Naslov
Zagier's sporadic sequences and Atkin and Swinnerton-Dyer congruences

Autori
Kazalicki, Matija

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Skup
Joint meeting UMI-SIMAI-PTM

Mjesto i datum
Wrocław, Poljska, 17.09.2018. - 20.09.2018

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Podatak o recenziji nije dostupan

Ključne riječi
modular forms for non-congruence subgroups

Sažetak
In 1979, in the course of his proof of the irrationality of $\zeta(2)$ Robert Ap\'ery introduced numbers $b_n = \sum_{; ; k=0}; ; ^n {; ; n \choose k}; ; ^2{; ; n+k \choose k}; ; $ that are surprisingly integral solutions of recursive relations $$(n+1)^2 u_{; ; n+1}; ; - (11n^2+11n+3)u_n-n^2u_{; ; n-1}; ; = 0.$$ Zagier performed a computer search on first 100 million triples $(A, B, C)\in \Z^3$ and found that the recursive relation generalizing $b_n$ $$(n+1)u_{; ; n+1}; ; - (An^2+An+B)u_n + C n ^2 u_{; ; n- 1}; ; =0, $$ with the initial conditions $u_{; ; -1}; ; =0$ and $u_0=1$ has (nondegenerate i.e. $C(A^2-4C)\ne 0$) integral solution for only six more triples (whose solutions are so called sporadic sequences) . Stienstra and Beukers showed that the generating function of Ap\'ery's numbers $b_n$ is a holomorphic solution of Picard-Fuchs differential equation of elliptic surface $\mathcal{; ; S}; ; :X(Y-Z)(Z-X)-t(X-Y)YZ=0$. Using this connection they proved that for prime $p\ge 5$ \begin{; ; equation*}; ; b_{; ; (p-1)/2}; ; \equiv \begin{; ; cases}; ; 4a^2-2p \pmod{; ; p}; ; \textrm{; ; if }; ; p = a^2+b^2, \textrm{; ; a odd}; ; \\ 0 \pmod{; ; p}; ; \textrm{; ; if }; ; p\equiv 3 \pmod{; ; 4}; ; .\end{; ; cases}; ; \end{; ; equation*}; ; Recently, Osburn and Straub proved similar congruences for all but one of the six Zagier's sporadic sequences (three cases were already known to be true by the work of Stienstra and Beukers) and conjectured the congruence for the sixth sequence (which is a solution of recursion determined by triple $(17, 6, 72)$. In this talk we will prove that remaining congruence.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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