| 1. |
- Biggs, Kirsti, 1989
(författare)
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Efficient congruencing in ellipsephic sets: the general case
- 2023
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Ingår i: International Journal of Number Theory. - 1793-0421 .- 1793-7310. ; 19:1, s. 169-197
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we bound the number of solutions to a general Vinogradov system of equations x(1)(j) + ... + x(s)(j) = y(1)(j) + ... + y(s)(j), (1 <= j <= k), as well as other related systems, in which the variables are required to satisfy digital restrictions in a given base. Specifically, our sets of permitted digits have the property that there are few representations of a natural number as sums of elements of the digit set - the set of squares serving as a key example. We obtain better bounds using this additive structure than could be deduced purely from the size of the set of variables. In particular, when the digits are required to be squares, we obtain diagonal behavior with 2k(k + 1) variables.
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| 2. |
- Conley, C. H., et al.
(författare)
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Harmonic Maass-Jacobi forms of degree 1 with higher rank indices
- 2016
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Ingår i: International Journal of Number Theory. - : World Scientific Pub Co Pte Lt. - 1793-0421 .- 1793-7310. ; 12:7, s. 1871-1897
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Tidskriftsartikel (refereegranskat)abstract
- We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of order 4. In ranks exceeding 1, the notions of H-harmonicity and semi-holomorphicity are the same.
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| 3. |
- Baba, Srinath, et al.
(författare)
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Quaternionic modular forms and exceptional sets of hypergeometric functions
- 2015
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Ingår i: International Journal of Number Theory. - Singapure : World Scientific Publishing. - 1793-0421. ; 11:2, s. 631-643
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Tidskriftsartikel (refereegranskat)abstract
- We determine the exceptional sets of hypergeometric functions corresponding to the(2, 4, 6) triangle group by relating them to values of certain quaternionic modular formsat CM points. We prove a result on the number fields generated by exceptional values, and by using modular polynomials we explicitly compute some examples.
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| 4. |
- Baier, Stephan, et al.
(författare)
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ON PRIMES IN QUADRATIC PROGRESSIONS
- 2009
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Ingår i: International Journal of Number Theory. - : World Scientific. - 1793-0421. ; 5:6, s. 1017-1035
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Tidskriftsartikel (refereegranskat)abstract
- We verify the Hardy-Littlewood conjecture on primes in quadratic progressions on average. The results in the present paper significantly improve those of a previous paper by the authors [3].
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| 5. |
- Bamunoba, Alex Samuel, et al.
(författare)
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A search for c-Wieferich primes
- 2021
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Ingår i: International Journal of Number Theory. - 1793-0421. ; 17:07, s. 1599-1616
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Tidskriftsartikel (refereegranskat)abstract
- Let q be a power of a prime number p, F-q be a finite field with q elements and G be a subgroup of (F-q,+) of order p. We give an existence criterion and an algorithm for computing maximally G-fixed c-Wieferich primes in F-q[T]. Using the criterion, we study how c-Wieferich primes behave in F-q[T] extensions.
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| 6. |
- Conley, C. H., et al.
(författare)
-
Harmonic Maass-Jacobi forms of degree 1 with higher rank indices
- 2016
-
Ingår i: International Journal of Number Theory. - 1793-0421. ; 12:7, s. 1871-1897
-
Tidskriftsartikel (refereegranskat)abstract
- We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of order 4. In ranks exceeding 1, the notions of H-harmonicity and semi-holomorphicity are the same.
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| 7. |
- Dofs, Erik, et al.
(författare)
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The equation x(1)/x(2) + x(2)/x(3) + x(3)/x(4) + x(4)/x(1) = n
- 2022
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Ingår i: International Journal of Number Theory. - : World Scientific Pub Co Pte Ltd. - 1793-0421. ; 18:01, s. 75-87
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Tidskriftsartikel (refereegranskat)abstract
- It is a subtle question as to when the Diophantine equation of the tittle has solutions in positive integers. Here, we show that the equation in the title does not have solutions in positive integers in the case that n is of the form n = 4q, where q(2) - 1 = 2(h)q(1), with h, q(1) is an element of Z(+), 2 vertical bar h, h >= 4, and 8 vertical bar q(1) + 1. We do this by explicitly calculating a Brauer-Manin obstruction to weak approximation on the elliptic surface defined by the title equation.
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| 8. |
- Duncan, John F. R., et al.
(författare)
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Modular forms on the double half-plane
- 2020
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Ingår i: International Journal of Number Theory. - : World Scientific Publishing Co Pte Ltd. - 1793-0421. ; 16:9, s. 1989-2003
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Tidskriftsartikel (refereegranskat)abstract
- We formulate a notion of modular form on the double half-plane for half-integral weights and explain its relationship to the usual notion of modular form. The construction we provide is compatible with certain physical considerations due to the second author.
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| 9. |
- Granath, Håkan, 1969-, et al.
(författare)
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Orthogonal Systems of Modular Forms and Supersingular Polynomials
- 2011
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Ingår i: International Journal of Number Theory. - : World Scientific. - 1793-0421. ; 7:1, s. 249-259
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Tidskriftsartikel (refereegranskat)abstract
- We extend a construction of Kaneko and Zagier to obtain modular forms which, modulo a prime, vanish at the supersingular points. These modular forms arise simultaneously as solutions of certain second order differential equations, and as an orthogonal basis for an inner product on the space of modular forms.
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| 10. |
- Jones, Rafe, et al.
(författare)
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Eventually stable rational functions
- 2017
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Ingår i: International Journal of Number Theory. - : WORLD SCIENTIFIC PUBL CO PTE LTD. - 1793-0421. ; 13:9, s. 2299-2318
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Tidskriftsartikel (refereegranskat)abstract
- For a field K, rational function phi is an element of K(z) of degree at least two, and alpha is an element of P-1(K), we study the polynomials in K[z] whose roots are given by the solutions in K to phi(n)(z) = a, where fn denotes the nth iterate of phi. When the number of irreducible factors of these polynomials stabilizes as n grows, the pair (phi, alpha) is called eventually stable over K. We conjecture that (phi, alpha) is eventually stable over K when K is any global field and a is any point not periodic under f (an additional non-isotriviality hypothesis is necessary in the function field case). We prove the conjecture when K has a discrete valuation for which (1) f has good reduction and (2) facts bijectively on all finite residue extensions. As a corollary, we prove for these maps a conjecture of Sookdeo on the finiteness of S-integral points in backward orbits. We also give several characterizations of eventual stability in terms of natural finiteness conditions, and survey previous work on the phenomenon.
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