Publications

This is a list of my academic publications.
Return to Main Site. You might also be looking for my teaching website or my misc notes website.

• A. Babei, A. Fiori, C. Franc, Families of ϕ-congruence subgroups of the modular group, Mathematika (2023), Volume 69, Issue 4, pp. 35.
  Journal Version
  arXiv Version
  We introduce and study families of finite index subgroups of the modular group that generalize the congruence subgroups. Such groups, termed ϕ-congruence subgroups, are obtained by reducing homomorphisms ϕ from the modular group into a linear algebraic group modulo integers. In particular, we examine two families of examples, arising on the one hand from a map into a quasi-unipotent group, and on the other hand from maps into symplectic groups of degree four. In the quasi-unipotent case we also provide a detailed discussion of the corresponding modular forms, using the fact that the tower of curves in this case contains the tower of isogenies over the elliptic curve y^2=x^3−1728 defined by the commutator subgroup of the modular group.
  
  
• A. Fiori, H. Kadiri, J. Swidinsky, Sharper bounds for the error term in the Prime Number Theorem, Research in Number Theory (2023), Volume 9, Article Number 63, pp. 19.
  Journal Version
  arXiv Version
  We provide very effective methods to convert both asymptotic and explicit numeric bounds on the prime counting function Psi(x) to bounds of the same type on both Theta(x) and Pi(x). Applying these to our explicit results for Psi(x) we obtain new unconditional bounds for |Pi(x)-Li(x)| valid for x>2.
  
  
• A. Fiori, H. Kadiri, J. Swidinsky, Sharper bounds for the Chebyshev function psi(x), Journal of Mathematical Analysis and Applications (2023), Volume 527, Issue 2, pp. 28.
  Journal Version
  arXiv Version
  We obtain strong unconditional bounds on the error terms in the prime number theorem. The results here concern the Psi function, we shall adapt the results to other prime counting functions in follow up work.
  
  
• A. Fiori, Numerical Verification of the Least Prime in the Chebotarev Density Theorem Appendix To H. Kadiri, P. Wong Primes in the Chebotarev density theorem for all number fields Journal of Number Theory (2022), Volume 241, p.700-737.
  Journal Version
  arXiv Version
  Numerical verification of bounds on first prime in the Chebotarev density theorem for number fields of small discriminant.
  The main article of Kadiri-Wong establishes unconditional bounds on the first prime in the Chebotarev density theorem.
  
  
• C. Cunningham, A. Fiori, N. Kitt, Appearance of the Kashiwara-Saito singularity in the representation theory of p-adic GL16. Pacific Journal of Math (2022), Volume 321, Number 2, p. 239-282.
  Journal Version
  arXiv Version
  We study the appearance of the Kashiwara-Saito singularity in a certain moduli space of L-parameter for GL16. This singularity leads to a ABV packet with two representations and a number of other unexpected features..
  
  
• C. Cunningham, A. Fiori, Q. Zhang, Toward the endoscopic classification of unipotent representations of p-adic G2. Submitted (2020), pp. 53.
  arXiv Version
  We study ABV packets for all unipotent representations of p-adic G2 as well as introduce a geometric endoscopic transfer.
  
  
• A. Fiori, C. Franc, The unbounded denominator conjecture for the noncongruence subgroups of index 7. Journal of Number Theory (2022), Volume 240, p.611--640.
  Journal Version
  Preprint
  We establish the unbounded denominator conjecture for all non-congruence subgroups of index 7. We also investigate the associated Eisenstein series.
  
  
• R. Baillie, A. Fiori, S. Wagstaff, Strengthening the Baillie-PSW Primality Test. Mathematics of Computation (2021), Volume 90, Number 330, p. 1931-1955.
  Journal Version
  Preprint
  We study various aspects of the Lucas pseudo-primes and propose a refinement of the BPSW Primality Test.
  
  
• C. Cunningham, A. Fiori, Q. Zhang, Arthur packets for G2 and perverse sheaves on cubics. Advances in Mathematics (2022), Volume 395, pp. 74.
  Journal Version
  Preprint
  We study the microlocal geometry of the space of homogeneous cubics in two variables equipped with an action of GL2. We apply the results to the study of the local Langlands correspondance for G2 over p-adic fields.
  
  
• A. Fiori, F. Scavia. Embeddings of maximal tori in groups of type F4. Pacific Journal of Math (2021), Volume 311, Number 1, p. 53-83.
  Journal Version
  Preprint
  We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type F4 and we prove a local global principal for the existance of embeddings of such tori.
  
  
• A. Fiori. Lower Bounds for the Least Prime in Chebotarev. Algebra and Number Theory (2019), Volume 13, Number 9, p. 2199-2203.
  Journal Version
  Preprint
  In this paper we show there exists an infinite family of number fields L, Galois over Q, for which the smallest prime p of Q which splits completely in L. has size at least ( log|D_L|)^{2+o(1)}. This gives a converse to various upper bounds, which shows that they are best possible.
  
  
• C. Cunningham, A. Fiori, J. Mracek, A. Moussaoui, B.Xu. Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples. Memoirs of the American Mathematical Society (2022). Volume 276, Number 1353, pp. 220.
  Journal Version
  Preprint
  arXiv Version
  We describe and generalize a conjecture of David Vogan which gives a geometric description of Arthur parameters under the local langlands correspondance.
  This paper is the first in a series.
  
  
• A. Fiori, A. Shallue, Average Liar Count for Degree 2 Frobenius Pseudoprimes. Mathematics of Computation (2020), Volume 89, Number. 321, p. 493-514.
  Journal Version
  Offprint
  We give upper and lower bounds on the number of quadratic Frobenius pseudoprimes less than x.
  
  
• A. Fiori, Sub-Shimura Varieties for type O(2,n). Journal de théorie des nombres de Bordeaux (2018), Volume 30, Number 3, p. 979-990.
  Journal Version
  Preprint
  We give a complete characterization, up to consideration of connected components, of the sub-Shimura varieties for the Shimura varieties attached to quadratic forms of signature (2,n).
  
  
• A. Fiori, Rational Conjugacy Classes of Maximal Tori in Groups of type D4. Journal of Algebra and Its Applications (2021), Volume 20, Number 4, pp. 52.
  Journal Version
  Preprint
  We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type DN including especially the groups of type D4. We discuss the structure of these tori for both simply connected forms and the `standard' forms.
  
  
• A. Fiori, A Riemann-Hurwitz Theorem for the Algebraic Euler Characteristic. Canadian Mathematical Bulletin (2017), Volume 60, p. 490-509.
  Journal Version
  Preprint
  We prove a generalization of the Riemann-Hurwitz theorem for computing Euler characterstics of pullbacks of coherent sheaves through finite maps of smooth projective varieties, subject only to the condition that the irreducible components of the branch and ramification loci have simple normal crossings.
  
  
• A. Fiori, On The j-Invariants of CM-Elliptic Curves Defined Over Zp. Functiones et Approximatio Commentarii Mathematici (2017). Volume 56, Number 2, p. 271-286.
  Journal Version
  Preprint
  We characterize the possible reductions of j-invariants of elliptic curves which admit complex multiplication by an order O where the curve itself is defined over Zp. In particular, we show that the distribution of these j-invariants depends on which primes divide the discriminant and conductor of the order.
  
  
• A. Fiori, Rational Conjugacy Classes of Certain Subgroups of G2 published on arXiv (2015), pp. 19.
  Preprint
  arXiv Version
  We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type G2 over number fields and p-adic fields. In the same context we characterize the rational conjugacy classes of simply connected A2 subgroups of G2. We relate the concrete classification to one which may be obtained through Galois cohomology.
  Due to simultaneous discovery and publication of similar results by other researchers (See Hooda and Beli-Gille-Lee) this preprint is not likely to be further revised or resubmitted.
  
  
• A. Fiori, Toroidal Compactifications and Dimension Formulas for Spaces of Modular Forms for Orthogonal Shimura Varieties published on arXiv (2016), pp. 40.
  Preprint
  arXiv Version
  This is an expository article on toroidal compactifications and dimension formulas for spaces of modular forms on locally symmetric spaces focusing specifically on the case of orthogonal locally symmetric spaces.
  This is mostly a revision of part of my Thesis. It contains some new results not appearing there. It is a work in progress, may contain errors and further revisions are likely as new results are obtained. There is no current timeline for preparing it for submission to a peer-reviewed journal as a complete solution to the problem is still far out of reach. None the less, this article already contains original work on the problem.
  
  
• A. Fiori, Transfer of Hermitian Lattices over p-adic Rings. Annales Mathématique du Québec (2018), Volume 42, p. 49 - 78.
  Journal Version
  Preprint
  We study the integral structure of lattices arising from transfer (restriction) between finite extensions of Qp. We focus on those quadratic forms arising from Hermitian forms and apply the results to studying arithmetic volumes of such lattices where the base ring is Zp.
  The same content appears in an extended form in my PhD Thesis.
  
  
• A. Fiori, Arithmetic Volumes of Lattices over p-adic Rings, Journal of Number Theory (2014), Volume 141, p. 343 - 374.
  Journal Version
  Preprint (rev. 2014)
  We develop formulas for the arithmetic volume, aka local densities, of orthogonal groups for arbitrary lattices over the maximal orders of arbitrary finite extensions of Qp, including especially the case p=2.
  The same content appears in an extended form in my PhD Thesis.
  
  
• A. Fiori, Questions in the theory of orthogonal Shimura varieties, PhD. Thesis, McGill University, under supervision of Prof. Eyal Goren (2013).
  Ph.D. Thesis (Official Final Submission Version)
  Ph.D. Thesis (Extended Version) (Contains more material than official version) (rev. 2014)
  Ph.D. Thesis (Dense Layout Extended Version) (rev. 2014)
  We discuss the problem of computing dimension formulas for spaces of modular forms on orthogonal Shimura varieties.
  (The extended version discusses also their toroidal compactifications)
  We discuss the problem of characterizing special fields for orthogonal Shimura varieties.
  We give a characterization of the algebraic tori in orthogonal groups.
  We develop formulas for local densities.
  We study the integral structure of lattices arising from transfer.
  See associated papers for more thorough description. The thesis contains more expository and background detail than the paper versions.
  
  
• A. Fiori, Characterization of special points of orthogonal symmetric spaces, Journal of Algebra, Volume 372 (2012), p. 397 - 419
  Journal Version (revised. 2012)
  Ph.D. Thesis Version (revised. 2013)
  We give a characterization of the algebraic tori in orthogonal groups. The approach taken relates etale algebras with involution and their reflex algebra to the problem.
  This research is a follow-up to my masters thesis and forms part of my Ph.D. thesis.
  
  
• A. Fiori, Characterization of special points on orthogonal symmetric spaces, Masters Thesis, McGill University under the supervision of Prof. Eyal Goren (2009)
  Masters Thesis (Revised Dense Layout Version)
  We introduce the background theory of algebraic groups and orthogonal locally symmetric spaces which we need. We study the problem of classifying tori in orthogonal groups, both from the group cohomology perspective and a more concrete approach.