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bio website math.lsa.umich.edu/~zieve
location University of Michigan
age
visits member for 1 years, 7 months
seen 9 hours ago

I am a professor at the University of Michigan. In 2013-2014 I am on leave from Michigan, and I am at Shing-Tung Yau's Mathematical Sciences Center attached to Tsinghua University.


Apr
17
comment Advice for number theory library
I also suggest Serre's "A Course in Arithmetic", Ireland-Rosen, Cassels' "Local Fields" (which is great fun), Serre's "Local Fields", Washington's "Cyclotomic Fields", Serre's "Lectures on Mordell-Weil" and "Topics in Galois Theory", Samuel's "Algebraic Theory of Numbers", Lemmermeyer's "Reciprocity Laws", Koblitz's "p-adic numbers, p-adic analysis and zeta functions", Bombieri-Gubler, Cassels-Frohlich, Silverman's various books, etc.
Apr
17
comment Advice for number theory library
Dickson's "History" is indeed great, it basically includes all of number theory that was known at that time, and it is well-organized and readable. It's a fantastic reference for elementary number theory. It is also extremely inexpensive (published by AMS/Chelsea).
Apr
5
comment Branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$
I assume this will be migrated to math.stackexchange, which seems more suitable. Anyway, the reason to require the map to be defined over $\overline{\mathbb{Q}}$ is that otherwise you could have examples like $f:\mathbb{P}^1\to\mathbb{P}^1$ given by $z\mapsto z^2+\pi$, which has the non-algebraic branch point $\pi$.
Mar
20
comment Discriminant of a compositum of number fields, a bound?
Your questions are answered in a paper by Toyama cited in mathoverflow.net/questions/137557/…
Mar
16
revised On the maximum cardinality of the image of a non-onto polynomial function on finite fields
added 11 characters in body
Mar
16
answered On the maximum cardinality of the image of a non-onto polynomial function on finite fields
Mar
8
awarded  Disciplined
Mar
6
answered The proof of Belyi theorem by Lando and Zvonkin
Feb
26
comment A curious identity related to finite fields
I don't see why the factor $q-1$ is explained through the action of the affine group. While most orbits of that group on triples $(a_1,a_2,a_3)$ have size $q(q-1)$, not all do. All orbits have size divisible by $q$ though. Anyway, while making your observed factor of $q-1$ mysterious, perhaps this suggests that for the remaining classes of $q$ mod $24$ you might modify what you're counting in order to take account of non-regular orbits of the affine group.
Feb
26
comment A curious identity related to finite fields
Just an observation: your summation counts the number of monic degree-8 polynomials over $\mathbf{F}_q$ which have eight distinct roots in $\mathbf{F}_q$ and which can be written as $g(h(x))$ where $g,h\in\mathbf{F}_q[x]$ have degrees $2$ and $4$, respectively. I'm not sure whether this will help solve the problem, but it helped me understand what was being asked.
Feb
26
comment Function field Towers of larger depth of recursion
Interesting. In light of that paper, and the earlier paper by Bassa, Beelen, Garcia, and Stichtenoth, it seems that an interesting research direction in this subject is towers of Drinfeld modular curves and towers of curves on Drinfeld modular varieties.
Feb
26
answered Function field Towers of larger depth of recursion
Feb
26
answered A special curve with points of order 3(or 6)
Feb
26
answered Consecutive square values of cubic polynomials
Feb
21
comment When are arithmetic and geometric monodromy equal?
Oops -- at the end of my first comment, I meant the group of permutations in $S_n$ which preserve the partition of $\{1,2,\dots,n\}\times\{1,2,\dots,n\}$ which corresponds to the decomposition of the fibered product into components.
Feb
21
comment When are arithmetic and geometric monodromy equal?
[cont.] the decomposition of the fibered product into components. This is most useful if there are no intermediate curves between $X$ and $Y$, since then the arithmetic monodromy group is primitive but not doubly transitive, which is a big constraint on the group. For instance, if the fibered product has just three components then the arithmetic group is primitive of rank three, and such groups have been classified.
Feb
21
comment When are arithmetic and geometric monodromy equal?
Another way to get bounds is to determine the irreducible components of the fibered product $Y\times_X Y$. If the diagonal and its complement are both geometrically irreducible, then the geometric monodromy group is doubly transitive (and then often the local monodromies will force the group to be the full symmetric group). If the fibered product has more than two irreducible components over $\mathbf{F}_q$ then one gets an upper bound on the arithmetic monodromy group, namely the group of permutations in $S_n$ which preserve the partition of the $n$ letters which corresponds to...[cont.]
Jan
16
comment When do tamely ramified Belyi maps exist in characteristic p?
Osserman's result gives necessary and sufficient conditions for the existence of a genus-$0$ curve $C$ with $C\to\mathbf{P}^1$ having prescribed ramification. So this answers the stated questions in case the $\lambda_i$ have a combined total of $d+2$ parts.
Jan
14
comment Doubly primitive groups with simple socle
In light of the conjecture in Atkinson's paper "Doubly transitive but not doubly primitive permutation groups, II", I infer from your answer that the Ree groups must act as groups of automorphisms of a block design on $\Omega$ in which $\lambda=1$. This seems surprising because the Suzuki groups don't act in this way. Is there a simple explanation for why there is this difference between the Ree and Suzuki groups?
Jan
9
answered Irreducibility of a class of polynomials