formal subschemes of modular varieties over which the canonical subgroup exists.   These sheaves generalize the integral powers, $\omega^k$, of the sheaf $\omega$ of relative differentials on a modular curve.   Global sections of $\Omega^\kappa$ provide geometric realizations of overconvergent automorphic forms of non-integral weight.  Applications of this approach to the theory of $p$-adic Hilbert modular forms will be given.   This is joint work with Fabrizio Andreotti and Adrian Iovita.

We apply the theory of the radius of convergence of a $p$-adic connection to the special case of the direct image of the constant connection via a finite morphism of compact $p$-adic curves, smooth in the sense of rigid geometry. We show that a trivial lower bound for that radius implies a global form of Robert's $p$-adic Rolle theorem. The proof is based on a widely believed, although unpublished, result of simultaneous semistable reduction for finite morphisms of smooth $p$-adic curves. We also clarify the relation between the notion of radius of convergence used in our previous work and the more intrinsic one used by Kedlaya. (The paper is available athttp://arxiv.org/abs/1209.0081)

For a proper semistable curve over a DVR of mixed characteristics we re prove the ``invariant cycles theorem'' with trivial coefficients  by Chiarellotto i.e. that the group of elements annihilated by the monodromy operator on the first de Rham cohomology group of the generic fiber coincides with the first rigid cohomology group of the special fiber, without the hypothesis that the residue field is finite. This is done using the explicit description of the monodromy operator on the de Rham cohomology of the generic fiber with coefficients convergent F-isocrystals given in a work of Coleman and Iovita. We apply these ideas to the case where the coefficients are unipotent convergent F-isocrystals defined on the special fiber: we show that the invariant cycles theorem does not hold in general in this setting. Moreover we give a sufficient condition for the non exactness. It is a joint work with B. Chiarellotto, R. Coleman and A. Iovita.

In 1986 Mazur, Tate and Teitelbaum came up with a $p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer for elliptic curves over the rationals. In this talk I will report on joint work with Jennifer Balakrishnan and William Stein on a generalization of this conjecture to the case of modular abelian varieties and primes $p$ of good ordinary reduction. I will discuss the theoretical background that led us to the formulation of the conjecture, as well as numerical evidence supporting it in the case of modular abelian surfaces and the algorithms that we used to gather this evidence.

We extend the following theorem of H. Imai in several ways: If  $A$  is an abelian variety with potentially good reduction over a finite extension  $K$  of  $\mathbf{Q}_p$, then it has only finitely many rational torsion points over the maximal $p$-cyclotomic extension of  $K$. In particular, we prove the finiteness over $K(K^{1/p^\infty})$.

The overconvergence of the canonical subgroup of the universal abelian variety is one of the key ingredients of the theory of overconvergent modular forms. In this talk, I will show the overconvergence of the canonical subgroup with expected properties via the Breuil--Kisin classification, including the case of $p=2$.

I will explain recent joint work with Matthew Emerton and David Savitt, in which we relate the geometry of various tamely potentially Barsotti--Tate deformation rings for two-dimensional Galois representations to the integral structure of the cohomology of Shimura curves. As a consequence, we establish some conjectures of Breuil regarding this integral structure.
The key technique is the Taylor--–Wiles--–Kisin patching argument, which,when combined with a new, geometric perspective on the Breuil–--Mezard conjecture, forges a tight link between the structure of cohomology (a global automorphic invariant) and local deformation rings (local Galois-theoretic invariants).