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 Events
Université Laval
Mon 5 Mar 2018, 11:00am SPECIAL
Number Theory Seminar / PIMS Seminars and PDF Colloquiums
ESB 4127
Introduction to Supersingular Iwasawa Theory of Elliptic Curves (talk 1)
ESB 4127
Mon 5 Mar 2018, 11:00am-12:00pm

Abstract

Let E/Q be an elliptic curve. In Iwasawa Theory, we study the behaviours of E over a tower of number fields. For example, it is known that the Mordell Weil ranks of E over all p-power cyclotomic extensions of Q are bounded when p does not divide the conductor of E. Surprisingly, the techniques required to show this are very different depending on the number of points on the finite curve when we consider E reduced modulo p. The easier case is when E has "ordinary" reduction at p and the more difficult case is when E has "supersingular" reduction at p. I will review the Iwasawa-theoretic tools used to study the behaviours of E over cyclotomic fields in these two cases. I will also discuss some recent developments on the Iwasawa theory of elliptic curves over quadratic extensions of Q.

This is talk 1 of 3 by the speaker, and part of the PIMS Thematic Events on "Galois groups in arithmetic".
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Université Laval
Wed 7 Mar 2018, 11:00am SPECIAL
Number Theory Seminar / PIMS Seminars and PDF Colloquiums
ESB 4127
Introduction to Supersingular Iwasawa Theory of Elliptic Curves (talk 2)
ESB 4127
Wed 7 Mar 2018, 11:00am-12:00pm

Abstract

Let E/Q be an elliptic curve. In Iwasawa Theory, we study the behaviours of E over a tower of number fields. For example, it is known that the Mordell Weil ranks of E over all p-power cyclotomic extensions of Q are bounded when p does not divide the conductor of E. Surprisingly, the techniques required to show this are very different depending on the number of points on the finite curve when we consider E reduced modulo p. The easier case is when E has "ordinary" reduction at p and the more difficult case is when E has "supersingular" reduction at p. I will review the Iwasawa-theoretic tools used to study the behaviours of E over cyclotomic fields in these two cases. I will also discuss some recent developments on the Iwasawa theory of elliptic curves over quadratic extensions of Q.

This is talk 2 of 3 by the speaker, and part of the PIMS Thematic Events on "Galois groups in arithmetic".
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Université Laval
Fri 9 Mar 2018, 11:00am SPECIAL
Number Theory Seminar / PIMS Seminars and PDF Colloquiums
ESB 4127
Introduction to Supersingular Iwasawa Theory of Elliptic Curves (talk 3)
ESB 4127
Fri 9 Mar 2018, 11:00am-12:00pm

Abstract

Let E/Q be an elliptic curve. In Iwasawa Theory, we study the behaviours of E over a tower of number fields. For example, it is known that the Mordell Weil ranks of E over all p-power cyclotomic extensions of Q are bounded when p does not divide the conductor of E. Surprisingly, the techniques required to show this are very different depending on the number of points on the finite curve when we consider E reduced modulo p. The easier case is when E has "ordinary" reduction at p and the more difficult case is when E has "supersingular" reduction at p. I will review the Iwasawa-theoretic tools used to study the behaviours of E over cyclotomic fields in these two cases. I will also discuss some recent developments on the Iwasawa theory of elliptic curves over quadratic extensions of Q.

This is talk 3 of 3 by the speaker, and part of the PIMS Thematic Events on "Galois groups in arithmetic".
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Concordia University
Thu 15 Mar 2018, 3:30pm SPECIAL
Number Theory Seminar / PIMS Seminars and PDF Colloquiums
ESB 4127
p-Adic modular forms
ESB 4127
Thu 15 Mar 2018, 3:30pm-5:15pm

Abstract

Abstract: p-Adic modular forms have first been defined by J.-P. Serre as q-expansions and have later been interpreted geometrically by N. Katz as sections of certain modular line bundles over the ordinary locus of the relevant modular curves.
Katz also defined overconvergent modular forms of integer weights as overconvergent sections of the modular line bundles of that weight. Many years later H. Hida and respectively R. Coleman defined ordinary, respectively finite slope overconvergent modular forms of arbitrary, p-adic weight as q-expansions and using these Coleman and Mazur constructed at the end of the 90's the famous eigencurve. Recently, together with Andreatta, Pilloni and Stevens we have been able to geometrically redefine the overconvergent modular forms of Hida and Coleman and so we were able to generalize these constructions to Hilbert and Siegel modular forms.

This talk is part of the PIMS Thematic Events on "Galois groups in arithmetic".
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