About

Adebisi Agboola is a faculty member in the Department of Mathematics at the University of California, Santa Barbara. His specialization is in Number Theory. He is based in South Hall, Room 6607, and his office hours are Monday through Friday from 9 am to 12 pm and 1 pm to 4 pm. His contact email is agboola@math.ucsb.edu, and his phone number is (805) 893-3844. His research focus is on Number Theory, and he is actively involved in the academic community at UCSB, contributing to the department's educational and research missions.

Research topics

• Combinatorics
• Mathematics
• Discrete mathematics
• Art
• Geometry
• History
• Pure mathematics
• Aesthetics

Selected publications

• On the square root of the inverse different

  Canadian Journal of Mathematics · 2023 · 1 citations

  1st authorCorresponding
  • Mathematics
  • Combinatorics
  • Discrete mathematics

  Abstract Let $F_{\pi }$ be a finite Galois-algebra extension of a number field F , with group G . Suppose that $F_{\pi }/F$ is weakly ramified and that the square root $A_\pi $ of the inverse different $\mathfrak {D}_{\pi }^{-1}$ is defined. (This latter condition holds if, for example, $|G|$ is odd.) Erez has conjectured that the class $(A_\pi )$ of $A_\pi $ in the locally free class group $\operatorname {\mathrm {Cl}}(\mathbf {Z} G)$ of $\mathbf {Z} G$ is equal to the Cassou–Noguès–Fröhlich root number class $W(F_{\pi }/F)$ associated with $F_\pi /F$ . This conjecture has been verified in many cases. We establish a precise formula for $(A_\pi )$ in terms of $W(F_{\pi }/F)$ in all cases where $A_\pi $ is defined and $F_\pi /F$ is tame, and are thereby able to deduce that, in general, $(A_\pi )$ is not equal to $W(F_\pi /F)$ .

  PublisherOA PDFDOI

• On anticyclotomic variants of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -adic Birch and Swinnerton-Dyer conjecture

  Journal de Théorie des Nombres de Bordeaux · 2022 · 3 citations

  1st authorCorresponding
  • Mathematics
  • Combinatorics
  • Pure mathematics

  We formulate analogues of the Birch and Swinnerton-Dyer conjecture for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -adic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -functions of Bertolini, Darmon, and Prasanna attached to elliptic curves <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>/</mml:mo> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> </mml:math> at primes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> of good ordinary reduction. Using Iwasawa theory, we then prove, under mild hypotheses, one of the inequalities predicted by the “rank part” of our conjectures, as well as the predicted leading coefficient formula, up to a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -adic unit. Our conjectures are very closely related to conjectures of Birch and Swinnerton-Dyer type formulated by Bertolini and Darmon in 1996 for Heegner distributions, and as application of our results we also obtain the proof of an inequality in the rank part of their conjectures.

  PublisherOA PDFDOI

• No Feed, No Animal - The Parable of the Twelve Baskets

  Academia Letters · 2021 · 1 citations

  1st authorCorresponding
  • Art
  • Aesthetics
  • History
  PublisherDOI

• On anticyclotomic variants of the $p$-adic Birch and Swinnerton-Dyer conjecture

  arXiv (Cornell University) · 2019-10-19

  preprintOpen access1st authorCorresponding

  We formulate analogues of the Birch and Swinnerton-Dyer conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna attached to elliptic curves $E/\mathbf{Q}$ at primes $p$ of good ordinary reduction. Using Iwasawa theory, we then prove under mild hypotheses one of the inequalities predicted by the rank part of our conjectures, as well as the predicted leading coefficient formula up to a $p$-adic unit. Our conjectures are very closely related to conjectures of Birch and Swinnerton-Dyer type formulated by Bertolini-Darmon in 1996 for certain Heegner distributions, and as application of our results we also obtain the proof of an inequality in the rank part of their conjectures.

  PublisherOA PDFDOI

• On the square root of the inverse different

  arXiv (Cornell University) · 2018-03-26 · 1 citations

  preprintOpen access1st authorCorresponding

  Let N/F be a finite, normal extension of number fields with Galois group G. Suppose that N/F is weakly ramified, and that the square root A(N/F) of the inverse different of N.F is defined. (This latter condition holds if, for example, G is of odd order.) B. Erez has conjectured that the class (A(N/F)) of A(N/F) in the locally free class group Cl(ZG) of ZG is equal to the Cassou-Nogues-Frohlich root number class W(N/F) attached to N/F. We establish a precise formula for (A(N/F)) - W(N/F) in terms of the signs of certain symplectic Galois-Gauss sums whenever N/F is tame and (A(N/F)) is defined. We thereby show that, in general, (A(N/F)) is not equal to W(N/F).

  PublisherOA PDFDOI

• On the relative Galois module structure of rings of integers in tame extensions

  Algebra & Number Theory · 2018-12-04 · 4 citations

  articleOpen access1st authorCorresponding

  Let [math] be a number field with ring of integers [math] and let [math] be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group [math] of [math] that involves applying the work of McCulloh in the context of relative algebraic [math] theory. For a large class of soluble groups [math] , including all groups of odd order, we show (subject to certain mild conditions) that the set of realisable classes is a subgroup of [math] . This may be viewed as being a partial analogue in the setting of Galois module theory of a classical theorem of Shafarevich on the inverse Galois problem for soluble groups.

  PublisherOA PDFDOI

• CLASS INVARIANTS AND p-ADIC HEIGHTS

  2014-01-01

  article1st authorCorresponding

  Let F be a number field with ring of integers OF, and let E/OF be an abelian scheme of arbitrary dimension. In this paper, we study the class invariant homomoprhisms onE with respect to powers of a primep of ordinary reduction ofE. Our main result implies that if the p-adic Birch and Swinnerton-Dyer conjecture holds for E, then the kernels of these homomorphisms are of bounded order. It follows from this that (under the same hypotheses), ifL 2 Pic 0 (E) is a (rigidified) line bundle on E, thenL is determined up to torsion by its restriction to the p-divisible group scheme of E/OF.

  Publisher

• On counting rings of integers as Galois modules

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 2011-06-03 · 5 citations

  article1st authorCorresponding

  Let K be a number field and G a finite abelian group. We study the asymptotic behaviour of the number of tamely ramified G-extensions of K with ring of integers of fixed realisable class as a Galois module.

  PublisherDOI

• On Rubin’s variant of the p-adic Birch and Swinnerton–Dyer conjecture II

  Mathematische Annalen · 2010-06-24 · 1 citations

  articleOpen access1st authorCorresponding

  Let E/Q be an elliptic curve with complex multiplication by the ring of integers of an imaginary quadratic field K. In 1991, by studying a certain special value of the Katz two-variable p-adic L-function lying outside the range of p-adic interpolation, K. Rubin formulated a p-adic variant of the Birch and Swinnerton–Dyer conjecture when E(K) is infinite, and he proved that his conjecture is true for E(K) of rank one. When E(K) is finite, however, the statement of Rubin’s original conjecture no longer applies, and the relevant special value of the appropriate p-adic L-function is equal to zero. In this paper we extend our earlier work and give an unconditional proof of an analogue of Rubin’s conjecture when E(K) is finite.

  PublisherOA PDFDOI

• On Rubin's variant of the p-adic Birch and Swinnerton-Dyer conjecture II

  arXiv (Cornell University) · 2009-07-24

  preprintOpen access1st authorCorresponding

  Let E be an elliptic curve over Q with complex multiplication by the ring of integers of an imaginary quadratic field K. In 1991, by studying a certain special value of the Katz two-variable p-adic L-function lying outside the range of $p$-adic interpolation, K. Rubin formulated a p-adic variant of the Birch and Swinnerton-Dyer conjecture when $E(K)$ is infinite, and he proved that his conjecture is true for E(K) of rank one. When E(K) is finite, however, the statement of Rubin's original conjecture no longer applies, and the relevant special value of the appropriate $p$-adic L-function is equal to zero. In this paper we extend our earlier work and give an unconditional proof of an analogue of Rubin's conjecture when E(K) is finite.

  PublisherOA PDFDOI

Recent grants

• Galois structure, Iwasawa theory and arithmetic geometry

  NSF · $105k · 2004–2008

Frequent coauthors

• Benjamin Howard

  CCI Reprographics (United States)

  6 shared

• David Burns

  King's College London

  5 shared

• Francesc Castella

  2 shared

• G. Pappas

  Michigan State University

  2 shared

• Luca Caputo

  2 shared

• E. A. Iyayi

  1 shared

• Yu Kuang

  1 shared

• B. R. O. Omidiwura

  1 shared