Anwesh Ray

Explicit Reciprocity Laws and Iwasawa theory for Modular forms

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Abstract: A conjecture of Mazur and Tate predicts that analytic theta elements of modular forms, which encode special values of L-functions, should lie in the Fitting ideal of their Selmer groups over cyclotomic extensions. In this talk we outline a proof of this conjecture (up to scaling) for p-power cyclotomic extensions in the case that the modular form is non-ordinary at p. The key tool is a general local construction of cohomology classes via the p-adic local Langlands correspondence. This is joint work with Matthew Emerton and Robert Pollack.

Class Groups and fine Selmer Groups

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Abstract: Starting from a result by Lim-Murty relating classical Iwasawa invariants of fine Selmer groups and p-class groups over the cyclotomic Zp-extension, we investigate generalizations of this results for multiple Zp-extensions and unfirom p-adic Lie- extensions. If time allows we will also discuss a density result for the weak Leopoldt conejcture. This is joined work with Sören Kleine.

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Video Recording Link

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L-invariants attached to the symmetric square representation

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Abstract: The p-adic L-function attached to the symmetric square of a modular form vanishes at certain critical twists, even though the complex L-function does not. We'll survey what is known about the first derivative of this p-adic L-function, and then describe an algorithm to compute the first derivative for non-CM elliptic curves. This talk should hopefully be accessible to graduate students in number theory.

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Video Recording Link

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Iwasawa Invariants of Modular Forms with a_p=0

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Abstract: Mazur-Tate elements provide a convenient method to study the analytic Iwasawa theory of p-nonordinary modular forms, where the associated p-adic L-functions have unbounded coefficients. The Iwasawa invariants of Mazur-Tate elements are well-understood in the case of weight two modular forms, where they can be related to the growth of p-Selmer groups and decompositions of the p-adic L-function. At higher weights, less is known. By constructing certain lifts to the full Iwasawa algebra, we compute the Iwasawa invariants of Mazur-Tate elements for higher weight modular forms with a_p=0 in terms of the plus/minus invariants of the p-adic L-function. Combined with results of Pollack-Weston, this forces a relation between plus/minus (and Sprung's sharp/flat) invariants at weights 2 and p+1.

Video Recording Link 

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Fine Selmer groups and duality

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Abstract: In Iwasawa Theory of p-adic Representations (1989),  R. Greenberg developed an Iwasawa theory for p-ordinary motives. In particular, he showed that the p-Selmer group over the cyclotomic Zp-extension satisfies an algebraic functional equation. In the intervening years, this strategy has been extended by several authors to prove functional equations in other settings. After discussing the history of these results, I will report on joint work with J. Hatley, A. Lei, and J.Ray where we take the first steps in trying to prove an algebraic functional equation for the fine Selmer group.

Video recording link

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On refined conjectures of Birch and Swinnerton-Dyer type in the Rankin Selberg setting

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Abstract: In the late 80’s, Mazur and Tate proposed conjectures on the structure of the Fitting ideals of Selmer groups over number fields of elliptic curves over Q. These conjectures are aimed to refine Birch and Swinnerton-Dyer type conjectures over number fields as well as the Iwasawa main conjectures over the cyclotomic Zp-tower of Q. Results in this direction have been obtained by Kim and Kurihara, who studied the Fitting ideals over finite sub-extensions of the cyclotomic Zp extension of Q. In this talk, I will describe results analogous to theirs on the Fitting ideals over the finite layers of the cyclotomic Zp-extension of Q of Selmer groups attached to the Rankin–Selberg convolution of two modular forms f and g. In the case where f corresponds to an elliptic curve E/Q and g to a two-dimensional odd irreducible Artin representation with splitting field F, I will explain how our results give an upper bound of the dimension of the g-isotypic component of the Mordell-Weil group of E over the finite layers of the cyclotomic Zp-extension of F.  This is joint work with Antonio Lei.

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Selmer classes on CM elliptic curves of rank 2 

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Abstract: Let E be an elliptic curve over Q, and let p be a prime of good ordinary reduction for E. Following the pioneering work of Skinner (and independently Wei Zhang) from about 8 years ago, there is a growing number of results in the direction of a p-converse to a theorem of Gross-Zagier and Kolyvagin, showing that if the p-adic Selmer group of E is 1-dimensional, then a Heegner point on E has infinite order. In this talk, I'll report on the proof of an analogue of Skinner's result in the rank 2 case, in which Heegner points are replaced by certain generalized Kato classes introduced by Darmon-Rotger. For E without CM, such an analogue was obtained in an earlier work with M.-L. Hsieh, and in this talk I'll focus on the CM case, whose proof uses a different set of ideas. 

A Shimura-Shintani correspondence for rigid analytic cocycles

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Abstract:  In their paper Singular moduli for real quadratic fields: a rigid
analytic approach, Darmon and Vonk introduced rigid meromorphic cocycles,
i.e. elements of H^1(SL_2(Z[1/p]), M^x) where M^x is the multiplicative
group of rigid meromorphic functions on the p-adic upper-half plane.
Their values at RM points belong to narrow ring class fields of real
quadratic fiends and behave analogously to CM values of modular functions
on SL_2(Z)\H. In this talk I will present some progress towards
developing a Shimura-Shintani correspondence in this setting.

 Asymptotic formula for Tate-Shafarevich groups of p-supersingular elliptic curves over anticyclotomic extension

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Abstract:   Let p>= 5 be a prime number and E/Q an elliptic curve with good supersingular reduction at p. Under the generalized Heegner hypothesis, we investigate the p-primary subgroups of the Tate-Shafarevich groups of E over number fields contained inside the anticyclotomic Z_p$-extension of an imaginary quadratic field where p splits. This is joint work with Meng Fai Lim and Katharina Mueller.

Kolyvagin's conjecture, bipartite Euler systems, and higher congruences of modular forms

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Abstract:   For an elliptic curve E,  Kolyvagin used Heegner points to construct special Galois cohomology classes valued in the torsion points of E. Under the conjecture that not all of these classes vanish, he showed that they encode the Selmer rank of E. I will explain a proof of new cases of this conjecture that builds on prior work of Wei Zhang. The proof naturally leads to a generalization of Kolyvagin's work in a complimentary "definite" setting, where Heegner points are replaced by special values of a quaternionic modular form. I'll also explain an "ultrapatching" formalism which simplifies the Selmer group arguments required for the proof.

Musings on mu

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Abstract: 

Iwasawa showed how to produce examples of Z_p-extensions where the mu-invariant (for class groups) is nonzero, and this method also yields extensions

where the \ell-part of the class group is unbounded, where \ell is a prime different from p. I'll review this construction and some related computations and then discuss

some ideas on whether this completely accounts for the behavior of the class number.

Partial classicality of Hilbert modular forms

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Abstract: 

Modular forms are global sections of certain line bundles on the modular curve, while p-adic overconvergent modular forms are defined only over a strict neighborhood of the ordinary locus. The philosophy of classicality theorems is that when the p-adic valuation of Up-eigenvalue is small compared to the weight (called a small slope condition), an overconvergent Up eigenform is automatically classical, namely it can be extended to the whole modular curve. In the case of Hilbert modular forms, there are the partially classical forms which are defined over a strict neighborhood of a “partially ordinary locus”. Modifying Kassaei’s method of analytic continuation, we show that under a weaker small slope condition, an overconvergent form is automatically partially classical.

Overconvergent Eichler--Shimura morphism for families of Siegel modular forms

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Abstract: 

Classical results of Eichler and Shimura decompose the cohomology of certain local systems on the modular curve in terms of holomorphic and anti-holomorphic modular forms. A similar result has been proved by Faltings' for the etale cohomology of the modular curve and Falting's result has been partly generalised to Coleman families by Andreatta--Iovita--Stevens. In this talk, based on joint work with Hansheng Diao and Ju-Feng Wu, I will explain how one constructs a morphism from the overconvergent cohomology of GSp_2g to the space of families of Siegel modular forms. This can be seen as a first step in an Eichler--Shimura decomposition for overconvergent cohomology and involves a new definition of the sheaf of overconvergent Siegel modular forms using the Hodge--Tate map at infinite level. If time allows it, I'll explain how one can hope to use higher Coleman theory to find a complete analogue of the classical Eichler--Shimura decomposition in small slope.

Arithmetic statistics for 2-bridge links

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Abstract: 

Let p be a fixed odd prime number. A famous theorem due to Iwasawa gives a formula for the rate of growth of the p-class group when the fields vary in a Z_p-extension of a number field. In this talk, based on joint work with Anwesh Ray, we study the topological analogue of Iwasawa theory for knots or, more generally, for links which are disjoint union of knots. In this setting, one can show that the lambda-invariant associated to a Z_p-cover of a link with at least 2 components is always greater than 0. We give explicit formulae to detect when the case mu=0 and lambda=1 do occur, at least in the case of 2 and 3-components links. We then study the proportion of 2-components links for which mu=0 and lambda=1 when the links are parametrized in Schubert normal form. Backed by numerical evidence, we conjecture that mu=0 for 100% of such links.

On some theorems in graph theory analogous to past results in Iwasawa theory

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Abstract: 

In the 1950s, Iwasawa proved his celebrated theorem on the growth of the p-part of the class number in Zp-extensions of number fields. The growth of the q-part, where q is another rational prime distinct from p, was studied by Washington and Sinnott among others. In this talk, we will explain our work in obtaining analogous results in graph theory for the number of spanning trees in some infinite towers of graphs analogous to Zp-extensions of number fields. Part of this work is joint with Kevin McGown and part of this work is joint with Antonio Lei.