[seqfan] Re: Reference Help?

Sat Jun 3 18:54:00 CEST 2023

Dear Brad and all,

The remark about the link to Beauville is somewhat tangentially related to the OEIS sequence, and I would say that it is not very helpful or enlightening for most OEIS users. Also, the paper by Beauville in the OEIS page is

Arnaud Beauville, Les familles stables de courbes sur P_1 admettant quatre fibres singuli¨¨res<http://gallica.bnf.fr/ark:/12148/bpt6k5543443c/f31.item>, Comptes Rendus, Acad¨¦mie Sciences Paris, Vol.. 294 (May 24 1982), pp. 657-660,

which concerns elliptic fibrations (families of elliptic curves) and classifies the possible singular fibres, but makes no mention of the Franel numbers that I could find, so again does not strike me as being a useful reference to include in A000172. I think the connection is via the work of Beukers, who found a neat way of reproving Apery's result on the irrationality of zeta(3), using integrals; see
Beukers, F.: A note on the irrationality of ¦Æ(2) and ¦Æ(3). Bull. Lond. Math. Soc. 11, 268¨C272 (1979).
The sequence of Franel numbers is reminiscent of sums of products of binomial coefficients which appear in Apery's proof - the best place to start reading about this is van der Poorten's article "A proof that Euler missed". Then the further connection with modular forms and families of elliptic curves is developed in this later article by Beukers:
http://www.numdam.org/article/AST_1987__147-148__271_0.pdf

Hopefully this helps as a starting point. It would be good to go through all the references to A000172 and figure out which ones are really appropriate. I'm travelling at the moment, but will try to look at it again more carefully next week.

All the best,
Andy

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From: SeqFan <seqfan-bounces at list.seqfan.eu> on behalf of sven-h.simon at gmx.de <sven-h.simon at gmx.de>
Sent: 03 June 2023 13:21
To: 'Sequence Fanatics Discussion list' <seqfan at list.seqfan.eu>
Subject: [seqfan] Re: Reference Help?

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Hello Brad,
the topic is far from the reach of my rather old mathematical background.
But I found an email of a Stephan Herfurtner. He is still living at the
exact same address as the one in his thesis. He seems to be leading a small
IT tech, which has the following email:

kontakt at karu-it.de
I hope it is ok to write his email here, otherwise the moderator could
delete it, as I  send it cc directly to Brad.

Perhaps he does know something more on the topic.

Sven

-----Urspr¨¹ngliche Nachricht-----
Von: SeqFan <seqfan-bounces at list.seqfan.eu> Im Auftrag von brad klee via
SeqFan
Gesendet: Samstag, 3. Juni 2023 00:17
An: seqfan at list.seqfan.eu
Cc: brad klee <bradklee at proton.me>
Betreff: [seqfan] Reference Help?

Hiya seqfans,

It was recently brought to my attention that a few sequences in OEIS have
missing representations as elliptic integrals, for example:

https://eur01.safelinks.protection.outlook.com/?url=https%3A%2F%2Foeis.org%2FA000172&data=05%7C01%7CA.N.W.Hone%40kent.ac.uk%7C1b42900d69da4795621a08db642d2a2c%7C51a9fa563f32449aa7213e3f49aa5e9a%7C0%7C0%7C638213917374808035%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=HXjG4OBGuDKLM9IqZjA24%2B7v%2Fv4%2Bw%2FAfJt5pVFBBH4w%3D&reserved=0<https://oeis.org/A000172>

Perhaps it's obvious to some of us, but not to me, so I had to read the
Weierstrass data from Stephan Herfurtner's thesis:

g2 = 3/64 (1 + 12 x + 30 x^2 - 228 x^3 + 9 x^4)
g3 = 1/512 (1 + 18 x + 99 x^2 + 684 x^3 + 3159 x^4 - 1566 x^5 - 27 x^6) (try
searching OEIS for "herfurtner" to find the reference)

Unfortunately I can't read French and German very well, so haven't tracked
back Schmickler-Hirzebruch paper:

https://eur01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fmathoverflow.net%2Fquestions%2F220290%2Flooking-for-schmickler-hirzebruch-&data=05%7C01%7CA.N.W.Hone%40kent.ac.uk%7C1b42900d69da4795621a08db642d2a2c%7C51a9fa563f32449aa7213e3f49aa5e9a%7C0%7C0%7C638213917374808035%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=oVFB1zJ7AUVFFuUjkJhAqIYtkoaGOoZIpYs%2BUGmbPQQ%3D&reserved=0<https://mathoverflow.net/questions/220290/looking-for-schmickler-hirzebruch->
monograph-on-elliptic-surfaces

Many of the related OEIS sequences also refer cryptically to Beauville:

>This is the Taylor expansion of a special point on a curve described by
Beauville.

Does anyone know if this means "period integral" or something else? I also
have not been able to track back the Matthijs Coster reference (and don't
even know where to start):

> Matthijs Coster, Over 6 families van krommen [On 6 families of
> curves], Master's Thesis (unpublished), Aug 26 1983.

From an editorial point of view, ambiguity and missing references are
problematic.
But more in general I'm curious about publication history before and after
the Herfurtner thesis.

The Herfurtner thesis contains some beautiful results, which I think Picard,
Goursat, Klein etc. would have paid attention to, had those results been
known in previous lifetimes.
How is it that Stephan Herfurtner's thesis has met with so little
recognition and fanfare?

One possibility is that contextual information is buried in French or German
language articles not immediately accessible to an English language
audience. Another possibility is that contextual information is part of an
academic secret-keeping culture, which I think many of us would stand
against. A third and most exciting possibility is that we simply have been
slow to understand these rarefied tables, and are only now over the last few
years finding contextual information to increase their relevance and value.

Anyone with relevant information is welcome to write me. I'm especially
petitioning native French and German speakers who have better skills than I
do for this sort of research.

In the meantime I will update some more of our references with data that I
have from Herfurtner thesis, and probably add a few missing sequences as
well.

Thanks again,

--Brad

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