Welcome to the *Algebra and its Applications* seminar
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The seminar is organized by Ettore Aldrovandi. Please send an email to contact me.

The seminar meets on Thursdays, 2:00-3:15pm in 104 LOV

January 13 | Amod Agashe | Visibility and the Shafarevich-Tate group |
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The Shafarevich-Tate group of an elliptic curve is an important invariant, measures the failure of the local-to-global principle. The second part of the Birch and Swinnerton-Dyer conjecture gives a conjectural value of the order of the Shafarevich-Tate group. In recent joint work with Loic Merel, we used the theory of visibility to give some theoretical evidence towards this conjecture. We shall describe this using an example, after introducing all the necessary concepts.

January 20 | Organizational meeting |
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January 27 | Meeting canceled |
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February 3 | Ettore Aldrovandi, FSU | Modular Forms I |
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This is the first of a series of talks devoted to an introduction to the basic facts on Modular Forms.

February 10 | Sa'ar Hersonsky | Maximal cusps on boundaries of deformations of hyperbolic 3-manifolds |
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Let M be a compact 3-manifold with non empty boundary. The space of allhyperbolic structures on the interior of M (i.e. Riemannian metrics with curvature equals -1), is wellunderstood. Its boundary is a topic of much research. We show that a certain type of hyperbolic manifolds (which are easy to understand) are dense on the boundary. This is a joint work with Richard D. Canary (Ann-Arbor).

February 17 | Ettore Aldrovandi, FSU | Modular Forms II |
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This is the second of a series of introductory talks on Modular Forms. After we are finished talking about the fundamental domain of the modular group, topics to be covered are: lattices, elliptic functions, and their relations to modular forms.

February 24 | Ettore Aldrovandi, FSU | Modular Forms III |
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We hace seen how a lattice function of weight
*2k* (a function on the space of lattices in
**C** of weight *2k*) corresponds to a
modular function of corresponding weight.
Therefore we need a good supply of lattice
functions in order to produce modular forms. These
are provided by the Eisenstein series. In this part
we'll show how to produce them in terms of elliptic
functions and elliptic curves. In particular, we
will obtain them as Laurent coefficients of
Weierstraß ℘ function near *z=0*
in **C**.

March 3 | Ettore Aldrovandi, FSU | Modular Forms IV |
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We have seen the definition and regularity properties
of the Eisenstein series associated to a lattice
in **C**. In this part, carrying over from last
meeting, we'll show how to produce them in terms of
elliptic functions and elliptic curves. In particular,
we will characterize Eisenstein series as Laurent
coefficients of Weierstraß's ℘ function
near *z=0* in the complex plane.

If time permits, we'll start looking into theorems
characterizing the ring *M*_{k}(Γ(1)) of
modular forms of weight *2k*—dimension, ring
structure, etc.

March 17 | Postponed |
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March 24 | Ettore Aldrovandi, FSU | Modular Forms V |
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In this talk we will compute the dimension of
*M*_{k}(Γ(1)), the vector space of
modular forms of weight *2k*, and characterize the
ring structure of the direct sum *M*(Γ(1)).

March 31 | Postponed |
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April 7 | Ettore Aldrovandi, FSU | Modular Forms VI |
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More about the structure of the
ring *M*(Γ(1)) and the modular
invariant *j*.

April 14 | Ettore Aldrovandi, FSU | Modular Forms VII |
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We will look at a few classical expansions of modular forms for the modular group Γ(1). After having covered these classical results, we will start looking at the (compactified) quotient of the upper half plane by (a subgroup of) the modular group, and introduce the appropriate Riemann surface structure.

April 21 | Ettore Aldrovandi, FSU | Modular Forms VIII |
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Introduction to the modular curves *X*(*N*) for the
congruence subgroups Γ(*N*) of the Modular
group Γ(1).