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For my math friends: The Journal of Number Theory is delighted to announce the publication of an expository article by Professor Amanda Folsom entitled: "Perspectives on mock modular forms". This is temporarily freely available and may be found at: http://www.sciencedirect.com/science/article/pii/S0022314X17300653?dgcid=raven_sd_via_email

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The classical product formula for number fields is a fundamental tool in arithmetic. In 1993, Pierre Colmez published a truly inspired generalization of this to the case of Grothendieck's motives. In turn, this spring Urs Hartl and Rajneesh Kumar Singh put an equally inspired manuscript on the arXiv devoted to translating Colmez into the theory of Drinfeld modules and the like. Underneath the mountains of terminology there is a fantastic similarity between these two beautiful papers and I have created a blog to bring this to the attention of the community. Please see:

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Like all number theorists I am fascinated (to say the least) with the functional equation of

classical L-series. Years ago, I came up with a simple characterization of functional equations basically using only complex conjugation. This point being that, via a canonical change of variables (going back to Riemann), such L-series are, up to a nonzero scalar, given by real power series with the expectation that the zeroes are also real. In characteristic p the best one can hope is also that the zeroes will be as rational as the coefficients (though this statement needs to be modified to take care of standard factorizations as well as the great generality of Drinfeld's base rings A).

If you are interested, please follow the attached link:

classical L-series. Years ago, I came up with a simple characterization of functional equations basically using only complex conjugation. This point being that, via a canonical change of variables (going back to Riemann), such L-series are, up to a nonzero scalar, given by real power series with the expectation that the zeroes are also real. In characteristic p the best one can hope is also that the zeroes will be as rational as the coefficients (though this statement needs to be modified to take care of standard factorizations as well as the great generality of Drinfeld's base rings A).

If you are interested, please follow the attached link:

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In studying function field arithmetic, one runs into the binomial coefficients (like one does most everywhere in mathematics); or rather the coefficients modulo a prime p. The primary result related to the reduction of binomial coefficients modulo p is, of course, the congruence of Lucas. In function field arithmetic one seems to be unable to avoid the group obtained by permuting p-adic (or q-adic) coefficients of a number. I recently discovered a congruence using these permutations and the product of two binomial coefficients that I decided to blog about. The proof is an indirect consequence of Lucas and is perhaps more interesting than the result itself. One is then led to look for something related with the Carlitz polynomials, which are the function field analog of the binomial coefficients. You can find a pdf of this here:

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Dear Colleagues: The Journal of Number Theory is delighted to publish a special issue in honor of the work of Winnie Li! The papers will be freely available till the end of May. For more information, please visit

http://www.journals.elsevier.com/journal-of-number-theory/news/honoring-of-the-lifelong-work-of-wen-ching-winnie-li/

http://www.journals.elsevier.com/journal-of-number-theory/news/honoring-of-the-lifelong-work-of-wen-ching-winnie-li/

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This is for those colleagues interested in the approach of Alain Connes and Katia Consani to the Riemann hypothesis via noncommutative geometry and related concepts (the field with one element, semirings, hyperrings, monoïds and what not). Connes and Consani have a new paper to appear in JNT and a great associated Video Abstract which you can now view....

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ABC Day 3

We started with Ben-Bassat and Czerniawska on Frobenioids, which we learned yesterday were not needed in full generality but we saw a special case we do need, the so-called model Frobenioids. This is a category formed out of a variety V over a field and a collection of (possibly ramified) covers of V that, at the level of function fields give all finite subextensions of a (possibly infinite) Galois extension of the function field of V. The objects, in addition to the covers, carry information about the monoids of effective Cartier divisors of the covers and a few other things. Again, mostly definitions.

Szamuely then spoke on Anabelioids. These are close to the fundamental group of a variety and come from etale covers and have a bit more structure. A complex hyperbolic curve has a fundamental group that embeds in PSL(2,R) (as the universal covering is the upper half plane). The point of anabeliods is to construct, group-theoretically, a group that likewise contains the fundamental group of p-adic curves. This only works for curves not "isogenous" (isogeny here is a kind of an etale correspondence) to Shimura curves. This has to do with commensurable subgroups of arithmetic groups and these need to be avoided. We broke for lunch and Szamuely needed more time to finish after lunch.

Here I confess I took a little break from the conference and missed both Szamuely's afternoon talk and Lepage's talks. I was told Lepage defined tempered fundamental groups of p-adic curves, these are subgroups of the etale fundamental group that are not closed in the profinite topology, complete to the whole etale fundamental group and serve as a substitute to the topological fundamental group in the complex case. In some ways its elements can be measured.

Kucharczyk was scheduled to speak on Hodge-Arakelov theory but couldn't come but provided notes. Yamashita gave a short talk in his place. Apparently this theory is not needed for IUT but motivates a construction (the theta link). Yamashita tried to explain this motivation. He said that the whole work is "locally trivial" but that eventually the reason for these many definitions will become clear. It hasn't yet.

Kedlaya spoke on the etale theta function. He used the tempered fundamental group of a Tate elliptic curve over a local field and some covers of it to construct some cohomology classes in the spirit of Kummer theory. As in Stix's talk, cohomology classes can be used to recover the function field of these covers. The ones constructed produce a theta function with a power series expansion identical to the Jacobi theta function similarly to what happens in Tate's work.

Tomorrow might be the day that more substantial stuff happens. The "Hodge theaters" will be introduced. I was told that "theater" refers to "theater of war" where some action happens and not a place where plays are staged. Let's hope that's the case and that we won't see instead a comedy or a tragedy.

We started with Ben-Bassat and Czerniawska on Frobenioids, which we learned yesterday were not needed in full generality but we saw a special case we do need, the so-called model Frobenioids. This is a category formed out of a variety V over a field and a collection of (possibly ramified) covers of V that, at the level of function fields give all finite subextensions of a (possibly infinite) Galois extension of the function field of V. The objects, in addition to the covers, carry information about the monoids of effective Cartier divisors of the covers and a few other things. Again, mostly definitions.

Szamuely then spoke on Anabelioids. These are close to the fundamental group of a variety and come from etale covers and have a bit more structure. A complex hyperbolic curve has a fundamental group that embeds in PSL(2,R) (as the universal covering is the upper half plane). The point of anabeliods is to construct, group-theoretically, a group that likewise contains the fundamental group of p-adic curves. This only works for curves not "isogenous" (isogeny here is a kind of an etale correspondence) to Shimura curves. This has to do with commensurable subgroups of arithmetic groups and these need to be avoided. We broke for lunch and Szamuely needed more time to finish after lunch.

Here I confess I took a little break from the conference and missed both Szamuely's afternoon talk and Lepage's talks. I was told Lepage defined tempered fundamental groups of p-adic curves, these are subgroups of the etale fundamental group that are not closed in the profinite topology, complete to the whole etale fundamental group and serve as a substitute to the topological fundamental group in the complex case. In some ways its elements can be measured.

Kucharczyk was scheduled to speak on Hodge-Arakelov theory but couldn't come but provided notes. Yamashita gave a short talk in his place. Apparently this theory is not needed for IUT but motivates a construction (the theta link). Yamashita tried to explain this motivation. He said that the whole work is "locally trivial" but that eventually the reason for these many definitions will become clear. It hasn't yet.

Kedlaya spoke on the etale theta function. He used the tempered fundamental group of a Tate elliptic curve over a local field and some covers of it to construct some cohomology classes in the spirit of Kummer theory. As in Stix's talk, cohomology classes can be used to recover the function field of these covers. The ones constructed produce a theta function with a power series expansion identical to the Jacobi theta function similarly to what happens in Tate's work.

Tomorrow might be the day that more substantial stuff happens. The "Hodge theaters" will be introduced. I was told that "theater" refers to "theater of war" where some action happens and not a place where plays are staged. Let's hope that's the case and that we won't see instead a comedy or a tragedy.

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Wonderful quote from this article: " “In England it was enough that Newton was the greatest mathematician of the century,” wrote Jean d'Alembert, a French philosopher and mathematician; “in France he would have been expected to be agreeable too.” "

Still, I never met Grothendieck, but in all my experience nobody ever described him as "agreeable"...

Still, I never met Grothendieck, but in all my experience nobody ever described him as "agreeable"...

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Bruno Anglès and Lenny Taelman published a great paper on special values in finite characteristic recently in the Proceedings of the London Math. Society. Many remarkable results are given here which fit in beautifully with classical cyclotomic theory (with some possible differences also). I have written a longish review for Math Reviews/MathSciNet which I have also put on a blog (together with a url for a pdf copy) with the express consent of MathSciNet. If interested it is here:

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For those colleagues interested in the emerging function field arithmetic and, especially, in the analogs of Fourier series in finite charcteristic (and their relationships)....

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